Acoustics from A to Z - sandv.com · Acoustics from A to Z ... against which to judge the measured val-ues ... Wiggling small hair cells that sense tones

8 SOUND AND VIBRATION/JANUARY 2002

Acoustics from A to ZEric E. Ungar

During the March 1999 joint meeting ofthe Acoustical Society of America withthe European Acoustics Association,held at the Technical University of Ber-lin, I had the opportunity to visit the In-stitute of Technical Acoustics of that uni-versity. Prominently posted on a bulletinboard in a corridor, I found an eleven-page 1958 article which was coauthoredby Lothar Cremer, the late former direc-tor of the institute.

This unusual publication, whose titlemay be translated as “The ABC of theAcoustics of Buildings,” consists of briefverses. There is one verse for each letterof the alphabet; each deals lightheartedlywith some aspect of acoustics, each is il-lustrated by a cartoon structured aroundthe letter, and each is followed by a briefsummary of related facts. I was so in-trigued with this article’s approach thatI resolved to translate it and asked thedirector of the Institute, Professor Mi-chael Möser, to send me a copy. Soon,after he kindly complied with my re-quest, it became clear that translating thearticle without losing its basic spirit wasbeyond me. So, I decided to give up andstart from scratch, developing my ownverses and discussions, but maintainingthe spirit. Here’s the result.

ACOUSTICS deals with sound and noise;One may be pleasant, one annoys.How sound’s produced and propagated,And put to use, attenuated,And how perception plays its part –It is part science and part art.

The story is told about George Wash-ington Carver, the famous African-

American scientist who studied peanutsand developed many uses for them, thatas a young man he prayed that God revealto him the secrets of the universe. Be-cause God replied that “the peanut ismore your size,” Dr. Carver focused on amore limited field.

Many of us who work in acoustics alsohad ambitions to know all about acous-tics, but we soon learned that the field ismuch too diverse. Acoustics, as Ira Dyerof MIT has said, deals with “anythingthat moves and many things that don’t.”

That statement may be a little far-fetched,but it does convey the breadth of this artand science. To get an idea of this breadthone merely needs to look at the Journalof the Acoustical Society of America, forexample. The topics covered there rangefrom physics and engineering – aero-acoustics, underwater sound, ultrasonics,transduction, vibration, signal processing– through physiological and psychologi-cal acoustics – including speech produc-tion and perception, as well as humanand animal bioacoustics – to noise effectsand noise control, architectural acous-tics, music and musical instruments.Many other fields are closely tied toacoustics – sound systems, audiology,acoustic oceanography and ultrasonicinstrumentation to name a few. You canundoubtedly think of many others.

Clearly, the science of acoustics is welldeveloped, and research is progressingon many fronts. The news lately has beenrife with talk about such things as acous-tic microscopy, acoustical refrigeratorswith no moving mechanical parts, andcochlear implants that enable people thathave severely damaged hearing to hearagain. Often, however, more than scienceis needed. In cases involving the noiseexposure of communities or work areas,human relations and politics also playmajor roles. And, the artful application ofjudgment is usually needed to solve prac-tical problems. Typically, they involvetradeoffs between conflicting require-ments.

. . . . . . . . . . . . . . . . . . . .In BUILDINGS where we have employment,Or want to sleep or have enjoyment,We need to stop intruding noiseFrom streets, TVs, and neighbors’ toys,From footfall impacts, other shocks,Most anything that squeaks or knocks.

One person’s music is anotherperson’s noise. It all depends on

what we want to hear. Achievement ofthe desired acoustical environment in aroom involves providing envelope struc-tures that block noise intrusion from ad-jacent areas and adding acoustical ab-sorption to avoid the build-up of noise

that gets through the envelope.In addition to dealing with audible

“air-borne” noise in adjacent areas, onealso needs to address “structure-bornenoise” – that is, noise that results frommechanical vibrations of the envelopestructures. The walls, floor and ceiling ofa room tend to act somewhat like loud-speaker membranes whose vibrations inthe audio frequency range radiate sound.Structurally radiated noise in rooms mayresult from people walking or chairsscraping upstairs and from vibratingequipment (refrigerator compressors,unisolated plumbing or the legs of pi-anos, for example) in contact with wallsor floors.

What can you do to deal with noisefrom the neighbors? Ask them to avoidannoying activities at times when youdon’t want to hear them. Get them to in-stall thick rugs or, better yet, a floatingfloor. Get vibrating equipment isolated.Build an isolated ceiling and secondarywalls, so that you in effect have an iso-lated room within a room. Start livingwith permanent earplugs. Or, move else-where. . . . . . . . . . . . . . . . . . .CRITERIA for floor vibrationOr sound (that is, air oscillation)Depend on what is really needed;Intended usage must be heeded.If set too tight, there’s undue cost;If set too loose, there’s function lost.

In the index of any book on acoustics oron noise and vibration control you will

find a profusion of listings under criteria.You will find criteria for environmentalnoise from aircraft and surface vehicles,for the prevention of hearing damage inindustrial settings, for acceptable condi-tions in dwellings and offices, for effi-cient speech communication, for goodlistening conditions in class rooms andauditoriums. Also for acceptable vibra-tion environments in buildings, surfacevehicles and ships (sea-sickness effectsare considered in standards being devel-oped), for vibrations exerted on workers’hands and arms by machine tools, and forevaluation of the quality of rotating ma-

935th ANNIVERSARY ISSUE

chinery. I apologize if I have omitted yourfavorite among the many other criteriathat are available.

Many criteria that have been developedon the basis of extensive studies havegained broad acceptance, have been setforth in international and national stan-dards, have been the basis of ordinancesand regulations, and have been cited incases involving litigation. But how firmare these criteria and standards? Stan-dards are meant to represent the consen-sus of experts, and they do – to the extentthat the experts participating in develop-ment of the standards can agree. Unfor-tunately, the number of specialists in-volved in this development processtypically is small, some may have limitedranges of interest and concern, and somemay have certain prejudices. Therefore,standards tend to reflect only the smallamount of information on which the par-ticipants in the development process canagree. And even then, the consensus isnot always one hundred percent. Somestandards only address how measure-ments should be made, leaving criteria –the (usually controversial) magnitudesagainst which to judge the measured val-ues – to appendixes that are not officialparts of the standard.

Some criteria, for example those limit-ing the exposures of sensitive equipment,are set forth by the equipment suppliers.These criteria often are more stringentthan necessary, perhaps because the sen-sitivity of the equipment is not wellknown or – as a suspicious person mayfeel – to give the supplier the opportunityto blame the noise and vibration environ-ment for the occasional less than optimalperformance of his equipment.

Equipment criteria often are written bynon-specialists in acoustics and vibra-tions. This has led to problems with in-appropriately specified noise spectrumweightings, with confusion between vi-bratory displacements and acceleration,and with omission of measurement dura-tion and bandwidth requirements, amongothers. I have spent much time explain-ing to suppliers of optical equipment thatrelative displacements of the opticalcomponents are important, and not theabsolute vibratory displacements of theequipment’s support points. And I haveoften tried to convince clients that onecannot limit the displacement ampli-tudes of buildings to very small values ina range of frequencies that extends downto zero, relying on the argument that themoon induces tidal motions in the earthmuch as it does in the oceans and we asyet don’t have the technology to hold themoon still.

. . . . . . . . . . . . . . . . . . . .DAMPING’s poorly understood.It doesn’t always do much good.Although it may speed wave decay,It makes few problems go away.Damping does mean dissipation,But may not yield attenuation.

According to my dictionary, the verb“to damp” means “to make damp;

moisten” or “to check or retard the energyof” or “to stifle or suffocate, extinguish.”Similarly, the verb “to dampen” is de-fined as “to make damp, moisten” or “todull or depress.” In acoustics and vibra-tions, however, ‘damping’ has nothing todo with moisture. Although ‘damping’sometimes is used to mean attenuation,in precise technical language its usagepreferably should be confined to pro-cesses involving energy dissipation.

I have long advocated that ‘dampening’not be used instead of ‘damping,’ because‘dampening’ primarily means ‘moisten-ing.’ Nevertheless, the writers who pro-duced my former employer’s annual re-port a few years ago proudly proclaimedthat “we dampen submarines.” I found itdifficult to explain how we can ask to bepaid for that.

Most vibration textbooks deal onlywith viscous damping – that is, withdamping due to a force that opposes themotion and is proportional to velocity.The primary reason for focusing on thistype of damping is not that viscous damp-ing necessarily represents the real world(although it fortunately is a reasonableapproximation in many cases), but thatthe assumption of this sort of dampinggives us linear differential equations withconstant coefficients, which we knowhow to solve comparatively easily. So, infact, we act much like the proverbialdrunk who lost a silver dollar in themiddle of the block, but looks for it nearthe corner because the street lamp’s lightis better there – that is, we solve an easierproblem rather than the real one.

Of course, much can be learned fromthe textbook problems, and generally theanswers we get from viscous dampinganalysis are reasonable as long as thedamping is not too large. But, watch out!Most texts and handbooks, as well asmuch sales literature for vibration isola-tors, show equations and curves that in-dicate that the isolation that a spring-damper combination provides above asystem’s natural frequency is severelycompromised by large damping. This re-sult, derived for viscous damping, tendsto overstate markedly this detrimentaleffect of damping for metal or rubber iso-lators, in which the damping is not of theviscous type.

Contrary to some misguided commer-cial claims, damping is not a cure-all.

Basically, damping has a significant ef-fect only on motions that are controlledprimarily by energy dissipation. Thesemotions include steady responses at andnear resonance, freely decaying vibra-tions, and freely propagating waves; theydo not include vibrations due to steadyexcitation at frequencies that are not neara system’s natural frequency. Space limi-tations and my desire not to bore those ofyou who are not particularly interested inthis topic keep me from further preach-ing here. Those of you who are curious tolearn more may find my “StructuralDamping” chapter1 useful, though lessentertaining than this brief discussion.

1. Chapter 12 of Noise and Vibration ControlEngineering, edited by Leo Beranek andIstvan Ver, John Wiley & Sons, NY, 1997.. . . . . . . . . . . . . . . . . . . .

The membrane of the human EAR

Responds to sound so we can hear.Its motion vibrates tiny bones,Wiggling small hair cells that sense tones.Their nerve cells connect to the brainFrom which we information gain.

The first book of Caesar opens withGallia omnes in partes tres divisa est

– all of Gaul is divided into three parts –as I remember (surprisingly) from myhigh school Latin. Anatomists similarlyconsider the human ear in terms of threeparts: the outer, inner and middle ear. Isuspect that both of these somewhat ar-tificial divisions into three parts weremade for the same reason: to divide acomplex entity into smaller parts that canbe discussed more easily.

The outer ear consists of the fleshyappendage attached to the head, calledthe pinna. This Latin word means ‘sail’and undoubtedly was chosen by someonewith large protruding ears who lived neara windy beach. The pinna and the earcanal with which it communicates chan-nel sound waves to the canal’s termina-tion, the eardrum or “tympanic mem-brane.”

The middle ear works somewhat likeThomas Edison’s phonograph: a mem-brane, set into motion by sound waves, isconnected to a mechanical amplificationsystem that communicates the amplifiedmotions to the next stage. In the ear themechanical amplification is achieved bya set of tiny bones or ‘ossicles,’ whichconnect to the so-called oval window.The ossicles make this window move in


and out much like a piston and thesemotions are transmitted via the innerear’s essentially incompressible fluid tothe basilar membrane and the organ ofCorti. This organ is not a musical instru-ment; rather, it is essentially a membranethat supports a forest of about 20,000 haircells of different types and lengths,which respond differently to sounds atdifferent frequencies. These hair cells areconnected to nerve cells that communi-cate with the brain, which does most ofthe difficult data processing.

Hearing loss may result from damage toany of the conductive mechanisms orfrom damage to the neurological ele-ments. My wife’s hearing loss was re-duced by surgical freeing of the ossiclesthat had become locked together andcould not transmit the sound-inducedvibrations well. Noise-induced hearingloss most often results from damage to thehair cell structures, which deteriorate,break off and are not regenerated. ‘Pres-bycusis’ – the hearing loss we experienceas we get older – begins with loss of thehair cells responsible for hearing thehigher frequencies. In more ways thanone, the hairless hear less.

. . . . . . . . . . . . . . . . . . . .The FREQUENCY of oscillationsTells us how many fluctuationsUp and down from mean are reckonedPer unit time (minute or second).The unit ‘Hertz’ is now preferred;Cycles-per-second’s been interred.

Although I have the utmost respect forthe German physicist Heinrich Hertz

(1857-1894) after whom the unit of fre-quency is named that used to be called“cycles per second,” I wish the worldwould have stayed with the old designa-tion. I have never had to explain what“cycles per second” means or how manycycles per minute correspond to a givennumber of cycles per second, but the un-initiated often need to be told what‘Hertz’ (Hz) means.

According to such eminent referencesas Cyril Harris’ Handbook of AcousticalMeasurements and Noise Control, the fre-quency of a periodic phenomenon is de-fined as (a) the number of times the phe-nomenon repeats itself in one second or(b) the reciprocal of the period, where theperiod is the time it takes for the phenom-enon to repeat itself. These definitions,however, are not entirely precise. Visual-ize a simple sinusoidal trace that goes

through one cycle each second. It clearlyrepeats itself once per second, but it alsorepeats itself once every two seconds,once every three seconds and so on toinfinity. So, its frequency would be notonly 1 Hz, but also 1/2 Hz, 1/3 Hz, etc.Thus, at least for simple stationary sig-nals it may be more precise to define thefrequency as equal to the reciprocal of theshortest time it takes for any portion ofthe signal to repeat itself.

And what about a signal that never re-peats itself – as is the case for most sig-nals in the real world? The usual spec-trum analysis is done by sampling asignal over a selected time interval andassuming that the sample repeats itselfforever. So, if we apply the foregoing defi-nition to this repeated sample signal, wefind that its frequency corresponds to thearbitrary length of the sample we took –implying that the signal’s frequency isarbitrary. If the signal indeed is random,so that it never repeats itself, then its pe-riod would be infinite and its frequencywould be zero.

Spectrum analyzers fortunately are notbothered by these definitional dilemmas.They typically process data sampled overspecified intervals on the basis of the as-sumption that the samples are repeatedindefinitely, fit the sum of a series of (in-finitely extended) sinusoids to the data,and report the magnitudes of these sinu-soids as a function of their frequencies.(The frequency of the sinusoids is de-fined as suggested at the end of the firstparagraph above.) This so-called Fouriertransform process allows one to representa time-varying signal sample in terms ofa series of frequency-dependent values.

It has been told that the French math-ematician Fourier,2 who invented thetransform, used to take quite a long timeto work out the necessary integrals, whilehis younger brother took about half aslong. Consequently, the older brothercame to be known as Slow Fourier andthe younger sibling came to be known asFast Fourier. In recent years the latterachieved fame posthumously by lendinghis name to the Fast Fourier Transform(FFT) algorithm that is implemented inmodern digital spectrum analyzers.

2. Fred Schloss of Wilcoxon Research sent methe following interesting footnote to history:“Jean Baptiste Joseph Fourier was orphanedat a young age and later debarred from theArmy on account of his lowness of birth andpoverty. However, he went into the ‘State ofde Nile’ with Bonaparte, 1798, and becamegovernor of half of Egypt. He took an impor-tant part in preparation of the famous ‘De-scription de l’Egypte.’ His mathematical dis-coveries were the result of his interest in theconduction of heat.”

. . . . . . . . . . . . . . . . . . . .G’s refer accelerationTo our earth’s own gravitation.Displacements may be very smallFor g’s that aren’t small at all.And so, in many situations,Velocity’s used to mark vibrations.

Nowadays it is not much of a trick tomeasure accelerations of 1 milli-g

(mg) at 1 kHz. The corresponding dis-placement amplitude turns out to be ofthe order of a mere 10-8 inches, which isequal to about 0.0002 micrometers or 2Ångström units. Compare this to the 40micrometer typical diameter of a humanhair and to the wavelength of visiblelight, which is in the 4000 to 8000Ångström range!

It is interesting to explore what hap-pens if I drop an accelerometer onto afloor from a height of 10 cm, for example.If the floor were to cause the accelerom-eter to decelerate uniformly, making itcome to rest within 10-3 cm, then the ac-celerometer would experience a mere10,000 g and probably would be dam-aged. That’s one reason for being carefulwith accelerometers.

In many vibration situations one ob-serves relatively small accelerations andrelatively large displacements at low fre-quencies. At high frequencies, the situa-tion tends to be reversed. That is whydisplacement sensors are better at lowfrequencies and accelerometers are betterfor measurements at high frequencies.Velocity is often used to characterize orspecify vibrations, because it tends toexhibit mid-range magnitudes over largerranges of frequency. Among the criteriathat are stated primarily in terms of ve-locity are those for judging the quality ofrotating machines, for determining theperceptibility of vibrations and for as-sessing the suitability of a floor for vari-ous types of sensitive equipment.

. . . . . . . . . . . . . . . . . . . .

The HEARING threshold, it is known,Is six dB for a pure tonePrecisely at one thousand Hertz,Agreed upon by most experts.The smallest pressure we can hear?A billionth of an atmosphere.


Inc., 1997.)4. Engineering Noise Control, D. Bies and C. H.

Hansen, Unwin Hyman Ltd., 1988.5. “Damage Risk Criteria for Hearing and Hu-

man Body Vibration,” H. E. von Gierke andC. W. Nixon, Chapter 16 of Noise and Vibra-tion Control Engineering, L. L. Beranek andI. L. Ver, Eds., John Wiley & Sons, Inc. 1992.. . . . . . . . . . . . . . . . . . . .

INSTRUMENTS for measuring soundFor many years have been around.Recent years have seen improvementIn sensing sound, and also movement.Some systems are so fast and smallThey almost aren’t there at all.

Ted Schultz, with whom I worked atBBN for many years, liked to tell the

story of how he made measurements ofairframe noise in one of the earlier Dou-glas transport aircraft. He used the onlysuitable filters available then, namely ananalog system that permitted him to mea-sure the noise level in one octave-band ata time. So, he had the pilot climb to acomfortable altitude and shut off the en-gines so that engine noise would notdrown out the airframe noise, and then hemeasured the noise in one octave-bandduring a gliding descent. Then the pilotwould restart the engines and repeat theprocess until Ted got data in all elevenoctave-bands.

With today’s instrumentation, oneshort glide would have sufficed to ac-quire all the data, analyze it in octave,one-third-octave, or narrow bands, dis-play it and even print it out. And, today’sequipment would weigh at most only akilogram or two, where Ted’s ‘portable’system was portable by perhaps twopeople or a mule.

Modern microelectronics and digitaltechnology have made possible all sortsof compact and energy-efficient signalprocessors and recorders, with featurestoo numerous to mention. Similar tech-nology also has led to accelerometerswith built-in processing chips that con-dition (e.g., amplify, filter, limit, inte-grate) the acceleration signal. It also hasgiven rise to accelerometers that weighonly a few carats, where a carat (equal to0.2 grams) is the unit in terms of whichthe weight of precious stones is usuallystated.

Modern technology also has led to la-ser systems that allow one to measure thevibrations of objects without attachinganything to them. These systems enableone to measure the motions of a given

point on a vibrating object over a widerange of frequencies. Some such systemscan even scan the surface of an object andgenerate a plot of its amplitude distribu-tion. These systems have at least onedrawback, in addition to their cost – theyonly work if the gross motion of the testobject relative to the laser is small.

. . . . . . . . . . . . . . . . . . . .JETS make noise from mixing flowOf their exhausts with air that’s slow.Their turbines may act siren-likeTo generate a spectrum spike.Bypass fans give quiet thrust,For modern aircraft they’re a must.

Sir James Lighthill, who died in July1998 at age 74, is credited with devel-

opment of the theory of jet noise. (Youmay have read that he succumbed whileattempting a nine-mile swim around oneof the islands in the English Channel – aswim that he had done earlier at least adozen times.) One of his students, JohnE. Ffowcs Williams, tried to explain thistheory to some of my colleagues and mewhile we worked together at Bolt Beranekand Newman some time ago. He showedus the basic equation, which covered anentire blackboard which wrapped aroundthe room, and he discussed the meaningand implication of each term. Althoughhe was unsuccessful in making me under-stand everything, he later went on to highacademic positions at prestigious Britishestablishments and was responsible formuch work related to control of noise ofthe Concorde supersonic transport.

According to Lighthill’s eighth-powerlaw, the sound power produced by a jetmixing with the ambient air varies as theeighth power of the jet velocity. So, aslower jet should be a lot quieter. This isprecisely what is behind the relativequiet of fan-jet engines, which in essenceproduce a wider, slower air jet than dopure jet engines, yet provide the samethrust. In the newer large-diameter high-bypass-ratio turbofan engines, jet mixingnoise usually is not the dominant compo-nent; so-called core noise generatedwithin the engines (due to combustionand density inhomogeneities) and thesiren-like noise from fans, compressorsand turbines take on more prominentroles.

Powell and Preisser6 reviewed the ad-vances in aircraft noise reduction: “Whennormalized to total engine thrust, today’snew transports are about 20 dB quieterthan those introduced in the 1950s . . .This reduction resulted from major en-

To be precise, according to the American National Standards Institute’s

specification for audiometers, the thresh-old of perception (measured at the ear) ofa pure tone at 1 kHz is 6.5 dB relative tothe standard 2 × 10-5 Pa. This sound pres-sure level corresponds to an acousticpressure of 4.3 × 10-10 atmospheres. Thethreshold generally is greater at frequen-cies that deviate from 1 kHz. For ex-ample, at 20 Hz the threshold is about 90dB higher and at 20 kHz it is about 50 dBhigher than that at 1 kHz. Although thefrequency range of human hearing is gen-erally stated as extending from 20 Hz to20 kHz, the human ear actually is sensi-tive to a wider range.3

The standard threshold of perceptionapplies to “young persons with no oto-logical irregularities.” As we get older,our hearing sensitivity at frequenciesabove a few kHz decreases, causing olderpeople to have increasing difficulty dis-tinguishing ‘t’ from ‘p’ and ‘s’ from ‘f’sounds, for example. Unfortunately,much information content of spokensounds lies in this range. To quote Biesand Hansen,4 “Old folks may not laugh asreadily at jokes, not because of a jadedsense of humor, but rather because theymissed the punchline” – I assume theymeant due to a hearing loss.

The threshold of pain due to sound inthe audio frequency range is about 145dB, corresponding to a sound pressure ofthe order of 0.004 atmospheres. In theinfrasound range, below 20 Hz, the painthreshold is higher. As discussed by vonGierke and Nixon5 intense infrasoundtypically is more felt than heard. It maycause dizziness, coughing, breathingproblems and localized pain, but gener-ally has no effect on hearing. Intense ul-trasound (above about 17 kHz) can causeheadaches, tinnitus (a spontaneous ring-ing sensation in the ears) and malaise, butits detrimental effect on hearing has notbeen generally demonstrated. These ad-verse effects on ultrasound occur only inpeople who can hear these high-fre-quency sounds; those of us who are olderare safe.

3. Peter Narins, Professor of Physiological Sci-ence at UCLA, took me to task – and right-fully so – regarding the statement I made tothe effect that the threshold of hearing cor-responds to 6.5 dB. He pointed out that thehearing of about 1.5 million visitors wastested at a Bell Telephone exhibit at the 1938New York World’s Fair and it was deter-mined that at 1.0 kHz the average soundpressure to evoke a threshold sensation was0.0002 dynes per square centimeter. Thisvalue has since been used as the standardreference sound pressure, in reference towhich the hearing threshold corresponds to0 dB. This is correct, of course. The 6.5 dB Icited corresponds to the threshold soundpressure measured at the eardrum. The ex-ternal ear provides amplification, enablingus to perceive at the eardrum a pressure thatcorresponds to an external sound pressureof 0 dB. (For more details see Chapter 123,“Hearing Thresholds” by W. A. Yost and M.C. Killion, in the Encyclopedia of Acoustics,edited by M. J. Crocker, John Wiley & Sons,


gine cycle changes that improved fuelefficiency and incremental efforts thatrequired careful optimization to preservethrust and efficiency. Low-bypass-ratioturbofan engines introduced in the 1960sprovided greater propulsive efficiencyand lower noise . . . But with jet exhaustno longer the primary noise source, fur-ther improvements in total engine noiserequired reductions in fan-generatednoise. These resulted mainly from elimi-nation of inlet guide vanes, a decrease inthe number and rotational speed of fanblades and improved blade aerodynamicdesign. A major breakthrough was the fanblade passage frequency ‘cutoff’ designconcept . . . in which the BPF tone doesnot propagate outside the engine nacelle.In addition, advances . . . allowed acous-tic treatments to be designed or tuned forenhanced absorption of the fan tones.”

Active noise cancellation is also in theworks but hasn’t quite been reduced topractical installations, as far as I know.Eventually, only noise due to flow overthe airframe itself will be left. This air-frame noise should be relatively benignin general; in tests some years ago re-searchers at Wright Field were unable tomeasure noise from aircraft in unpow-ered flight past a microphone array attimes when crickets were active.

6. “Research for quieter skies,” C. A. Powelland J. S. Preisser, Aerospace America, Au-gust 1999.

. . . . . . . . . . . . . . . . . . . .KINETIC is the energyThat always works in synergyWith energy that is potential.Both of them are quintessentialFor unforced periodic motion.We sometimes fail to grasp this notion.

I have often expressed amazement abouthow much one can understand about

vibrations from studying the simplest ofall the conceptual vibrating systems –namely, a mass connected to a spring,possibly with an added damping ele-ment. Of course, some of the utility of themass-spring model stems from the factthat the behavior of any mode of a dy-namic system corresponds to that of anequivalent mass-spring system. But whyis a simple mass-spring assemblage an ap-propriate representation of anything thatcan vibrate?

The answer is that such an assemblageincorporates an element that can store ki-

netic energy and one that can store poten-tial energy – and an interchange betweenpotential and kinetic energy is at the coreof any vibration. Let’s consider the simplestsituation of free (unforced) vibration of aspring-mass system. If the mass is deflectedfrom equilibrium and released with zerovelocity, then it initially has no kinetic en-ergy, but there is potential energy stored inthe spring. The spring accelerates the mass,giving it kinetic energy, but losing some ofits potential energy in the process. Thisgoes on until the mass reaches the equilib-rium position, where the spring’s potentialenergy storage is zero and all of the energyof the system is kinetic. As the mass movesfurther, it deflects the spring, causing en-ergy to be stored in it, but giving up a cor-responding amount of kinetic energy. Andso on.

A few years ago Sound and Vibration andI offered a prize to that reader who wouldgive me the best physical explanation (thatis, without the use of mathematics) of whya simple undamped mass-spring system hasa definite natural frequency. Although S&Vgave away the prize, I was not entirely sat-isfied with the explanation. So, here ismine. Can you challenge or improve uponit?

A system has a natural frequency if, as itvibrates in absence of external forces, ittakes the same time to complete each cycle.Because the total energy (the sum of thekinetic and the potential energy) in an un-damped freely vibrating system is constant,the instantaneous magnitude of the poten-tial energy determines the correspondinginstantaneous magnitude of the kinetic en-ergy. This now implies that whenever themass passes a given location (measured interms of displacement from equilibrium) itdoes so with the same velocity. This veloc-ity establishes the time interval it takes forthe mass to move from one point to theneighboring one. Therefore, the mass al-ways takes the same time to move from onepoint to any other point and it always takesthe time for the entire round-trip it makesin a cycle. In other words, all cycles havethe same period and thus the same fre-quency.

The foregoing argument, incidentally, isnot limited to linear springs, but also ap-plies to springs with nonlinearites. For non-linear springs, the natural frequency de-pends on the amplitude, as determined bythe initial displacement and velocity. Whyis the natural frequency of an undampedmass-spring system with a linear spring (aspring whose deflection is proportional tothe applied force) independent of the am-plitude? Borrowing from the language oftextbooks: this is left as an exercise for thereader.

. . . . . . . . . . . . . . . . . . . .The LOUDNESS of a sound we hearTells how intense it may appear.For simple or for complex tones,One measures it in phons or sones.But how we do perceive a soundDepends on what else is around.

Loudness refers to the subjectiveevaluation of a sound. A 3 dB

change in sound pressure level (whichcorresponds to a doubling or halving ofthe sound power) results in a barelyperceptible change in the perceivedloudness. A 10 dB increase in thesound pressure level (which corre-sponds to a tenfold increase in thesound power) is judged as doubling theloudness. According to Bies and Han-sen,7 if one started with 100 tromboneplayers behind a screen, all doing theirbest, and if 99 of them leave, the audi-ence would perceive a loudness reduc-tion by a factor of four. Advertisementsthat claim a 99% noise reduction forsimilar scenarios “are written by theuninformed for the ignorant.”

One ‘sone’ is defined as the loudnessof a 1-kHz tone at a sound pressurelevel of 40 dB. A 1-kHz tone at n sonesis n times as loud as this 40-dB tone. A10 dB increase in the sound pressurelevel results in doubling of the loud-ness in sones. Plots of the frequency-variations of the sound pressure levelsthat correspond to a given loudness arecalled “equal-loudness contours,”which are labeled by ‘phon’ numbers.All points on such a contour corre-spond to the same perceived loudness;thus, a 40 phon tone at 60 Hz soundsjust as loud as a 40 phon tone at 8000Hz, even though the related soundpressures may be quite different. Forpure tones, the sone and phon mea-sures are simply related, but for morecomplex sounds the situation becomesmore phoney.8 Methods for estimatingthe loudness of sounds that are notpure tones are discussed by Small andGales,9 for example.

‘Masking’ – interference in the per-ception of one sound by the presenceof another sound – may make commu-nication difficult. And, it may consti-tute a critical safety issue, for example,where construction noise may mask analarm signal or where a pedestrian’searphones may mask the sound of anoncoming car. Sound masking may alsohave beneficial effects, some of whichare realized by the installation ofsound masking systems in open-planoffices in order to eliminate the distrac-tions caused by neighboring conversa-tions. Unfortunately, the maskingneeded to cover up the rock music froma neighboring apartment would have tobe so loud that it would lead to more


tics of Flight Vehicles: Theory and Practice;Volume 1, Noise Sources, H. H. Hubbard,Ed., NASA Reference Publication 1258,1991.

13. “On the Edgetone” A. Powell, Journal of theAcoustical Society of America 33, April1961, pp. 395 - 409.

14. The Theory of Sound, J. W. S. Rayleigh,1896. Republished by Dover Publications,New York, 1945. Volume II, Section 322i.

. . . . . . . . . . . . . . . . . . . .A noisy NOISE annoys an oysterAnd quiet noise annoys a cloister.Annoyance from a sound, it’s trueDepends on what one wants to do.Shaped noise can be used to maskA sound that complicates a task.

I must admit that I have no idea whetheroysters can perceive any sound at all; I

was just carried away by the cadence ofthe words. But I do remember hearing apaper concerned with sound perceptionby fleas that was presented at an AnimalBioacoustics session of the AcousticalSociety of America a few years ago. Atthat time ultrasonic flea collars for dogswere being advertised aggressively and astudy was carried out to determine theirefficacy. This study, which was not spon-sored by manufacturers of flea collars,found that: (1) fleas cannot perceivesound; (2) ultrasound emitted by the fleacollar would be blocked and absorbed bythe dog’s fur so that little sound wouldreach any fleas; and (3) in a comparisoninvestigation, dogs wearing ultrasonicflea collars harbored a somewhat greaternumber of fleas than dogs without suchcollars.15 I don’t recall whether anyoneconcluded that dogs might be driven todistraction by the ultrasound.

According to a paper presented byDouglas Barret of HMMH at the 1999Summer Meeting of the TransportationResearch Board, nuns objected to con-struction of a highway near their convent,insisting that quiet and serenity were es-sential to their work. They protested,even though the noise at the site was pre-dicted to increase by a mere 10 dBAabove the present 45 dBA. They may nothave realized how valid their objectionswere. Highway noise levels typically arestated in terms of the energy-average lev-els observed during a day’s loudest one-hour period – and changes in this noiselevel clearly do not account for thegreater interruption of the evening quietby short-duration loud noise intrusionsfrom passing trucks.

Quite a different situation exists in the“Land of the Rising Decibels,” as de-

insanity than the music itself.

7. Engineering Noise Control, D. Bies and C. H.Hansen, Unwin Hyman Ltd., 1988.

8. David Towers of HMMH, an experiencedpunster, felt the need to top my “phon num-ber” and “phoney” wordplay. His comment:“To each his sone!”

9. “Hearing Characteristics,” A. M. Small, Jr.,and R. S. Gales; Chapter 17 of Handbook ofAcoustical Measurements and Noise Con-trol, C. M. Harris, Ed.; Mc-Graw-Hill, Inc.,3rd Edition, 1991.. . . . . . . . . . . . . . . . . . . .

MACHINERY of any kindBrings two main types of noise to mind.One’s from surface radiationDue to structural vibration.The other comes from pressured airAs air is moved from here to there.

Intake and exhaust noise usually pre-dominates in engines or compressors,

because intakes and exhausts are acous-tic monopole sources that radiate soundefficiently. What is left when intakes andexhausts are quieted often is called “cas-ing noise” – that is, noise radiated fromthe machine’s structural envelope as theresult of its vibrations. In machinerywhose internal components do not com-municate directly with the ambient air,casing noise is all there is.10

Noise-radiating casing vibrations mayresult, for example, from internal pres-sure pulses, from hydraulic systems,from imbalance of rotating parts, from re-ciprocating elements, and from impactsand other interactions of mechanicalcomponents, such as those of bearingsand gears. The latter have some particu-larly interesting aspects.

At a small lunchtime conference sometime ago, one of my colleagues pulled aball taken from a ball bearing from hispocket, rolled it across the table andasked why this shiny, smooth ball shouldmake so much broadband noise as it rollsacross a polished wood surface. Try it;you’ll be surprised how noisy it is! Whenwe later repeated the same experiment ona flat glass mirror, we again observed con-siderable noise. I don’t know whether anyresearch has been done on this problem,but I conjecture that the tiny asperities onthe surfaces interact, possibly producinglocal surface deformations and causingthe surfaces to vibrate and thus to radi-ate sound. We didn’t try lubricated balls

or balls with resilient covers, but I betthese would have produced a lot lessnoise.

Combustion noise, which is respon-sible for the roar of furnaces and the “corenoise” of jet engines, is due to nonuni-form combustion, where there in effectoccur local hot spots that behave asacoustic monopoles and thus radiatesound well. Temperature and density in-homogeneities behave as dipoles whenaccelerated in nonuniform flow.11,12

Some flow-related acoustic phenomenainvolve feedback, such as those associ-ated with edge tones and also somewhistles.13 There also may occur thermal-acoustic feedback phenomena exempli-fied by the Rijke tube, which first wasreported in 1859. As Lord Rayleigh de-scribes it:14 “When a piece of fine metal-lic gauze, stretched across the lower partof a tube open at both ends and held ver-tically, is heated by a gas flame placedunder it, a sound of considerable powerand lasting for several seconds is ob-served almost immediately after removalof the flame.” As he goes on to explain,the air column in the tube is driven atresonance by periodic transfer of heatfrom the gauze to the air, with appropri-ate phasing resulting from the combina-tion of convection with the acoustic pres-sure oscillations. This phenomenondiffers from that of “singing flames,” inwhich acoustic pressures interact withthe combustion process.

And, why do transformers make noise,where these have no intakes or exhausts,nor internal moving parts? The answer ismagnetostriction – slight changes in thedimensions of iron or steel componentsresulting from changes in the magneticfields acting on these components. Prac-tical and economic constraints make itdifficult to reduce the noise produced bylarge power transformers at the source.But I’m glad that power companies havenot let this get in the way of providing ourhomes with electricity; otherwise, wewould have to watch TV in the dark.

10. Frank Kushner of the Elliott Company haspointed out that a particular type of what Ihave called casing noise is left when intakeand exhaust noises are removed and inter-nal components of machinery do not com-municate directly with the outside air. Heindicated that piping noise tends to be animportant component, for example in thecentrifugal compressors with which he isconcerned. As he stated, sound waves atfrequencies above the cutoff frequencies ofthe attached pipes travel easily along thepipes, exciting cross-modes in the gas andstructural modes of the pipes. These pipesusually are much thinner and more compli-ant than the machinery casings. When nodissipative silencers are used in the gasstreams, external treatment of the pipesneeds to be given priority.

11. “Noise of Gas Flows,” M. S. Howe and H.D. Baumann, Chapter 14 of Noise and Vi-bration Control Engineering, L. L. Beranekand I. L. Ver, Eds. John Wiley & Sons, Inc.,New York, 1992.

12. “Combustion and Core Noise,” J. R. Mahanand A. Karchmer, Chapter 9 of Aeroacous-


. . . . . . . . . . . . . . . . . . . .An ORGAN pipe emits a toneWhen air into one end is blown.Its pitch is given by its lengthAnd somewhat by the blowing strength.In many instruments, indeed,The tone’s established by a reed.

Every introductory text on acousticstalks about organ pipes and how their

lengths are related to the wavelengthsand frequencies of the tones they pro-duce. However, discussions of howsteady blowing into a pipe produces os-cillations generally are left to specializedtexts. Clearly, if the injected airflow wereentirely smooth, no oscillations wouldoccur.

There are two basic types of organpipes: flue pipes and reed pipes. In theformer the incoming air stream passesthrough a narrow passage formed by a‘flue’ and then impinges on the edge of athin plate ‘lip.’ The resulting flow turbu-lence generates a rather broad band offrequencies in the pipe’s air volume,which responds predominantly at itsnatural frequencies. The resulting oscil-lations then interact with the turbulentjet to stabilize both that jet and the acous-tic oscillations. Reed pipes, as the nameimplies, use a vibrating brass reed tomodulate the injected air, with the pipeand reed generally tuned to the same fre-quency. More details can be found in theexcellent books by Rossing, Strong andPlitnik.17,18

The mechanism by which flutes pro-duce sound is somewhat different: it isthe same mechanism as that responsiblefor the whistle one hears as one blowsacross a bottle. A flute player essentiallyblows across a hole in the flute, resultingin some turbulence, which generatesstanding waves in the flute’s volume andthus produces tones. A piccolo is a wood-wind instrument that is about half thesize of a standard flute. Its developmenthas an interesting history. Professor Pe-ter Schickele reports that a hungry Ital-ian constructed the first piccolo bysautéing an ordinary flute in a frying panuntil it had shrunk about fifty percent.This event later came to be known as theMediterranean Flute Fry.19

scribed in a recent newspaper article.16

According to this article, the “Japaneseare subjected to a variety of clatter thatis perhaps unlike anywhere else in theworld.” Not only do their vending andATM machines talk to customers withelectronic voices and escalators tell themto watch your step, but there also aredemonstrators with bullhorns every-where. Even in rural towns one canhardly escape from the ubiquitous pub-lic address systems which spew forthmessages at all hours of the day andnight. Some public address proclama-tions quoted in the article include, “Chil-dren, go home, it’s getting dark.” “Don’tuse too much water, it hasn’t rained inrecent days.” “Make sure the stove is offbefore you go to bed.” On trains, passen-gers are instructed to turn off their cellphones, with announcements that aremuch louder and more annoying than thetelephones themselves.

Although there is much quiet objection(pun intended) to this noise pollution, acitizens’ group organized about a decadeago to fight this pollution reportedly hashad little success, largely because someof their cultural attitudes prevent theJapanese from expressing their discon-tent publicly.

We’ve all heard that one person’s mu-sic is another person’s noise. But quietmay not be the optimum situation andwhat is noise to one person may be mu-sic to another. I’ve been in noisy officesof plant managers who were happy tohear the production machinery; they feltthat they were making money as long aseverything was running, and quiet was anindication of trouble. Some plant person-nel could even identify problems fromchanges in the noise they heard in theiroffices and they objected strongly to anyproposal to give them more quiet.

In a recent survey of workers in cu-bicles in open-plan offices, about 70%reported that noise was the number onedistraction, with conversational noiseand the lack of acoustical privacy as theleading cause of acoustical dissatisfac-tion and stress. The most practical solu-tion here consists of making more noise– adding a ‘masking’ noise to reduce theinformation content of the total noiseperceived by a listener. The installationof sound masking systems has becomemore prevalent in recent years and stud-ies have shown that use of such systemshas resulted in significant productivityimprovements.

15. Dr. William Murphy of the National Insti-tute for Occupational Health and Safetysent me the following e-mail: “. . . Dr.Glenis Long presented (this) work at the St.Louis ASA meeting in 1990. . . My favoritephotograph from the study was the one inwhich more fleas were sitting on the ultra-sonic transducer of the activated collarthan on the inactive collar.”

16. “Now, the Land of the Rising Decibel,” S.Moshavi, The Boston Sunday Globe, 23September 2000.

17. The Science of Sound, T. D. Rossing, Addi-son-Wesley Publishing Co., New York, 2ndEdition, 1990.

18. Music Speech Audio, W. J. Strong and G.R. Plitnik, Soundprint, Provo, UT, 1992.

19. From “Bach to the Future,” Diane Cyr, U.S.Airways Attaché, July 1999.

. . . . . . . . . . . . . . . . . . . .In PROPAGATION through the airIndoors, outdoors, everywhere,Sound waves that spread out from asourceTake energy away, of course.Pressure by spreading is abatedAnd some by losses dissipated.

The basics of sound propagation in theatmosphere have been understood

ever since the wave nature of sound hasbeen recognized. Sound pressure de-creases with increasing distance from asource because the energy injected by thesource is spread over larger and largerareas at locations further away from thesource, and also because acoustic energyis dissipated as sound passes through theair. The attenuation due to dissipation ismore pronounced at high frequencies andwhen the humidity is high. As PhilipMorrison20 puts it: “Energy loss in soundtransport is the result of internal diffu-sion that wipes away the contrast be-tween compressed crests and rarefiedtroughs as any pressure wave advances.The longer the wavelength of the sound,the farther it can go.”

One can easily visualize that windswhose speeds increase with increasingaltitude tend to refract sound propagat-ing in the windward direction toward theground, because here the sound travelsfaster (with respect to the ground) in ar-eas of higher wind-speed. Because thespeed of sound in air is proportional tothe square-root of the absolute tempera-ture, an atmospheric temperature profilemarked by increasing temperature withincreasing altitude has a similar effect.Thus, wind (and temperature) gradientscan result in focusing of sound down-wind from a source and the formation ofquiet “shadow zones” upwind. In the1960s, test firing of large rockets atNASA’s Marshall Space Flight Center inAlabama was found occasionally to dam-age some buildings in downtown Hunts-ville several miles away as the result ofatmospheric focusing of the rockets’ in-tense low-frequency sound. NASA even-tually instituted the use of meteorologi-cal balloons to measure the wind andtemperature profiles before each planned


Circle 106 on Inquiry Card


To let us sleep and be content.Some helps ensure that we can hearWhen our retirement draws near.

According to some recent statistics,more than 20 million Americans are

exposed to hazardous sound levels on aregular basis. There are approximately 28million Americans who have some de-gree of hearing loss; about one third ofthese – more than 9 million – have beenaffected, at least in part, by exposure toexcessive noise.23

Workers in some industries in theUnited States have routinely been receiv-ing compensation for hearing loss at thetime of their retirement. This situationchanged with industry’s compliance withthe noise exposure limits promulgated bythe Occupational Safety and Health Ad-ministration (OSHA). These limits in es-sence require a worker’s daily eight-hournoise exposure not to exceed 90 dBA,with a 5 dBA increase in the permissiblenoise level for each halving of the dailyexposure period, but with no exposurepermitted above 115 dBA. For employeesexposed to noise at different levels dur-ing a day, OSHA prescribed calculationof the ratios of the actual to the allowedexposure durations for the different noiselevels and summing the resulting frac-tions – with the sum required not to ex-ceed unity.

OSHA’s regulations are based on thesimplified distillation of the consensus ofsome experts. The fact that the formulaadopted in these regulations is somewhatarbitrary, as evident from other agencies’adoption of other exposure limits, did notprevent me and my colleagues from de-veloping a precise estimate of the num-ber of overexposed miners in the UnitedStates. We relied on incomplete data onthe noise levels to which personnel asso-ciated with various machines were ex-posed, assumed certain probability distri-butions of the exposure durations andnoise levels and used only the availablepartial census of mine workers.24 In spiteof all this handwaving and smoke, ourestimation approach was adopted byother agencies and other countries.

I was reminded of the tenuous basis ofour estimate when I read a recent articlethat described how a group of psycholo-gists developed a happiness factor rang-ing from 0 to 1 and evaluated the averagehappiness of the populations of variouscountries. They also multiplied the hap-piness factor by the average life expect-ancy and came up with the average num-ber of equivalent happy years per person.In case you’re wondering, the Nether-lands had the highest happiness factor(0.797) and the second highest number(61.66) of happy years, with Iceland hav-ing the most happy years (62.04) and thesecond highest happiness factor (0.793).The United States came in tenth, with ahappiness factor of 0.760 and 57.76happy years, below the Scandinavian

test firing and postponed the testingwhen calculations indicated a potentialfor focusing in built-up areas.21

Since sound that propagates from anatmospheric layer with a lower tempera-ture to one with a higher temperature isrefracted back toward the cooler layer,sound can be “trapped” in a “sound chan-nel” formed by a cooler layer that is lo-cated between two warmer layers. Suchtrapped sound decreases with distancemuch less than freely propagating sound,sometimes enabling acoustic phenomenato be detected at very large distances. Inthe earth’s atmosphere there exists a rela-tively permanent sound channel at highaltitude, and it has been reported that the1883 Krakatoa volcano eruption could beheard on the other side of the world. Ihave some doubts about that, because theaudible components of the sound wouldhave been attenuated over such long dis-tances. However, the low-frequency com-ponents of the sound produced by the1991 Mount Pinatubo eruption and bysome nuclear explosions were detectedby instruments thousands of miles away.Similarly, in the “Sofar Channel,” asound channel in the deep portions of theoceans, sound signals have been trans-mitted and received over great dis-tances.22

As long as we are on the subject of at-mospheric refraction, it is also interest-ing to consider wind blowing along theground and across a noise barrier. Thewind needs to accelerate to get over thebarrier, resulting in a wind profile thatmay refract the sound toward the groundbeyond the barrier – thus reducing its ef-fectiveness.

20. “Wonders: Double Bass Redoubled,” P.Morrison, Scientific American, May 1998.

21. Rudy Volin wrote: “. . .During a visit to theMarshall Space Flight Center (Huntsville,Alabama) in February 1967 my hosts askedme if I wanted to visit a static test firing ofa Saturn second stage . . . I declined . . .Sure enough, the motel began to shake, fol-lowed by a brief low-frequency rumblingnoise . . . I also heard, and I forget where,that occasionally focusing caused the noisefrom rocket firings at the Marshall SpaceCenter to be transmitted to Birmingham,AL, which is a little over 100 miles south-east of Huntsville.”

22. Sound Waves and Light Waves, W. E. Kock,Anchor Books, New York, 1965.

. . . . . . . . . . . . . . . . . . .

The quest for QUIET in the nationHas led to useful regulation.Some deals with the environment

countries, Belgium, Switzerland, Austra-lia, Ireland, and Canada. The worst? Bul-garia (0.443 and 31.57 years), followed byNigeria, Belarus, and Russia (0.510 and34.48 years).25 Do these numbers suggestthat we in the U. S. would do well tomove to Iceland to collect an extra coupleof happy years? And, given the choice,should one opt for living five miserableyears with a happiness factor of perhaps0.2 or for living one glorious year with ahappiness factor near 1.0?

23. “Hearing Loss Statistics,” Anon., The OlderAmerican, p.15, September 2000.

24. “The Noise Exposure of Operators of Mo-bile Machines Used in U.S. Surface CoalMines,” Bolt Beranek and Newman Inc.,Report No. 3688, November 1977. Themethodology is described in “Statistical es-timation of percentage of overexposedworkers,” E. E. Ungar and C. B. Cruik-shank. Jr., Journal of the Acoustical Soci-ety of America, 64(1), July 1978, pp. 331-332.

25. “Happy hunting,” Gareth Cook, The BostonGlobe, October 11, 2000, p. A1. Note theprecision of the quoted numbers!

. . . . . . . . . . . . . . . . . . . .RECIPROCITY, it’s strange,Says that if we interchangeThe point at which acts excitationWith that of motion observation,The force-to-motion rate once moreWill be the same we had before.

Good old Maxwell’s theorem! If yourmemory works like mine, you may

remember its content, even though youmay have forgotten its name. I had to lookit up when I wanted to revisit the idea ofreciprocity. According to this theorem, ifa force FA acting on the beam at point Acauses a deflection DAB at point B, and ifa force FB acting at B causes a deflectionDBA at A, then DAB/FA=DBA/FB. In otherwords, the ratio of the deflection to theforce remains unchanged if the force ap-plication point and the deflection obser-vation point are interchanged. This theo-rem can be quite useful for checkingexpressions for the deflections of beamsor of other structural elements and it alsopermits one often to substitute an easieranalysis for a more complicated one.26

The reciprocity principle in essence isa generalization of Maxwell’s theorem.This principle states that the ratio of theexciting force to the observed velocityremains the same if the excitation pointand the velocity observation point areinterchanged, provided that the directionin which the force acts in each case is thesame as that in which the velocity is mea-sured in the other case. Lord Rayleigh


Circle 107 on Inquiry Card


than technical intuition to recognize thatthe maximum strain in a simple struc-tural element is proportional to the ratioof the greatest vibration velocity in the el-ement to the speed of sound (that is, thespeed of longitudinal waves) in its mate-rial – and that the constant of proportion-ality generally is less than 2.0. This facthas been demonstrated for beams andplates vibrating in bending,31 and it alsoapplies to longitudinal vibrations of rods.

This “Mach number relation” is usefulfor estimating maximum vibratory strainsand stresses. For example, consider aplate-like concrete floor on which onemeasures a maximum vibration velocityof 0.8 in./sec (2 cm/sec) – no matter atwhat frequency this occurs. Since thelongitudinal wavespeed in concrete(given by , where E is the modulusof elasticity and ρ the density of concrete)is about 1.25×105 in./sec (3.2×105 cm/sec), we find that the greatest strain doesnot exceed 2(0.8)/1.25×105 ≈ 1.3×10–5. Ifwe multiply this by the modulus of elas-ticity of typical concrete, 3.4×106 psi(23,400 Mpa), we find that the maximumvibratory stress in this concrete floor doesnot exceed 45 psi (0.3 Mpa) – a value thatgenerally is negligible from the stand-point of structural integrity. You’ll haveto agree that estimation of the maximumstructural strain couldn’t be done withless mental strain!

26. For instance, one can easily derive (or lookup) the deflection y at any point x along acantilever that carries a load F at its tip. Ifx is measured from the root (the built-inend) of the cantilever, then

where L denotes the beam’s length and EIits flexural rigidity, E represents Young’smodulus and I the moment of inertia of thecross-section. The same formula also givesthe deflection at the tip due to a force F ap-plied at a distance x from the cantilever’sroot.

27. The Theory of Sound, Vol. I, §§ 104 - 110,Dover Publications, 1945.

28. A good discussion may be found in “Inter-action of Sound Waves with Solid Struc-tures” by I. L. Vér, Chapter 9 of Noise andVibration Control Engineering, edited by L.L. Beranek and I. L. Vér, John Wiley & Sons,Inc., New York, 1992.

29. I couldn’t find the exact reference, butmuch of Prof. Theo Priede’s work is sum-marized in “Noise and Vibration Control ofthe Internal Combustion ReciprocatingEngine,” Chapt. 19 of Noise and VibrationControl Engineering, Ed. by L. L. Beranekand I. L. Vér, Wiley Interscience, NewYork, 1992.

30. For example, see Structure-Borne Sound,L. Cremer, M. Heckl, and E. E. Ungar,Spring-er-Verlag, Berlin, 1973, pp. 505 ff.

31. E. E. Ungar, “Maximum Stresses in Beamsand Plates Vibrating at Resonance,” Trans-actions of ASME, Journal of Engineering forIndustry, 84, pp. 149-155, 1962. Also seeAnnex B to ISO Standard 4866:1990(E),“Mechanical vibration and shock – Vibra-tion of buildings – Guidelines for the mea-surement of vibrations and evaluation oftheir effects on buildings.” This annex,entitled “Estimation of peak stress frompeak particle velocity,” refers to a paper byGasch, “Eignung der Schwingungsmes-

sung zur Ermittlung der dynamischenBeanspruchung in Bauteilen (Utility of vi-bration measurements for determination ofdynamic loads in structural elements),”Berichte aus der Bauforschung, 58, Wil-helm Ernst & Sohn, Berlin, 1968.

. . . . . . . . . . . . . . . . . . . .Indeed, TRANSMISSIBILITY

Gives one the possibilityOf quantifying isolation –But not in every situation.It may not show the right amountOf benefit due to a mount.

Everyone who has been concernedwith vibrations has read about trans-

missibility in terms of a mass-spring-damper system that is constrained tomove only along a line. The mass may betaken to correspond to a sensitive itemthat is to be protected from motion of asupport to which it is attached; transmis-sibility then is defined as the ratio of theamplitude of the mass to the amplitudeof the support. Or, the mass may be con-sidered to represent a machine that gen-erates an oscillatory force and the discus-sion focuses on the corresponding forcethat acts on the machine’s rigid support.Here transmissibility is defined as theratio of the amplitude of the force thatacts on the support to the amplitude ofthe force that acts on the mass.

The two aforementioned transmissi-bilities are fundamentally different. Inorder to distinguish between them, I liketo call the first one “motion transmissi-bility” and the second one “force trans-missibility.” It turns out that the math-ematical expressions one obtains forthese two different transmissibilities areidentical, at least for the case of thesimple mass-spring-damper system.Why? The textbooks I’ve seen don’t say.

The reason can be traced to the reci-procity principle, which I’ve discussedunder ‘R.’ The proof is rather simple and,in fact, leads to a more general conclu-sion. Consider a general linear system,such as one consisting of an arbitrary ar-ray of masses, springs and dampers. Ap-ply a vibration to its support and observethe resulting motion of any selected pointon the system to obtain the motion trans-missibility from the support to that point.Now, hold the support rigid, apply a forceat the selected point and observe theforce that acts on the support to obtainthe force transmissibility from that pointto the rigid ‘ground.’ These two transmis-sibilities can be shown to be identical, atleast if all points on the system are con-

(1842 – 1919), who developed the basisfor almost all of the acoustical theory inuse today, initially demonstrated that thereciprocity principle holds under certainlimited circumstances.27 More recentlyit has been shown that this principle ap-plies for any system whose differentialequation of motion is symmetric in thespatial variables. Fortunately, this re-quirement is satisfied in all of acoustics,as long as the associated processes aremathematically linear – in other words,the reciprocity principle is valid for vir-tually all practical acoustical problems –including those where both structuresand fluid volumes are involved.28

Interchanging of the excitation and re-sponse observation points can lead toexperimental simplifications if one ofthese points is more accessible than theother. An interesting example is providedby Professor Priede’s work on casingnoise of piston engines. When he wantedto determine the vibration of an enginesidewall due to a known force pulse act-ing on one of the engine’s pistons, directmeasurement would have required hav-ing a shaker act on the piston and observ-ing the sidewall’s motion with an acceler-ometer. It turned out to be much simplerto apply reciprocity to his reciprocatingengine (I couldn’t resist this juxtaposi-tion), to shake the sidewall, and to mea-sure the resulting vibrations of the pis-ton.29

Application of reciprocity to structuralvibration problems is straightforward,but for airborne sound one runs into ex-perimental complexities. For airbornesound, interchanging of the excitationand observation points is valid only if thesound source and the receiver have thesame directional characteristics – e.g., ifboth have spherical directivity.30 Never-theless, reciprocity permits one relativelysimply to relate acoustic radiation fromstructures to the responses of these struc-tures to acoustic fields.

. . . . . . . . . . . . . . . . . . .

The greatest STRAIN within a plateOr beam or rod that may vibrateIs very nearly equal toA sort of Mach Number, times two,In which velocity is foundAs fraction of the speed of sound.

One clearly would expect the vibra-tory stress in a structural element to

be proportional to the element’s displace-ment and to its vibration velocity at agiven frequency. However, it takes more

E /ρ

yFxEI

Lx= −

2

2 3

1935th ANNIVERSARY ISSUE Circle 108 on Inquiry Card


VISCOSITY, it is a factOn fluid flow has the effectOf slowing down the speed near thingsLike piping walls and airplane wings.It causes turbulence decayBy making eddies go away.

According to Sir Isaac Newton,36 “Theresistance arising from the want of

lubricity in the parts of a fluid is, all otherthings being equal, proportional to thevelocity with which the parts of the fluidare separated from one another.” Nowa-days, we would be more inclined to saythat in smooth flow with a simple geom-etry the shear stress in a fluid is propor-tional to the velocity gradient. The con-stant of proportionality, as we all learnedin Physics 101, is known as the viscosity.

Since liquids are more viscous thangases (water at room temperature beingabout 50 times as viscous as air) and lendthemselves to easier experimentation, itis no surprise that viscosity was firststudied in liquids. Jean L. M. Poiseulle(1799-1869) enabled experimenters toavoid the grief associated with directmeasurement of shear stress in a fluid bycoming up with a simple expression thatrelates the volume rate of laminar flowthrough a tube to the pressure differenceacross it, to the tube geometry and to theviscosity. In gratitude for this “Poiseulle’sLaw” the unit of viscosity in the interna-tional system (SI), dyne-sec/cm2, hasbeen named ‘poise.’

Not to be outdone by a Frenchman, theBritish mathematician and physicist, SirGeorge Stokes (1819-1903), showed thatone can also determine the viscosity of afluid by measuring the terminal velocityof droplets falling through the fluid. The‘stoke,’ the SI unit of kinematic viscosity,is named after him. It is interesting thatin 1913 Robert A. Millikan relied on SirStoke’s equation for determining thecharge of the electron from his famous oildrop experiment.

Just to obtain an idea of the wide rangeof viscosities that have been measured:the viscosity of hydrogen at 0° C is about10–4 poise, that of water at 20° C is about10–2 poise, that of typical engine oil isabout 1 poise at operating temperatures,and that of glass changes from 1013 to 107

as its temperature is changed from 400°C to 800° C.

The relative importance of viscous ef-fects in a flow regime generally is judgedby Reynolds’ number, named after Os-borne Reynolds (1842-1912). This num-ber is equal to the ratio of the inertia forceto the viscous force in the flow. At smallReynolds numbers, which typically cor-respond to low flow speeds, viscousforces predominate and tend to keep theflow laminar. At high Reynolds numbers,inertia effects overwhelm viscous effectsand the flow becomes turbulent. But still,it is viscosity that causes the turbulenteddies to decay. Some researchers, infact, consider laminar flow as a limiting

methods have been developed for thepurpose of detecting incipient structuralfailures. There exists a relatively maturetechnology for the detection of “acousticemissions” – that is, of the high-fre-quency vibrations produced in structuralcomponents as cracks begin to form orgrow. Recent patents describe an ultra-sonic inspection system that can be usedto predict the remaining fatigue life of thestructural element being inspected andan acoustic microscope for the full-depthinspection of welds as these are beingmade.

Modern medicine uses ultrasound rou-tinely for many purposes. The progress ofembryos developing in the womb is al-most always monitored with the aid ofultrasound and I would expect that mostof us have seen some of the resulting pic-tures. I found out about many other fas-cinating medical applications of ultra-sound when I listened to Larry Crum’stalk on “Recent Developments in Medi-cal Ultrasonics” at the December 2000meeting of the Acoustical Society ofAmerica. He described various tech-niques for obtaining more detailed im-ages and better contrast, enabling the lo-calization of damaged tissue, of tumorsand of blood flow. He also presentedvideo clips showing how contrast agents,consisting of stabilized microbubbles thatare injected into the bloodstream, can beused to determine the functioning of vari-ous parts of the heart. He discussed howsuch microbubbles that contain drugsmay be burst by an ultrasound pulsewhen they are in the appropriate locationto deliver the drugs to a selected site.Furthermore, he told of clinical trials inwhich high-intensity focused ultrasoundwas used to destroy tumors without in-vasive surgery and without harming ad-jacent tissues, and he described others inwhich sites of internal bleeding wereimaged by ultrasound and sealed. As heput it, “image-guided, transcutaneous,bloodless surgery devices are now underdevelopment and ‘Star Trek medicine’ isjust around the corner.” I, for one, amgrateful.35

35. I also am particularly grateful for the life-saving detection by ultrasound of cancer inone of my kidneys about ten years ago. Oth-erwise, the world would have been sparedmy rhymes.

. . . . . . . . . . . . . . . . . . . .

strained to move only parallel to a givenaxis.32,33

In many practical situations, transmis-sibility does not tell the whole storyabout the effect of isolation. For example,consider resilient rail fasteners – in ef-fect, spring-like elements placed betweenthe rails and the track bed in a railwaysystem. The transmissibility does not tellus by how much replacing the usual rigidsupports with resilient rail fasteners re-duces the vibrations that are transmittedto the track bed. The problem here is thatrail vibrations result from the interactionof irregularities on the contacting sur-faces on the wheel and the rails, so thatthe rails tend to vibrate more if they aresupported more resiliently. Thus, some ofthe isolation benefit implied by the re-duced transmissibility is negated by theincreased vibration input.

Transmissibility can tell the wholestory by itself only if the excitation doesnot change as the isolation is changed.Otherwise, one needs to account for thecharacteristics of the excitation – that is,for the way the source responds to ‘load-ing.’ Thorough discussions of these andother aspects of transmissibility and iso-lation effectiveness may be found inDenys Mead’s excellent book with therather unpretentious title of Passive Vi-bration Control.34

32. See “Equality of Force and Motion Trans-missibilities,” E. E. Ungar, Journal of theAcoustical Society of America, 30 (1), p.596, July 1991.

33. Dr. V. M. Ryaboy, now at the Newport Cor-poration, addressed this problem in thebook Elastic-Inertial Vibration IsolationSystems; Limiting Performance, OptimalConfigurations (M. D. Genkin and V. M.Ryaboy, Nauka Publishers, Moscow, 1988,in Russian) and indicated that this relationis valid also for more general conditions.

34. Passive Vibration Control, D. J. Mead, JohnWiley & Sons, 1998.

. . . . . . . . . . . . . . . . . . . .ULTRASOUND can cut or heal,Inspect for flaws, or make a seal,Or weld to make unsound things sound;Its helpful benefits abound.It’s used to image cracks in bonesAnd even break up kidney stones.

Nowadays probably everyone is famil-iar with some of the many applica-

tions of ultrasound – sound at frequen-cies above the range of human hearing.Ultrasonic welding of plastic parts iswidely used; ultrasonic inspection ofcritical components has become routine,and sophisticated ultrasonic monitoring


case of turbulent flow: “Larger whorlshave smaller whorls that feed on theirvelocity, and smaller whorls have lesserwhorls . . . and so on, to viscosity.”37

36. F. Cajori, Sir Isaac Newton’s MathematicalPrinciples of Natural Philosophy and HisSystem of the World, University of Califor-nia Press, Berkeley, 1934.

37. According to David Rice of Tulane Univer-sity, the English physicist, Lewis FryRichardson (1881-1953), wrote the follow-ing, which appears in Richardson’s bookWeather Prediction by Numerical Process,Cambridge University Press, 1922, p. 66,which book was reprinted in 1965 by Do-ver Publications:Big whirls have little whirls,That feed on their velocity;And little whirls have lesser whirls,And so on to viscosity.

As David and also Rudy Volin informed mefurther, this verse is a parody of one byAugustus de Morgan (1806-1871), whichmay be found in A Budget of Paradoxes,Longman, Green and Company, London1872, p. 377:Great fleas have little fleas upon their backs to bite ‘emAnd little fleas have lesser fleas, and so ad infinitum.And the great fleas themselves, in turn, have greater fleas to go on;While these again have greater still, and greater still, and so on.

Going back yet further, the foregoing sup-posedly was inspired by Jonathan Swift(1667-1745), who had writtenSo, naturalists observe, a fleaHas smaller fleas that on him prey;And these have smaller still to bite ‘em;And so proceed ad infinitum.

Rudy Volin indicated that the foregoingmay be found in On Poetry in the Works ofJonathan Swift, Volume XIV, Bickers & Son,London, 1883, Second Edition, p. 311. Inaddition to mentioning some of the forego-ing information, Albert George of CornellUniversity referred to F. Gilford, Jr., “Onthe Origin of Richardson’s Rhyme,” Bulle-tin of the American Meteorological Society,Vol. 53, No. 6, June 1972, and recom-mended reading Sydney Chapman’s intro-duction to the Dover reprint to learn howvery interesting a person Richardson was.Demonstrating that one can improve on aclassic, Cliff O’Hearne of Polymer Dynam-ics Inc., suggested adding the followingtwo lines to the original verse:To a state of particles discrete,In random motion known as heat.

David Rice pointed out that the classicsrefer to ‘whirls,’ not to ‘whorls,’ but I amcomforted by David Towers’ observationthat, “It’s a small whorl after all.”

. . . . . . . . . . . . . . . . . . . .

WAVES, like ripples in the sea,Transmit power as they flee.Though mass points near their homebase stay,They transport energy away.If waves spread as they propagate,Their amplitudes attenuate.

We who work in the fields of soundand vibration have a pretty good

idea of what a wave is. However, I havebeen unable to come up with a definitionthat is both simple and technically satis-factory. I felt a little better about this fail-ure when I turned to Professor Tolstoy’sdefinitive book on Wave Propagation38

and read in his first chapter that “variousdefinitions . . . can be offered. A verysimple definition can be given, once it isadmitted that energy degradation effects(e.g., attenuation due to viscosity, heatconduction, . . ., etc.) are secondary andthat waves are fundamentally conserva-tive phenomena. Once this is agreedupon, one may define a wave as nature’smechanism for transporting energy with-out degradation and without transportingmatter. . . . this most illuminating andgeneral definition . . . emphasizes the pri-mary role of waves as a means of energytransport . . .”

After we’ve thrown a pebble into apond, our eyes follow the spreadingripples that seem to tell us that the wateris moving outward from the point of im-pact. But if we focus on a small leaf or awood chip floating on the surface, we canobserve that these rock back and forthwithout going anywhere. If we are veryastute observers, we might note that theparticles move along little circles in thevertical plane. It was only fairly recentlythat sound was understood to propagatein air without the air particles travellingfrom the source to where the sound isheard. Lord Rayleigh, the father of mod-ern acoustics, pointed out that if soundpropagation involved gross motion of theair, rather than wave propagation, then acricket that is heard at a considerabledistance would need to move a tremen-dous mass of air all at once. I can’t findthe exact reference and don’t rememberthe details, but a simple calculationshows that a hemisphere of air with a 10-meter radius weighs roughly 2100 kg (or4500 lb.) – quite a lot for a little cricketto push to be heard at 10 meters.

In sound waves in fluids and in com-pressional waves in solids, the particlesoscillate in the direction of propagation.In shear waves and torsional waves, theparticles move in directions that are per-pendicular to the direction of propaga-tion. Of the various types of elastic wavesthe so-called Rayleigh waves, whichdominate the propagation of energy alongthe surfaces of solids, are of great inter-est. Such waves are typical of earthquakemotions along the surface of the earth andalso play significant roles in certain ul-trasonic inspection applications. Ray-

leigh waves are characterized by ‘retro-grade’ elliptical motion of the particles,where the planes of the ellipses arealigned with the direction of propagationand the vertical and where at its topmostposition a particle moves in the directionopposite to the direction of propagation.

The foregoing are the simplest cases,which are best known in the acousticscommunity, but many other types ofwaves have been studied. These includenot only nonacoustic waves, such as elec-tromagnetic ones, which involve thecomplexity associated with polarization,but also waves that are not governed byrelatively simple linear equations. In par-ticular, surface waves in the ocean, whichare affected by gravity and whose charac-ter depends on the depth and on the con-figuration of the bottom, are of great in-terest to oceanographers, ship designersand surfers. However, the latter are likelyto have limited interest in the underlyingscience, so we’ll just wave goodbye tothem unwaveringly.

38.Ivan Tolstoy, Wave Propagation, McGraw-Hill Book Co., New York, 1973.

. . . . . . . . . . . . . . . . . . . .XYLOPHONES owe their crisp timbreTo bars and mallets made of timber.The bars’ bright tones are amplifiedBy tuned tubes near their underside.A cousin is the glockenspiel,With bell-like sounds from bars of steel.

There aren’t many English words thatbegin with X, and I’m very happy

that one of the few that does is related toacoustics, so that I can complete this al-phabet. Although I have little firsthandknowledge of xylophones, I was able toglean some interesting information fromone of Thomas Rossing’s books39 and Ihave based some of the following discus-sion on this information.

Etymologically, the word xylophonecomes from the Greek xylon meaning‘wood,’ and phone meaning ‘sound.’ Itseems to be unrelated to “knocking onwood,” which in olden times was done toalert the spirits thought to reside inwood. At any rate, xylophones, marim-bas, vibraphones (or vibraharps) andxylomarimbas all consist of tuned barsthat vibrate and produce sound at their


The incredibly talented Thomas Young(1773-1829), who also made numer-

ous contributions in acoustics and op-tics,40 is credited with discovering thatthe elastic stress in a metal rod is propor-tional to the strain; the modulus thatbears his name is defined as the ratio ofthe stress to the strain. However, the ba-sic discovery that the elongation of a rodis proportional to the force applied to itwas made by Robert Hooke (1635-1703)before Young came into this world. Thesame Hooke, by the way, was first to sug-gest use of a rotating toothed wheel toproduce a tone of a desired pitch.41

The proportionality of force to elonga-tion or of stress to strain (or vice versa)is determined from experimental evi-dence and thus only is valid within theaccuracy of the measurements. It is for-tunate that this proportionality is a goodapproximation for metals, both in tensionand compression, for strains normallyencountered in practice. For analyses ofmetal structures, a constant Young’smodulus (also called modulus of elastic-ity) is taken to apply for stresses up to theso-called proportional limit. Beyond thatlimit, stress and strain are no longer pro-portional to each other to an acceptableapproximation. In polymers (plastics andelastomers – that is, rubbery materials),such as may be used for vibration isola-tion, loads and deflections are nearly pro-portional only for very small loads, andthe stress-strain curves in tension andcompression tend to be different, gener-ally with greater effective moduli of elas-ticity applying in compression than intension. This gives us the hint that con-ventional elastic analysis methods maynot work too well for polymeric materi-als.

If one pulls on a metal rod along itsaxis, the rod not only elongates along thisaxis but its cross-section shrinks. Simi-larly, if one compresses a solid rubberpad, the pad thickness decreases and itssides bulge out. The ratio of the strainperpendicular to the loading axis to thestrain along the axis is called Poisson’sratio, named after the French mathema-tician Siméon Denis Poisson (1781-1840). Poisson’s ratio is between about0.25 and 0.33 for most structural materi-als; for rubbery materials, it approaches0.5.

One can easily visualize that it shouldtake more force to achieve a given axialdeformation if the cross-section is pre-vented from shrinking or expanding thanit does for a rod or pad with no con-straints on the cross-section. In fact, fromelasticity theory42 one can readily deter-mine that constraining the cross sectionchanges the ratio of the axial stress to theaxial strain from E (Young’s modulus) toE(1-n)/((1+n)(1-2n)), where n stands forPoisson’s ratio. For a rubbery materialwith n near 0.5 this becomes a very largenumber indeed.

If a compressive load is applied to a

natural frequencies when they are struck.The bars of xylophones and marimbastypically are made of rosewood or of syn-thetic materials, whereas the bars of vi-braphones usually are made of alumi-num. Soft mallets are used to play themarimbas, giving them a rich, mellowtone, whereas a xylophone is played withhard mallets and has a crisper sound.

All of the aforementioned instrumentshave tubular resonators mounted near thestruck bars to amplify the bars’ tones andthe various resonances of these tubes alsoadd to the character of the instrument’ssound. Vibraphones have motor-drivendiscs at the top of their resonator tubesto open and close these, so as to producethe vibrato characteristics of these instru-ments. Because their aluminum bars ringfor relatively long times, vibraphonesalso are provided with pedal-actuateddampers that permit the musician to sup-press this ringing.

Xylophones and their cousins belongto the larger class of idiophones – definedas instruments that produce sound viatheir natural resonances when they arestruck, rubbed, plucked or shaken. Thisclass also includes gongs, bells, pianos,hollow logs and human skulls.

Idiophone also comes from the Greek.We already know what ‘phone’ means.The root of idio is the Greek idios, whichmeans peculiar or separate and indicatesindividuality or isolation – akin to theGerman eigen as in ‘eigen value’ and‘eigen frequency,’ referring to a character-istic value and characteristic (i.e., natu-ral) frequency. Yes, the English designa-tion idiot also comes from the Greekidiotos, meaning a private person or per-haps a loner – somewhat different fromthe meaning of the English word which Ishall refrain from applying to some of themodern so-called musicians.

39. The Science of Sound, Thomas D. Rossing,Addison-Wesley Publishing Co., New York,2nd Edition, 1990.

. . . . . . . . . . . . . . . . . . . .

YOUNG’S MODULUS, let me explain,Relates the axial stress to strainFor simple tension in a bar,As long as it’s not stretched too far.If the bar’s section can’t contract,Poisson’s effect gets in the act.

rubber pad that is confined between twometal plates, where there is excellent lu-brication between the rubber and metal,then the pad’s edges can move outwardwithout constraint, and the pad’s effec-tive stiffness is given essentially by Emultiplied by the pad’s area and dividedby its thickness. If the edges of the padare fully confined, then the compressivestiffness of the pad is much greater, given(at least for small strains) by the forego-ing expression with all those n terms.

If the top and bottom surfaces of thepad are bonded to the metal plates or ifthere is considerable friction between thepad and the plates, then the parts of thepad edges near these surfaces cannotmove outward, and the rest of the edgescannot move as freely as they could in theabsence of any edge constraint. In thiscase, the compressive stiffness of the padis between the two values we discussed,and it depends on the pad’s geometry aswell as on the value of Poisson’s ratio.The geometry effect is usually repre-sented in terms of the pad’s “shape fac-tor,” defined as the ratio of the pad’sloaded area to the area that is free tobulge. For example, the shape factor of arectangular pad of length L, width w, andthickness h, is equal to Lw/(2h(L+w)).Note that little bulging can occur arounda thin pad, for which the shape factor islarge – thus, the effective pad stiffness in-creases with increasing shape factor.43

The foregoing discussion of pad stiff-ness pertains only to solid pads, not topads of foam material. In foams, the ma-terial can bulge into the small interioropen spaces, so that the shape factorplays no significant role and the stiffnessof a pad of a given thickness dependsessentially only on its loaded area – a dis-tinct selection and design advantage.Suppliers of isolation pads offer padscleverly achieving the same advantage byproviding the pads with ribs, dimples,knobs, and the like, so as to obtain thesame free-to-bulge area per unit surfacearea for any (large enough) pad.

Even knowing the shape factor, we’restill not in good enough shape. A pad’sdynamic stiffness typically depends onthe static load and the amplitude of thevibration the pad experiences, and alsoon the temperature. After all, elastomersare viscoelastic materials whose stiffnessand damping properties vary with fre-quency and temperature. As such, theyalso are subject to creeping – that is, todeflecting continually under a givenstatic load.44 Much theoretical informa-tion about such materials is available, butso many factors affect any practical appli-cation that theory can only serve as aguide and development generally re-quires a great deal of empirical testing.

40. According to Robert Beyer (see next foot-note), “Young was a master of many lan-guages, both contemporary and ancient, apracticing physician and a student ofEgyptology. (He pioneered in translation


work on the hieroglyphics of the famousRosetta stone before Champollion.) He wasalso a first class physicist in optics andelasticity. He was elected a fellow of theRoyal Society of London at the age of 21,primarily for his work on the eye . . .”

41. Sounds of our Times , Robert T. Beyer,Springer-Verlag, New York, 1999.

42. For example, Applied Elasticity , C.-T.Wang, McGraw-Hill Book Co., Inc., NewYork, 1953. This old text, left over from mygraduate school days, still is one of the bestI’ve seen.

43. For example, see “Rubber Springs,” Will-iam A. Frye, Chapt. 35 of Shock and Vibra-tion Handbook, Ed. by Cyril M. Harris andCharles E. Crede, McGraw-Hill Book Co.,New York, 2nd Edition, 1976.

44. The Handbook of Viscoelastic Damping, D.I. G. Jones, John Wiley & Sons, Ltd., Chi-chester, England, 2001, contains a greatdeal of useful information about the behav-ior of viscoelastic materials – and not justabout damping.

. . . . . . . . . . . . . . . . . . . .

The number ZERO is uniqueAnd carries with it much mystique.It can mean there is no amountOr just: “Begin here to count.”If you divide by it you findAn answer that is undefined.

Ask the proverbial man in the streetabout zero, and he’ll tell you that

zero means “nothing” or that to him zeromeans nothing. We probably also canconfound him with the following more-or-less well-known syllogism:45 “Nothingis better than fresh bread. But stale breadis better than nothing. Therefore, stalebread is better than fresh bread.”

Zero certainly does not always indicatethe absence of something. If we are at zerolatitude, that does not mean we are no-where. It merely means that we are on theequator. A temperature of zero degreesdoes not imply that there is no tempera-ture and a sound pressure level of 0 dBdoes not mean that there is no sound.Zero is often taken to represent an arbi-trary datum from which deviations aremeasured, and the datum has no mean-ing other than that given it by its defini-tion.

It is interesting to observe that the earlynumber systems, including that of theRomans, represented numbers withoutusing zeros. For example, a Roman mighthave written that in just VII years fromthe year MMIII, we’ll be in the year MMX– in contrast to our using zeros and “po-sitional notation” to write that in just 7

years from the year 2003, we’ll be in theyear 2010. Recall that in our decimal sys-tem we use positional notation proceed-ing from right to left, going from 100 = 1to higher powers of 10, so that the num-ber 2003 means (3×100) + (0×101) + (0×102)+ (2×103). Note that in positional notationzero is not really a number, but merely a“place holder” indicating the absence ofan entry corresponding to a particularpower of 10.

However, mathematicians wanted totreat zero like a number, and that neces-sitated making up some special rules forzero, so as not to spoil the system ofmathematical operations. Because multi-plication of any finite number by zero isdefined as resulting in zero, one finds forexample that 7×0 = 3×0 = 0. Thus, if zerowere to behave like an ordinary number,we could divide through by zero and con-clude that 7 = 3 = 1. In order to avoiddestruction of the structure of arithmetic,division by zero is not permitted – or, inother words, the result of division by zerois undefined.

And, what does 60 mean, for example?We know that 62 = 6×6, that 63 = 6×6×6,and that 6n represents the result we getby multiplying 6 by itself n times. In or-der to preserve the usual rule that ab+c =ab×ac, which follows directly from thedefinition of an, with n representing anyinteger, one needs to require that ab×a0 =ab because b+0 = b. This requirement canonly be satisfied if a0 = 1 for all numbersa, resulting in the counter-intuitive state-ment that multiplication of any numberby itself 0 times (or, of multiplication ofa number by itself not at all?) results in1.

As we know, if we multiply any finitenumber by 0 we get 0. Clearly, 0×0 = 0,and 0n = 0 for all integers n. Now, here isany interesting question: if a0 = 1 for alla and 0n = 0 for all n, is 00 equal to 1 orto zero? If you want to pursue this ques-tion and learn more about the structureand philosophy of mathematics, as wellas about the origin and myths that sur-round development of the idea of zeroand the symbol 0, I recommend that youperuse the very erudite little book en-titled The Nothing That Is; a Natural His-tory of Zero.46 I have based much of theforegoing discussion on what I read inthat book and I close with its intriguingintroduction: “If you look at zero you seenothing; but look through it and you willsee the world. For zero brings into focusthe great, organic sprawl of mathematics,and mathematics in turn the complexnature of things . . .”

Now, at long last, this series of alpha-betical excursions into acoustics-relatedtopics and word play is finished. And wehave arrived at nothing.

45. A syllogism, as you may recall, is a logicalconstruct like: “If A is greater than B and Bis greater than C, then A is greater than C.”

46. By Robert Kaplan, Oxford University Press,1999.

AcknowledgmentAs I have mentioned, the idea for this

alphabetical series came from one pub-lished in 1958 by the late ProfessorLothar Cremer of the Technical Univer-sity of Berlin. I am greatly indebted toJack Mowry, the editor and publisher ofSound and Vibration, for his encourage-ment and enthusiastic support of my ABCeditorials, which are collected here. I alsoam grateful to Jack for his involving hisArt Director, Jerry Garfield, in this projectand to Jerry for his cartoons, which man-aged to capture just the right nuances ofmy essays.

I have particularly enjoyed all the com-ments and inputs I have received from somany of the readers of Sound and Vibra-tion and from my colleagues at AcentechIncorporated. I have had great fun writ-ing this series and am a bit sorry thatthere aren’t more letters in the English al-phabet.

The author can be contacted at: [email protected].

The proper usage and meaning ofwords is of vital importance in thetechnological world. We have an in-grained habit of misusing the wordaccuracy. We say that a measurementor a value has an accuracy of ±1%.What does this mean to you? I hopeyou say that it is terribly inaccurate.In that case, why boast about it? Inthe engineering world, if ±1% accu-racy is good, ±1/2% accuracy is evenbetter! Of course, we mean inaccu-racy or uncertainty. I prefer the lat-ter. How did this misuse ever begin?It is utter nonsense, but it is very dif-ficult to overcome. Please help.

Now let me point out a commongrammatical error. The term ‘data’pops up often in technical literature.Many people treat the word ‘data’ asa singular noun. But it is plural. It isa Latin word: datum is singular anddata is plural. We are so used to see-ing, “The data is good” that “Thedata are good” sounds improper.That is unfortunate. But we havemade more strides in overcomingthis error than we have with the mis-use of the word accuracy.

One more thing, please do not buyinto the popular argument that gram-matical errors and misuse of wordsare not important because the onlything that matters is that people un-derstand what you mean. That is ri-diculous. If you don’t believe it, aska lawyer.

Anthony J. Schneider, The LittleBrown Book

Misuse of‘Accuracy’ and ‘Data’

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Acoustics from A to Z - sandv.com · Acoustics from A to Z ... against which to judge the measured val-ues ... Wiggling small hair cells that sense tones