Physics 141 Wave Motion 2 Page 1

Wave Motion 2

If at first you don't succeed, try, try again. Then quit. No use being a damn fool about it.— W.C. Fields

Waves in two or three dimensionsOur description so far has been confined to waves in which the energy moves only along one line. For waves in a string this is good enough, but energy in sound, water and light waves generally spreads out in two or three dimensions.The main new concepts needed in more dimensions are these:The direction of a harmonic wave is specified by giving a wave vector k. Its direction is that of the energy flow; its magnitude is , as in the one-dimensional case. The wave function at a position r relative to a small source then takes the form

.The energy spreads out in space over larger and larger areas, so the intensity (power per unit area) must decrease with distance from the source. In the simple case where the energy spreads equally in all directions, the intensity at distance r from the source is equal to the power emitted by the source divided by the area of a sphere of radius r:

.

A wave of this type is a spherical wave. Since intensity is proportional to the square of the amplitude, the amplitude of a spherical wave must fall off with distance as 1/r.

We are often dealing with the situation where the waves are received by a relatively small detector (ear, eye, or whatever) which is at a large distance from the source. This detector samples only a very small part of the spherical wave, so the curvature of the wavefront is negligible. A good approximation in that case is to treat the waves as one dimensional, moving directly away from the source with constant amplitude. This is the “plane-wave” approximation.

Decibel scale of loudnessOur perception of loudness of a sound is based on the response of our ears to the intensity of the waves entering them. Sound intensities vary over a vast range, but fortunately our ears respond (approximately) to the logarithm of the intensity. For this reason, a logarithmic scale of intensities is commonly used for the loudness of sound.The standard scale is based on a unit called a decibel (db). An arbitrary reference intensity is chosen (one usually picks , approximately the faintest audible sound). The received intensity is then converted to the sound loudness , measured in db according to the definition:

k = 2π /λ

Φ(r ,t) = Acos(k ⋅r −ωt +δ )

I(r) = Psource4πr2

I0 I0 = 10−12 W/m2

β

Loudness (in db) β = 10log10II0

⎛⎝⎜

⎞⎠⎟

Physics 141 Wave Motion 2 Page 2

A sound of intensity is painful to most hearers. This is a loudness db. It is typical for the sound near the stage in a rock concert.

The decibel name comes from “deci”, meaning one-tenth, and “bel”, a unit named after A.G. Bell, the inventor of the telephone.

The decibel scale is also used to describe amplification or attenuation of signals in electronic equipment. In those cases the relevant variable is power, not intensity.

The Doppler effectConsider a source emitting harmonic waves along a straight line toward a receiver located on that line. We are interested in the effects of motion along the line of the source, receiver, or both.If the source is stationary relative to the medium, the waves it emits have equally spaced wave crests. However, if the source is moving the wave crests in front of it are crowded closer together, while those behind it are spaced farther apart. Relative to the medium, the wavelength is smaller in front of the source and larger behind it. A stationary receiver in front of the moving source will detect more wave crests per second than if the source had been stationary, so the frequency received will be higher than that of the source. A receiver placed behind the moving source will detect a lower frequency. If the source is stationary but the receiver is moving, similar things happen. A receiver moving toward the source receives more wave crests per second than a receiver at rest. The frequency received is thus higher. Conversely, if the receiver moves away from the source the frequency received is lower. These phenomena constitute the Doppler effect. It is a general property of waves.For waves in a medium (such as sound) there is a simple formula relating the received frequency to the source frequency :

Here v is the wave speed, is the velocity of the receiver, and is the velocity of the source (all relative to the medium). The drawing shows the case for which both and

are positive numbers (the source is chasing a fleeing receiver). Other cases are handled by making one or both of these speeds negative.Note that as the received frequency becomes infinite. The entire wave collapses into a single pulse, called a shock wave. The “bow wave” created by a boat moving faster than the speed of water waves is a familiar example, as is the “sonic boom” caused by an object traveling faster than the speed of sound.The formula given is for “mechanical” waves resulting from motion of particles in a medium. Light is different, in that there is no medium supporting the wave, only electric and magnetic fields. The Doppler effect still occurs, but the formula is somewhat

1 W/m2 β = 120

fR fSS RvS vR

Medium at rest

vR vSvR

vS

vS→ v

Doppler effect (waves in a medium) fR = fSv − vRv − vS

Physics 141 Wave Motion 2 Page 3

different. If the speeds of source and receiver are small compared to the speed of light waves (c), the formula given above can be used as an approximation.

It is through the Doppler effect for light from distant galaxies that we know the universe is expanding in a way consistent with the “big bang” model.

Wave groupsReal waves begin and end; they cannot be single harmonic waves, but must be a superposition of waves with different wavelengths and frequencies. A real wave thus does not have a precisely defined wavelength or frequency, but rather involves distributions of these, with an average value and a "spread" around the average. Of course, any wave also lacks a precise “position” at which it can be said to be located.Shown is a wave "group" centered at x = 0.

Call the spread in position Δx. To make such a wave requires superposition of harmonic waves with wave numbers having a spread of size Δk around the average value of k.To make a highly "localized" wave requires superposition of a large number of different wave numbers, so the more precisely we know where the wave group is, the less precisely we know its wave number (or wavelength). Conversely, to have a relatively well defined wave number, the wave must spread over many wavelengths. There is a mathematical theorem relating these "uncertainties" in position and wave number:

.

In quantum theory the momentum of an object is proportional to k, and in that context this inequality becomes the famous Heisenberg Uncertainty Principle.

The motion of such a wave group can be complicated. The individual peaks of the waves move with the wave speed:

.

The pattern as a whole moves with the group speed:

.

These speeds are not in general the same. The energy density is substantial only where the wave amplitude is large, so energy travels at the group speed.

Δx ⋅ Δk ≥ 12

Wave speed: vw =ω avkav

Group speed: vg =dωdk

Physics 141 Wave Motion 2 Page 4

It is possible for the wave speed to be smaller or larger than the group speed. Information transmitted by a wave is carried by its energy, and thus travels at the group speed. No information can travel faster than the speed of light in vacuo, called c. There is no such restriction on the wave speed.

BeatsA relatively simple case of wave groups occurs with superposition of only two harmonic waves, of slightly different wavelengths and frequencies.We use the general function for two superposed waves, Eq (1) on page 6 of Wave Motion 1 (with ). At we have:

.

The figure below shows this function for , Δk = 2.

The short wavelength oscillations are from the term, while the slowly varying term "modulates" the amplitude, giving the overall envelope.

As time goes on, the waves move to the right, changing the pattern. The individual peaks travel with the wave speed; the groups in the envelope travel with the group speed. These speeds are not necessarily the same.Now consider the number of large groups (called beats) that pass a given point per second. The length L of one group (the distance between points where the modulation factor is zero) is given by , so . Since the group moves at the speed

, the time it takes a group to pass a particular point is . The number of groups passing per second, the beat frequency, is 1/T, so we have:

The beat frequency is the magnitude of the difference between the two original frequencies. This simple formula has many applications, from tuning pianos to radar speed detectors.

Standing waves in a stringConsider again two harmonic waves with the same amplitude, wavelength and frequency, but now moving in opposite directions. We get the formula for the sum of these waves from Eq (1) on page 6 of Wave Motion 1, by the simple trick of replacing k by – k in the second wave. The result is

. (1)

δ = 0 t = 0y(x,0) = 2A ⋅cos 12 Δk ⋅x( ) ⋅cos(kavx)

kav = 50

cos(kavx)

cos 12 Δk ⋅x( )

12 Δk ⋅L = π L = 2π /Δk

vg = Δω/Δk T = L/vg = 2π /Δω

y(x,t) = 2A ⋅cos(kx −δ /2) ⋅cos(ωt −δ /2)

Beat frequency fbeat =Δω2π

= Δf

Physics 141 Wave Motion 2 Page 5

Note that the dependences on x and on t are now in separate factors. This kind of disturbance does not result in a net transport of energy in either direction, although there is energy in it. It is called a standing wave. The particles execute SHM with an amplitude and energy that varies from place to place.One can produce standing waves by reflection from a boundary between two media, such as where a string is attached to a wall. The standing waves arise from the superposition of the incident and reflected waves.

To get equal amplitudes in the two waves, the reflection coefficient must be 1, which we will assume as an approximation. As was noted earlier, for a string attached to a wall, R is very nearly equal to 1.

String fixed at both ends. Let the string have length L, fixed at walls located at and . A fixed end cannot move, so we must have at all times, which by Eq (1) gives . The wave function becomes:

.

We must also have at all times, so . This can hold only for those values of k that satisfy

Zero or negative values of L do not exist, of course, so only positive multiples of π appear.

This condition restricts the values of k; this means the wavelengths and frequencies of the standing waves can have only the following values:

Only standing waves obeying these restrictions can exist in the string fixed at both ends. The values of n give the various “modes” of oscillation of the string. They are usually called harmonics. The case , for example, is the 2nd harmonic. The 1st harmonic is also called the fundamental.Shown is a string of length vibrating in the 5th harmonic, at the times when :

x = 0x = L y(0,t) = 0

δ = πy(x,t) = 2A ⋅sin(kx) ⋅sin(ωt)

y(L,t) = 0 sin(kL) = 0kL = π , 2π ,3π ,…

n = 2

sinωt = ±1

At other times the string configuration lies between these limiting curves.Note that there are places where the string remains always at rest. These are nodes. Of course there are nodes at the fixed ends of the string, but for the nth harmonic there are also n−1 equally spaced nodes at points between the ends.Halfway between each pair of nodes is a point where the oscillation of the string has its largest amplitude. These are antinodes. There are n antinodes for the nth harmonic.An actual string usually oscillates in a complex way, with many harmonics contributing simultaneously, in a manner that depends on how the string was put into motion. But any possible oscillation can be analyzed as a superposition of the various allowed harmonics. This decomposition into constituent modes is called harmonic analysis.

Standing Sound Waves in Pipes

The above analysis can be applied to sound waves traveling back and forth in a cylindrical pipe, but there are some important differences:• An open end of a pipe, where the pressure is fixed at normal air pressure, is a node in

terms of the pressure variation; it is analogous to a fixed end of a string.• A closed end of a pipe is an antinode for pressure variation.A pipe open at both ends is thus like a string fixed at both ends. The same formulas apply for the standing wave modes.A pipe open at only one end has a node at one end and an antinode at the other. This changes the boundary conditions. Let y represent the pressure variation and let the open (node) end of the pipe be at x = 0 . Then y(0,t) = 0 , which requires δ = π as before.

But now we require an antinode at the closed end ( x = L ). This means sin kL = ±1 , so the restriction on the wave number k now becomes

kL = π

2, 3π

2, 5π

2,…

In terms of wavelengths and frequencies, the allowed modes are

Physics 141 Wave Motion 2 Page 6

String fixed at both ends

(or pipe open at both ends)

λn =2Ln

fn = n ⋅v

2L

⎫

⎬⎪⎪

⎭⎪⎪

where n = 1,2,3,...

Physics 141 Wave Motion 2 Page 6

At other times the string configuration lies between these limiting curves.Note that there are places where the string remains always at rest. These are nodes. Of course there are nodes at the fixed ends of the string, but for the nth harmonic there are also equally spaced nodes at points between the ends.Halfway between each pair of nodes is a point where the oscillation of the string has its largest amplitude. These are antinodes. There are n antinodes for the nth harmonic.An actual string usually oscillates in a complex way, with many harmonics contributing simultaneously, in a manner that depends on how the string was put into motion. But any possible oscillation can be analyzed as a superposition of the various allowed harmonics. This decomposition into constituent modes is called harmonic analysis.

Standing sound waves in pipesThe above analysis can be applied to sound waves traveling back and forth in a cylindrical pipe, but there are some important differences:• An open end of a pipe, where the pressure is fixed at normal air pressure, is a node in

terms of the pressure variation; it is analogous to a fixed end of a string.• A closed end of a pipe is an antinode for pressure variation.A pipe open at both ends is thus like a string fixed at both ends. The same formulas apply for the standing wave modes.Pipe closed at one end. A pipe open at only one end has a node at one end and an antinode at the other. This changes the boundary conditions. Let y represent the pressure variation and let the open (node) end of the pipe be at . Then , which requires as before. But now we require an antinode at the closed end ( ). This means , so the restriction on the wave number k now becomes

In terms of wavelengths and frequencies, the allowed modes are

This situation differs from the case of a pipe open at both ends in two important ways:• The fundamental frequency is half that of a pipe of the same length open at both

ends. • The even numbered harmonics are missing.These formulas apply as given only for cylindrical pipes. They also neglect, as an approximation, a slight dependence on the diameter of the pipe, and the fact that the pressure node at the open end occurs a small distance outside the actual end.

The formulas for a pipe open at only one end would also apply to a string fixed at only one end, but there are few practical cases where standing waves are important in such a string.

n−1

x = 0 y(0,t) = 0δ = π

x = L sin kL = ±1

kL = π2, 3π2, 5π2,…

Pipe open at only one endλn =

4Ln

fn = n ⋅v

4L

⎫

⎬⎪⎪

⎭⎪⎪

where n = 1,3,5,...

Physics 141 Wave Motion 2 Page 7

Musical instrumentsExcept for those that use electronic circuits to generate their sound, all musical instruments producing definite pitches (frequencies) operate by creating standing waves in some kind of mechanical system. The distribution of energy among the various harmonics gives the characteristic quality of sound (timbre) of the instrument. Here we will discuss the simplest aspects of those instruments using strings or pipes.The stringed instruments all make use of the standing wave patterns of a string fixed at both ends. They differ in the range of pitches covered, in how the standing waves are set into motion, and in how the vibrational energy of the string is transferred to a larger surface so as to produce a substantial sound wave in air.The piano and harp have one or more strings for every pitch played. The strings of the piano are set into vibration by striking them with felt-covered wooden hammers linked mechanically to the keyboard. The harp player plucks the strings with fingers.Because it is desirable to keep the string tensions nearly the same, other parameters are used to vary the pitch. In the upper part of the range the strings all have the same mass per unit length (thus the same wave speed), so the pitch is varied by changing the length of the string. For low pitches this would make the strings unmanageably long, so the wave speed (and thus the frequency) is lowered by increasing the mass per unit length of the strings, without changing their length. These factors account for the characteristic shape of the (grand) piano and the harp. Both instruments transmit the sound to air by means of a wooden “resonator” or “sound board” of substantial surface area, which is coupled mechanically to the strings and vibrates like a driven oscillator.Members of the violin and guitar families have only a few strings. For example, the violin has four, all of the same length and at approximately equal tension, but of different mass density. To play notes other than those produced by the “open” strings, the player shortens the length of the string by pressing it down against a wooden “fingerboard” with the fingers of the left hand.The strings are set into motion on the guitar by plucking with the right hand fingers. The violin is generally played by pulling the bow across the string. This device has taut horse hair coated with pine rosin, which alternately sticks and slips, allowing the production of a continuous sound. In both the violin and the (acoustic) guitar the sound is amplified by a resonator box. The detailed properties of this resonator largely determine the sound quality — and very likely the price — of the instrument.The woodwind instruments and the pipe organ produce standing sound waves in a pipe by blowing air in at one end, providing the driving force for a driven oscillator. The frequencies produced are the standing wave (resonant) frequencies of the pipe. Non-resonant frequencies have negligible power because there is little damping.The organ has one pipe (or more) for each musical pitch covered. The sound is generated by blowing air across a sharp edge at one end of the pipe. The pitch is determined by the length of the pipe, and whether it is open or closed at the other end.The woodwind instruments have only one pipe, but the effective length is changed by opening or closing holes in the side of the pipe.

Physics 141 Wave Motion 2 Page 8

The flute is much like an organ pipe. Sound is produced by blowing across an opening at one end, and the other end is open. All multiples of the fundamental frequency occur in the sound, but most of the energy is in the fundamental mode.The clarinet is a member of the “reed” family with a cylindrical pipe. A wooden strip (reed) clamped against a solid support can open slightly, and when the player forces air between the reed and the support the reed vibrates, alternately opening and closing the air supply. This end of the pipe is closed as far as standing waves are concerned. The other end is open, so the harmonics produced are only the odd multiples of the fundamental. The absence of the even harmonics gives the clarinet its characteristic “hollow” sound.The oboe and bassoon have a pair of reeds clamped together, providing the alternate opening and closing of the air supply. The pipe is thus closed at one end and open at the other. However, these pipes are not cylindrical, but flare out like a cone. This has the effect of allowing all multiples of the fundamental to be produced.In the brass instruments the lips of the player provide the effect of a double reed. The flare of these instruments is often quite complicated, and the series of harmonics correspondingly complex. The length of the pipe can be varied by valves that open or close (in the trumpet or horn), or by a sliding piece of the pipe (in the trombone). In most cases the player normally plays harmonics higher than the fundamental.