36 PHILIPS TECHNICAL REVIEW VOLUME 22

In October 1959, ProfessorDr. Balthasar van der Pol diedat Wassenaar at the a.ge of 70.In the years from 1922 to 1949,during which he acquired inter-national repute in the field ofradio science and engineering,he directed the work done in thutfield in Philips Research Labo-ratory at Eindhoven. In thesame period he also played aprominent part in internationalorgunizoiionol acttvttteS con-cerned with radio comnwnica-tions. After his retirement fromindustry he continued in theseactivities, principally as Directorof the C.C.I.R. (Comité Consul-tatif International des Radio-communicauonsy at Geneva, anoffice he held until a few yearsbefore his death. Amongst thenumerous distinctions conferredupon him lVere the Medal ofHonor of the Tnsiitute of RadioEngineers, of which he was forsome time vice-president, theVoldemor Poulseri Gold Medal,several honorary doctortues, andthe honorary presidency of theV.R.S.!. (V nion Radio Scien-tifi.que J nternationale).

The ob uuary notices thatappeared in journals both. inthe Netherlands and abroadwere obviously unable to go far

into the details of his scientijic work. lt. gives the editors pleasure to be able 1.0 publish liere a more comprehensive appreciation,written by Prof Bremmer, one of Vnn. der Pot's former and closest collaborators and probably the most familiar with his work. In0"" view such an ap preciauoti is a fiuing way of paying tribute to the memory of this remarkable man, and cl! the scrne time itgives the reader a. glimpse of many important aspects of the euolutum of radio science.

The utle photograph shows Van der Pal (right) in conuersat ion. with Prof. Holst, the founder and first director of this laboriuory .

THE SCIENTIFIC WORK OF BALTHASAR VAN DER POL

by H. BREMMER. 538.56.001.5 :621.37.001.5

In this article we shall try to review the many-sided scientific work done by the late Prof. Dr.Balthasar van der Pol, former member of the PhilipsResearch Laboratories, Eindhoven, who died on the6th October 1959. His numerous investigationswere concerned primarily with problems of radioscience and associated fields. The international namehe SOOD acquired was due no doubt in the first place

to his pioneering work on the propagation of radiowaves and on non-linear oscillations. We shall seelater how his study of these subjects led him totake an interest in many other problems, includingproblems of pure mathematics.

Van der Pol was one of those rare men who areequallyat home in physics, engineering and mathe-matics. This enabled him to deal personally with

~ ~~- ----~~~~~~~~~~~~~~~~~~~------:---___:--ï

1960/61, No. 2 BALTHASAR VAN DER POL 37

the theoretical-mathematical treatment of tech-nical problems, and also to meet professional mathe-maticians on their own ground and persuade themof the importance of a rigorous mathematicalapproach to such problems. He saw how difficultit often is to bring together the practitioners inthese three fields. His success in this respect was avaluable facet of his life's work.As a mathematician, Van der Pol showed a strong ~

preference for heuristic methods, by which, alongpartly intuitive lines, results are readily arrivedat that can only be verified later. To this extenthe resembled Heaviside. This brilliant man, wholived from 1850 to 1925, used mathematical methodswhich proved to he very fruitful, but generally leftit to others to demonstrate their validity. Heavisidewas nevertheless the first to get to the bottom ofsuch technical problems as the role of inductancesin long-distance telephone cables. Van der Pol wasa great admirer of this self-made scientific genius,as testified by his inaugural address on 8th December1938 as extra-mural professor at Delft 1). The addresswas entirely devoted to Heaviside, and underlinedthe significance of his work at a time when the.application of Maxwell's equations was not yet a'commonplace, The following sentence from Van derPol's address will serve to illustrate the part playedby Heaviside in the development of electrotech-nical concepts that are n~nv thoroughly familiar:

"Electrotechnology and physics are also indebtedto Heaviside for the concept and the word"impedance", not merely as understood in pre-sent-day alternating-current theory, but in a verymuch wider sense which includes on-off switchingeffects, a form which even today has still notbeen dealt with exhaustively."In all his work Van der Polowed much to the

inspiration of Heaviside. This appears directly fromhis preoccupation with the methods of operationalcalculus, which were created by Heaviside forapplication to electrical networks. No one hasexplained better than Van der Pol the merits ofoperator methods. In particular he demonstratedtheir value for the discovery of numerous relationsin widely diverse fields of mathematics. In onerespect, however, there was a marked constrastbetween Heaviside and Van der Pol. The formersuffered all his life from the difficulty of getting hisideas accepted, not least because of the almostillegible form of his writings. The latter's publi-cations, on the other hand, were models of carefulexposition and cl~rity of th0u.ght.

1) B. van der Pol, Oliver Heaviside (1850-1925), Ned. T.Natnurk. 5, 269-285, 1938.

Van der Pol's lucidity was also much in evidencein the lectures he gave and in the innumerablemeetings and discussions over which he presided.In this connection, mention must he made of hisconsiderable talents as an organizer. The ease withwhich he could present a matter, and his completefamiliarity with the whole field of radio, soon madehim a leading figure in international organizationsconcerned with radio science and engineering,especially in the U.R.S.L and the C.C.LR. Itwas no surprise when, after retiring from PhilipsResearch Laboratory, he was invited to he the firstdirector of the C.C.LR. upon the establishment ofits permanent secretariat. In the time that he heldthis office (from 1949 to 1956) he worked withdevotion to ensure that justice was done to purescientific research in the advice issued on the pre-paration of technical regulations for internationalradio communications. It is scarcely believablethat, with all his activities on international com-mittees, he should still have found the time forabstract investigations. He succeeded in this onlybecause his zest for work.was quite out of the or-dinary, and remained with him to the end of his life,

V.ander Pol maintained that only a few great menhave had a decisive bearing on the developmentof physics. He used to say that, faced with the enor-mous number of scientific investigations, it waseasy to forget that the really fundamental work iscontained in a mere handful of publications. Hehimself was therefore strongly drawn towards per-sonal contact with the leading figures in the worldof science. In that respect he was fortunate at thevery beginning of his career. After taking his degreecum laude at Utrecht in 1916, he did experimentalresearch in England unti11919. At first he workedin London under Fleming, the man who, in 1904,was granted the first patent on a diode. There fol-lowed a period of two years at Cambridge, where·he worked under Sir J. J. Thomson. On his returnto the Netherlands he was attached for three yearsto the Teyler Foundation at Haarlem, where heworked under Lorentz. Thus, Van der Pol was inclose association .with two scientists, one of whomhe himself wittily described as the "discoverer"of the electron (in 1897 Thomson had determinedthe ratio elm from electron orbits in a combinedelectrical and magnetic field) and the other as the·"inventor" of the electron (in 1896 Lorentz had,calculated elm from the Zeeman effect). In the yearsthat followed, Van der Pol was in close contact with.other distinguished scientists, including Appleton , •who was awarded the Nobel prize in 1947 for hispioneering research on the ionosphere.

3B PHILIPS TECHNICAL REVIEW VOLUME 22

Propagation of radio wavesIn England Van der Pol soon came up against a

problem which was to remain one of his great life-long interests. We refer to the effect of the earth onthe propagation of radio waves. In 1909 Sommer-feld 2) had put forward a theory which explainedthe observed phenomena on the assumption that theearth's surface could be regarded as flat. As earlyas 1901, however, Marconi had succeeded in sendingradio signals across the Atlantic Ocean. In 1915,after experience had been gained in the use of un-damped waves, it had even proved possible toestablish telephonic communication by radio acrossthe Atlantic. It was evident that the curvature ofthe earth must have some effect on the propagationof the waves, because linear propagation straightthrough the highly absorbent earth would attenuatethe signal beyond possibility of detection. It wasconceivable that, as a result of diffraction, the wavesmight follow the surface of the earth far beyond thehorizon as optically observed from the transmittingaerial. The existing theory on this idea was criti-cized by Poincaré, who showed in 19103) that, in thecase of diffraction, the field over great distancesmust decrease proportionally to exp (-{JDf},,!),where À. is the wavelength and D the distance fromthe transmitter to the receiver measured over theearth's surface. Unfortunately, it was not at thattime possible to test this theory, since there was noway of deriving a reliable numerical value for theconstant {J.This was roughly the situation when Van der Pol

first came into touch with the propagation problem.Unless the value of {Jwas extremely small, the fieldupon diffraction would be so strongly attenuatcdthat there could only be one other explanation forthe long range of the radio waves: the presenceof the ionosphere. The existance of conductinglayers at high altitudes in the atmosphere, knownas the ionosphere, was postulated in 1902 by Ken-nelly and Heaviside to account for the propagationofwaves over great distances. The radio waves wouldthen follow a zig-zag path, being reflected succes-sively from the ionosphere and from the surface ofthe earth.The mathematical difficulties of the pure dif-

fraction problem, assuming the absence of theionosphere, were bound up with the numerousBcssel functions occurring in the solution. Thisinduced Van der Pol to encourage the rnathema-tician Watson to study this problem, at a time when

2) A. Sommerfeld, Ann. Physik 28, 665, 1909.2) H. Poincaré, Jahrb. drahtl. Telegr. Teleph. 3, 445, 1910.

the latter was working on his well-known book onBessel functions, which appeared in 1922. Watson'sresults, which were published in 1918 4), provedbeyond doubt that the value of {J was too large(it would be 0.00376 km-s if the earth were a per-fect' conductor and the atmosphere completelyhomogeneous) to explain the long range of radiowaves without invoking the help of the ionosphere.The implications of Watson's calculations werediscussed by Van der Pol in a paper published in19195).

\

This did not mean, however, that further studyof the pure diffraction problem, leaving the iono-sphere out of account, was no longer necessary.In the first place the wave propagated along theearth, the "ground wave", makes an importantcontribution over short distances, particularlyduring the day, compared with the contributionfrom the "sky wave" which is propagated via theionosphere. Moreover, the ionosphere has littleeffect on the extremely short waves used in tele-VISIOn, frequency-modulated transmissions andradar. The further study of the ground wave wastherefore still of importance. In cooperation withK. F. Niessen, Van der Pol elaborated on Sommer-feld's theory 6), which could still be used for shortdistances, and made a special'study of the influenceof the height of the transmitting and receivingaerials above the ground. In later years the ground-wave theory was worked out by Van der Pol andthe present author for distances at which the earth'scurvature plays an essential part 7). A mathematicalmethod was developed for computing the fieldsof the ground wave for all topographical conditions.It was found that, even in the case of very shortwaves, the decrease of the field strength beyondthe transmitter horizon was very much more gradualthan had formerly been thought. This is illustratedin fig. 1,where the ratio of the actual field strengthE on the earth's surface to the field strength Epr

in free space is plotted as a function of the distancefrom transmitter to receiver for a given case, viz.a transmitter at a height of 100 metres above theearth's surface and surface conditions correspondingto dry ground. The various curves show this ratiofor four different wavelengths (differing succes-sively from the other by a factor of 10) and forthe limiting case À. --+ O. A distinct shadow effect,whereby the field strength is reduced by a factor

4) G. N. Watson, Proc. Roy. Soc.A 95, 83, 191B.6) B. van der Pol, Phil. Mag. 38, 365, 1919.6) B. van der Pol and K. F. Niessen, T. Ned. Radiogen. 7,

1, 1935.7) B. van der Pol and H. Bremmer, Phil. Mag. 24, 141 and

825, 1937.

1960/61, No. 2

'._-~~~-~----------;---------,----------,

BALTHASAR VAN DER POL 39

II

a« 10-137 1 h, = lOOm

h2=0 ê =42 \

,\ II

5I~

2 ~, ~ r--I

5~ <;~ :-- À=7m

1\'\ -, <, ----

5......

À-O I 7mm '\ 7cm <,0,7m

11 1\ 1'\ <,

Fig. 1. Attenuation of a radio wave propagated along the carved surface of the earth(ground wave) for various wavelengths J., calculated for dry ground and transmitter heighthl = 100 m, receiver height h2 = 0 m. The ratio of the field strength E on the earth to thefield strength Epr at the same distance In free space is plotted as a function of the distanceD. A distinct shadow effect at the optical horizon (in this case at D = 35.7 km) appearsonly in the case of millimetre waves.

of at least 4 within a distance of 5 km beyond thehorizon, does not appear until the wavelength hasdropped to about 7 mm.The numerical results of the ground-wave pro-

pagation theory are of particular importance atwavelengths smaller than about 10 m, in which casethe heights of the transmitting and receiving aerialsabove the earth's surface have a considerable

2

0,

0,

0,

0,0eEprop

10,00,00

0,002

0,00

0,000

0,0002

0,000 o 10 20 30 35,740

nfluence. During Van der Pol's term of office withthe C.C.LR. a wide programme of computationswas decided upon in order to get numerical dataon ground-wave propagation, including the effectof refraction in the lowest layer of the atmosphere.The results were published in the form of graphs 8),an example of which is shown in fig. 2; this refersto a wavelength of 5 m, again for representativeground conditions (conductivity of 0.01 mho/mand a relative dielectric constant of 10). The graphindicates, for example, the extent to which thefield increases with increasing height h2 of thereceiver for a fixed transmitter height ~ of 10 m.The results are the same if the transmitter andreceiver heights are interchanged.

8) Atlas of ground-wave propagation curves for frequenciesbetween 30 Mc/s and 300 Mc/s, C.C.LR. Resolution No. 11,Geneva 1955.

The rigorous theory on which such' data arebased is applicable to all problems ofwaves meetinga spherical obstacle. This situation also arises whenlight rays refracted in spherical drops of watergive rise to rainbow effects. These considerationsaccordingly led to a new treatment of the rainbowin terms of wave theory 7) *).We shall now return to those cases in which the

50 60 117-D

80 700km90

1762

sky wave conducted via the ionosphere is moreimportant than the ground wave. As shown byWatson's calculations, mentioned above, this is thenormal state in short-wave transmissions over greatdistances. The important role played by the iono-sphere, established beyond all doubt by Watson'gresults, was later emphasized by Van der Pol inthe following sentence from his above-mentionedinaugural speech:

"You know of that conductive layer, high inthe atmosphere, which makes radio communi-cation over long distances possible, and whoseexistence is just as important to radio tech-nology as the existence of iron in the earth is toelectrical engineering as a whole".

*) Editorial note: By coincidence, a part of this wave treatmentof the rainbow is to be seen scribbled on the blackboardin the title photograph.

40 PHILlPS TECHNICAL REVIEW VOLUME 22

dE l

~ À=Sm CT=1O-13-

~ ~hr=10m e =10

80 ~

~ ~"~ --_<, --, --_ -----~~ \'\ r-,<, --- --- -_r-, ~

--- -- --- --- -- ---r--,---<, -- 160

\'\~"~ t- I"-.... -I"-<, ï""-- I"-,~\:~"< i'---- ............... .....--r-, I'--- ...............r-, r-... I-..40

~ ~ ~<,

~-............ ........_ r--

~ I'---.. r-; r--t-, ...... r-...

~ ~ ~ <, 5(j>...~ r-, 2O(P r-; 5àèP -....... !rlOO'àin>-- ....................... <, I'..... <,

20 »~ ~ -:I"-.... ......... '--- ............... ...........__ r--.

t--... <,<, r---... I'-- ~ t--. r-;

h:J,{O ~ -;~ ::--....-~ I'-- ........--..........

........<, j"-......0 i"'---. <, j"-...... <, <, t--.

~~ t-<,

........._~

.........I'--

.......__-....... .............._

I'---i ,._', <, -'

............. r--. <, r-..... ......-....;;:

~ ~.........t--. I'--

...........__ I"- --r-. r---... I'-- I'--- -.......~2f)

Although the existence of the ionosphere was notdefinitely established until 1925, when direct re-flection measurements placed it beyond all doubt,Eccles and. Larmor had worked out a theory asearly as 1912 for wave propagation through a con-ductive gas, which could he immediately appliedto the ionosphere. It was found that a conductivegas behaves formally like a medium possessing a

700

o 50

time an apparent dielectric constant smàller thanunity. This represented experimental confirmationof the now generally accepted propagation mecha-nism for radio waves that are refracted in the iono-sphere. This' whole subject was clearly dealt within Van der Pol's thesis in 1920, entitled "The in-fluence of an ionized gas on the propagation ofelectromagnetic waves, and its applications in the

2

1025

2

105

2

5

100 150

20-1

200km-D 1753

Fig. 2. Vertical component of the field strength of the ground wave (in dB and in fLVfm)of a transmitter at height hl = 10m, computed for various distances D and receiver heightsh2, for Î. = 5 m and over slightly moist ground. The dashed curve would he obtained in theabsence of the earth. The point on each curve indicates the horizon distance when theline connecting the transmitter and receiver touches the earth.

certain conductivity and a relative dielectric con-stant smaller than unity. This means that the phasevelocity of radio waves in the ionosphere exceedsthe speed of light c in a vacuum. However, thecorresponding propagation velocity of disconti-nuities (group velocity) is smaller than c, as itmust be to accord with the principles of the theoryof relativity. By means of resonance measurementson lecher lines, Van der Pol determined, during hisperiod in the Cavendish Laboratory at Cambridge,the conductivity and dielectric constant of theplasma in glow' discharges; these experimentsclosely simulated the conditions in the ionosphere.After experimental difficulties had been overcome,it was possible in this way to measure for the first

field of wireless telegraphy and in measurementson glow discharges".

Non-lineal' circuits: relaxation oscillations

Weshall now consider another field of research inwhich Van der Pol was particularly active, that ofnon-linear oscillations. It was understandable thatin the 'twenties his attention should be drawn tothis subject, since the advances 'already made in theapplication of triode circuits made necessary adeeper understanding of oscillation phenomena inorder to round off the theory and to discern furtherpossibilities. The simple linear theory of oscillatoryeffects is arrived at when the ia-vg characteristic

• I

1960/61, No. 2 BALTHASAR VAN DER POL , 41

of the triode (anode current versus grid voltage) isapproximated by a straight line. The unsatisfactorypoint is that the amplitude ofthe excited oscillationsthen remains undetermined. The limitation of theamplitude is due to the decrease of the slope of thecharacteristic at either side of the operating point.

r--....----lI--~a--11---------.

Va .-------+----,----T

lR V

·------.___._.........._____ l_1764

Fig. 3. Circuit for investigating oscillations of a triode circuit,in particular for studying the limitation of the amplitude(leading to Van der Pol's differential equation).

Fig. 3 shows a representative circuit for investigat-ing these effects, in which M is the mutual in-ductance between the grid circuit and the LC circuit(with shunt resistance R) in the anode lead. Wetake the triode characteristic to be given by:

where g is the amplification factor of the triode.The non-linear function ifJ of the variable u =Va + gVg can be represented by a Taylor series in theneighbourhood of the point u = Ea:

The coefficients are normalized by the introduetionofthe parameter k = g(MIL) - 1.This circuit leadsto the following differential equation for the timevariation of the voltage v (see figure):

d2 (1) .C dt: +~R -a + 2(Jv + 3yv2 + ...

1+ LV = o.

~dv-+dt

Van der Pol showed in the first place that thecoefficient (J, which determines the first non-linearterm of the characteristic, so important for detec-tion, does not in itself establish a finite value of theamplitude. The effect of this term will merely beto cause the angular frequency of the LC circuit todiffer slightly from its value (LC)-t in the linear

theory. The term with the coefficient y is the firstterm to influence the amplitude. The followingterm~ determine less essential properties. Forexample the fifth-order term in (2) is necessary in .order to understand certain hysteresis effects, alsostudied by Van der Pol 9). In his amplitude inves-tigations he therefore confined himself. mainly tostudy the influence of the y term, taking (J= O.By neglecting all higher terms in the expansion ofthe characteristic, eq. (2), he obtained from (3)a differential equation which, after introducingreduced variables both for the voltage v and thetime t, assumed the form:

"

(4)

(1)

rzrtz:where e= l'LIC (a- IIR).This differential equation is known as "Van der

Pol's equation". Initial investigation of this equationfor small values of the dimensionless parameter eshowed that the solution, after a preliminary"time" Xo of the order of lie, approximates to anormal sinusoidal oscillation of amplitude equalto 2. Van der Pol wondered what the correspondingsolution would look like for larger values of s, atleast of the order of unity. The mathematical dif-ficulties involved led him to apply a graphicalprocedure, the method of isoclines 10). In the caseunder consideration this amounts to reducingequation (4) with the aid of the new variabledyldx = z to the following first-order differentialequation:

(2)

dz 2 Y- = e(l-y )--.dy z

(5)

(3)

Plotting z against y, this equation determines atevery point a slope dz/dy, which can be indicatedby a short dash. Drawing a continuous curve througha series of these dashes produces a curve whichrepresents a possible solution in the variables y and z.A start can be made at various points, and in thisway solutions constructed that satisfy the relevantinitial conditions. Further numerical integrationproduces from each curve a solution for y as a func-tion of x.This procedure is illustrated infig. 4, where various

solutions are given for the relation between y anddyldx, when e= 1. Fig. 5 shows the solution for yitself, corresponding to one of these, and alsosolutionsfor y for the cases e = 0.1 and s = 10, similarly

U) B. van der Pol.-Rhil. Mag. 43, 700,1922.10) B. van der Pol, Phil. Mag. 2, 978, 1926.

42 ' PHILIPS TECHNICAL REVIEW VOLUME 22

dydx

t

o 2_y 1765

-2

Fig. 4. Illustrating the isocline method of solving the dif-ferential equation (5), for s = 1.

obtained. It follows from fig. 5 that, as the timeincreases, the solution approaches asymptoticallyto a non-sinusoidal periodic function, which differsmore from a sinusoidal oscillation the larger e is;the larger the value of e the sooner is the asym-ptotic end state reached. Furthermore, the periodin this end state for large values of e is seen toapproach a value Llx which is of the order of s; itthus differs appreciably from the correspondingperiod for very small values of ê, which, in termsof the reduced time unit x, approaches 2n. Return-

ing to the original variables 1) and tin eq. (3), thismeans that the period for large values of e is of theorder of magnitude of L(a- 1fR). This is a relax-ation time, i.e. a time that determines the decayof an aperiodic process. (Better known is the caseoccurring in other circuits where a relaxation timeis determined by an RC product, see e.g. fig. 6.)For this reason, Van der Pol gave the name relax-ation oscillations to these oscillatory effects which,at large values of s, consritute the end state of pro-cesses defined by the differential equation (4.).Van der Pol first described these oscillations in

1926, after which he continued to investigate them,theoretically and experimentally, in cooperationwith J. van der Mark 11). A particular study wasmade of forced oscillations 12),' produced when anexternal alternating voltage is applied to a systemlike that in fig. 3. In the simplest cases the situation.can be described by substituting for the right-handside of Van der Pol's equation a term proportionalto cos wt. For small values of e this provided a betterunderstanding of already known properties ofoscillator circuits. This applied particularly to thesuppression of the free oscillation of a system thatoccurs when its frequency is close to that of theimposed external oscillation. The resultant syn-chronism with' the latter oscillation was found to beparticularly pronounced when, with increasing s,the first, approximately sinusoidal, free oscillation. gradually changes into a free relaxation oscillation.Another way of putting this is that the relaxationoscillations very readily assume the frequency of asuperposed alternating voltage if the latter does

11) B. van der Pol and J. van der Mark, Onde électrique 6,461, 1927.

12) B. van der Pol, Phil. Mag. 3, 65, 1927.

__~or-:-II-£ =t=zz=\=l:LI\ =/ :\1=/'=\ :n=Î Î=I1:\ r='\:II='\=ft:fP==*=@==S=\ =jl:\=I':\1 =/\=, ':'1\=, 1='~=I=\=j/=\=/I=\=01..: ~ V =J 1_7 11t-\/ t/ 1=- Il-ttt==~ \I tU 1:/ 1/: 1/ ~} If' 17_= I/i:·~Po~---+'ro~--~~~--~~~--~~'---~~~--~w'---'~~O----~OO'---"OO~--~roon---."~O--~!~~,---.,~k---~,h~.---~,~~--~~,w

:tmlo to

Fig. 5. Solutions of the differential equation (4.),obtained by integration from solutions ofeq. (5), for e = 0.1, e = 1 and s == 10. (The variable along' the abscissa, wrongly denotedby' t, is the reduced quantity x.) , '

"

1960/61, No. 2 BALTHASAR VAN DER POL 43

not differ too much from the frequency, determinedby a relaxation time, of the natural free relaxationoscillations. If the circuit he modified so as to reducethis relaxation time to about half its original value,fairly abrupt synchronization will occur with halfthe frequency of the external oscillation, and after afurther change with a third of this frequency, andso on, In other words, it is possible to generaterelaxation oscillations that can successively hesynchronized with an external oscillation and itssub-harmonics. These synchronization effects occurthe more readily the larger the value of s, or moregenerally, the more numerically important are thehigher terms in eq. (2). Whilst the frequency of arelaxation oscillation can thus very easily bp. in-fluenced, the amplitude is not nearly so readilycontrolled.This synchronization with sub-harmonics later

proved to be of great practical importance in theearly development of television 13). A relaxationoscillation is used in television for the scanning ofsuccessive lines of the picture. With suitable cir-cuitry a succession of sub-harmonics can be formed,finally producing a relaxation oscillation with asynchronized frequency of 25 (or 30) cis. This isthen used to effect the return to the first line afterthe complete scanning of each frame. A simplecircuit for demonstrating the frequency divisionof relaxation oscillations is shown in jig. 6, where

r R

'----{-v )-----<~--------'

Eosin 21ffo t 1:766

Fig. 6. Circuit for demonstrating the frequency division pro-duced by the synchronization of relaxation oscillations withthe sub-harmonics of an imposed voltage Eo sin2nfot.

the battery voltage E must be higher than theignition potential VI of the discharge tube B. Inthe absence of the alternating voltage Eo sin2nfot,the capacitor e charges up via the high resistanceR until its voltage V reaches VI; it then dischargesthrough the low resistance r of the ignited gasdischarge. The discharge is extinguished when thecapacitor voltage has dropped to a value V2, afterwhich the charging and discharging process isrepeated. Applying now the alternating voltageEo sin2nfot, and continuously reducing the fre-

13) J. van der Mark, An experimental television transmitter andreceiver, Philips tech. Rev. 1, 16-21, 1936. . .

quency of the free relaxation oscillations (which isproportional to I/Re) by gradually increasing thecapacitance e, it is easy to demonstrate the suddenoccurrence of synchronization with the successivesubharmonies having frequencies fo/2, fo/3, fo/4and so on. This is further illustrated in fig. 7, where

-c 1767

Fig. 7. Synchronization with sub-harmonics. The graph repre-sents the period lIJ of the forced relaxation oscillation of thecircuit in fig. 6 excited at a constant frequency fo while thecapacitance C is gradually varied.

the variable capacitance e is set out along the.abscissa. The stepwise changes in the period of theforced oscillation can be read from the ordinate.Subsequently, in collaboration with C. C. J.

Addink, Van der Pol entered upon the study of moregeneral synchronization phenomena. It provedpossible to obtain synchronization not only withfrequencies fo/2, 10/3, 10/4, etc., but also with fre-quencies (n/m)fo, where n and m are arbitraryintegers (n may even be> m). For the~e experi-ments the frequency of the imposed oscillation wasvaried instead of one of the parameters of therelaxation oscillator 14).

Characteristic of the operation of the circuit infig. 6 is the occurrence of an essentially aperiodicprocess (in this case the exponential charge anddischarge of the capacitor) which is periodicallyinterrupted and restarted at certain critical valuesof a relevant parameter. Such situations are fre-quently found in nature, and they always give riseto relaxation oscillations. Van der Pol was struckby their very number and variety. He drew up along list of examples, which included: the aeolianharp, the scratching of a knife on a plate, the peri-odic recurrence of epidemics and economic crises,the periodic density variation of two (or anothereven number) types of animals that Iive together

14) B. van der Pol, Actualités scientifiques et industriellesNo. 718, published by Hermann, Paris 1938, p. 69-80.

44 PHILlPS TECHNICAL REVIEW VOLUME 22

and of which one serves as food for the other, thesleep of flowers, phenomena associated with peri-odic rain showers after the passing of a meteoro-logical depression, and finally the beat of the heart.

The latter example led Van der Pol to a veryinteresting and instructive practical applicationof relaxation oscillations, namely an electrical modelfor simulating all the rhythmic movements of thehuman heart 15). With this system it was a simplematter to record "electrocardiograms", and tostudy the normal heart beat as weU as cardiac dis-orders. The importance of the system to medical

The development of the heart model out of thework on relaxation oscillators was a side-tracksimilar to that which led from considerations ofradio-wave propagation to a new wave-theoreticaltreatment of the rainbow. Van der Pol liked topoint out how radio problems were apt to drawattention to fields which, at first sight, seem to havevery little connection with radio 16).

Transient phenomena and operational calculus

Operational calculus was introduced by Heavisidewith the original object of providing a simple means

+18OV

L_----~------~----~----------~--------~------~----~----_OO1768

Fig. 8. Circuit simulating the functioning of the heart. Resistance values in kO, capacitancesin [l.F. The three circuits S, A and V, each of which can enter into relaxation oscillationsand which are coupled in a special way, represent respectively. the sinus venosus, the auriclesand the ventricles. R is a delay system which simulates the transit time of a stimulusthrough the bundle of His. The ignition of each of the neon lamps in S, A and V correspondsto a systole of that part of the heart. The coupling between auricles and ventricles can bevaried with the potentiometer H (Erlanger's experiment). Electrocardiograms can be madeby recording the current in one of the leads from point P or Q.

science will be obvious. The heart may be regardedas consisting of three systems: the sinus venosus, thetwo auricles and the two ventricles. The operationof each system can be simulated by a circuit of thetype in fig. 6. The unilateral effect of the sinus on theauricles, and of the auricles on the ventricles, wasrepresented by non-amplifying triodes. After otherdetails of the operation of the heart had been takeninto account the model illustrated in figs. 8 and 9was finally produced. The ignition of the gas dis-charges in the circuits simulating the auricles andventricles represents respectively an auricular and aventricular systole. The synchronizations in thismodel with subharmonies of the resultant domi-nant heart beat can be recognized in the humanheart in certain cases.

Fig. 9. Photograph of the heart simulator. The three neonlamps of the circuits S, A and V (see fig. 8) are mouuted ou thethree corresponding parts of the anatomical representation ofthe heart.

16) The examples mentioned and various others will be foundin: B. van der Pol, Proc. World Radio Convention, Sydney

15) B. van der Pol and J. van der Mark, Phil. Mag. 6, 763,1928. 1938.

- -----_.~~~~~~~~,,-',"'?:::_,,--. ~~~-~~--------------.

1960/61, No. 2 BALTHASAR VAN DER POL 45

of studying switching transients in electrical net-works. The methods devised were later developedinto a system of calculus which has proved to beextremely useful in mathematics as a whole. Noone saw this more clearly than Van der Pol, whodemonstrated by numerous examples the possibleapplications of operational methods to widely diversefields of study.

To illustrate the starting point of operationalcalculus we shall briefly discuss an aspect of thehistorical development of alternating current cir-cuit theory, often examined by Van der Pol. Atfirst the aim was to determine how a certain systemresponds to an externally applied harmonic vibra-tion cos on, or, more generally, exp jwt, wherecos cot is the real component. Investigations weremade to ascertain in what way the properties of thesystem depend on the frequency f = wf2n. As acase in point we shall consider an amplifier. Theratio G(jw) between the output voltage and inputvoltage were studied in the case 'where both wereproportional to exp jwt. If the function G(jw) isknown, we can calculate with the aid of Fourieranalysis the output voltage for any arbitrary timefunction of the input voltage. The case of theresponse to a discontinuous input voltage can alsobe treated in this way. In applying this procedureit was often overlooked, however, that the complexfunction G(jw) depends on two real (though mutuallyrelated) functions, and that it is consequently notenough to take account only, for example, of themodulus IG(jw)I. This was discussed some consider-able time ago in this journal by J. Haantjes 17).On the other hand, all properties can equally bederive~ from the knowledge of a single real function,such as the unit step function response Gs(t). Thisis defined as the output voltage resulting from theapplication at a given moment t = 0 of a directvoltage of unit magnitude to the input side, whenno voltage was present for t < O. (By giving theinput function unit voltage, Gs is a dimensionlessfunction of time.) All characteristics of the amplifiercan be described in terms of this unit function.It can lie shown that for a given arbitrary primaryvoltage vpr(t) at the input, the output voltage ofthe amplifier will be

+00

= !Gs(t -.) ~ vpr(.) d-r =d.vet)-00 +00r Gs(.) ~ vpr (t-.)d r.

. dt-00

. . (6)

17) J. Haantjes, Philips tech. Rev. 6, 193, 1941.

This representation is simpler than that where thesystem is described in terms of the above-mentionedcomplex function G(jw), which defines the behaviourof the system in the case of a continuous, harmonicinput voltage. If the latter function is used, theeffect of an arbitrary input voltage vpr(t) is given.by the expression:

+00

vet) =~ !G(jw)2n

+00

[!Vpr(.) eiW(t-T) d. ] dw, (7)

-00 -00

which is generally more difficult to handle for a non-periodic Vpr(t) than eq. (6).

The above shows that it is sometimes simpler to study thebehaviour of a system with reference to a discontinuous process(here the sudden application of a unit step voltage, giving thestep-function response Ga(I», than with reference to a continu-ous process (the continuous application of an alternating vol-tage exp jWI, from I = -00 to I = +00). An objection toworking with the unit-step function might be that, althoughmathematically easier to handle in such cases, true discon-tinuity cannot in reality be achieved. Itshould be remembered,however, that any discontinuous process can be regarded as thelimit of a physically realizableprocess, and that, strictly speak-ing, there are in fact no perfectly continuous processes either.Van der Pol once remarked that there Iseemed to have been along-standing aversion to the'study of discontinuous phenom-ena. He; described] this laversion!wittily as a)"horror discon-tinuitatis", resembling the "horror vacui"l assumed by 17thcentury physicists.

We have discussed these preliminaries to showhow useful Heaviside's operational methods canbe for the study of electrical systems. This appearsfrom the fact that eq. (6) - a so-called compositionproduct or convolution integral, in this case of thetwo functions Gs(t) and vpr'(t) - is particularlysuitable for treatment by the methods of operationalcalculus. In the version of operational calculusadopted by Van der Pol and the author 18), to anyarbitrary function het) (the "original") one mayallot a new function, the operational "image" fep),defined by the following Laplace integral:

+00

fep) = p !e-pt het) dt.

-00

(8)

18) B. van der Pol and H. Bremmer, Operational calculus,based on the two-sided Laplace integral, Cambridge Univ.Press 1950. - The version of the operational calculusdeveloped in this book is characterized by the fact that theintegral in eq. (8) has the lower limit -00, as opposed tothe more conventional definition used by Carson, Doetsch,Wagner el al.,where the lower limit is alwaysO. The advan-tages of using - 00 as the lower limit are explained in th,ebook. . i,'

46 PHILIPS TECHNICAL REVIEW VOLUME 22

The image of a convolution integral is simply theproduct of the images of the individual factors. Inthe case of eq. (6) we therefore find the operationalimage of the function vet) as the product of theimages of Gs(t) and vpr'(t). By finding the originalof this operational image we then have a completesolution of the problem. .The other expression for vet), eq. (7), which must

obviously be equivalent to eq. (6), has a form which·does not lend itself so well to an operational solu-tion. Since eq. (7) is precisely the form most suitedfor the treatment of periodic processes, we can seewhy the operational calculus is less useful for dealingwith such processes than with transients. Never-theless, eq. (7) must evidently also have some relat-ion to operational calculus; this relation becomesclear if we recall. that (7) represents a Fourier inte-gral which, after appropriate substitutions, can beput in the form of the Laplace integral (8). It isprecisely the linking of operational methods to anexponential integralof this kind (which is relatedto the familiar Fourier integrals) that has made thiscalculus into a rigourously grounded mathematicaltool.

In order to apply operational methods to thesolution of all kinds of concrete problems, such asdetermining the above-mentioned integral (6) withgiven explicit functions, it is evidently useful tohave a list of images available that correspond tooriginals given by elementary functions. Van derPol called such a list his "dictionary", whilst therules of operational calculus were the "grammar".One of these rules, for example, is the foregoingtheorem concerning the convolution integral.Armed with such a grammar and dictionary Van

der Pol studied numerous properties of electricalnetworks, and of filters in particular. He arrivedat theorems, often of very general application,which concerned for instance the relation betweenthe unit function response of a given 'filter and thatof a new filter derived from it; the latter was pro-duced from the original by replacing all inductancesby capacitances, and vice versa. General theoremsof this kind sometimes led to interesting specialcases. One notable example was the discovery thatthe charging of a system of capacitors from a D.e.source always takes place with an efficiency of50%, irrespective of the number of resistances andinductances in the circuit 19). This is to say thatthe final energy of the charged capacitors amountsto half the' energy which the source, after beingsuddenly switched on, must deliver in order to

lO} B. van der Pol, Physica 4, 585, 1937.

sustain the currents which must flowuntil the systemreaches its final state.Many of the mathematical relations discovered

by Van der Pol were arrived at quickly by theskilful manipulation of the methods of operationalcalculus. An example of such a relation in the caseof continuous functions is the remarkable andpreviously unknown identity;

for Re a> O. (9)

Van der Pol liked in particular to work on rela-tions for discontinuous functions. A representativeexample is the function A2(t), which indicates thenumber of pairs of integers (m,n) ~or whichm2 + n2 < t; operational methods led at once tothe surprising relation:

i {A2(t) -I} = [i]-[i] + [~J- [~J+ ..., (10)

where the symbol [] here represents the largestinteger ~ the number within the brackets.As a last typical example of the application of

operational calculus we mention the "movingaverage" or "sliding mean" theory, which wasextensively studied by Van der Pol. Starting froman arbitrary function get), the sliding mean isdefined by the new function

t+Ll

g*(t) = f g('i) dr . (ll)1-<1

In many cases we want to know the function ·g(t)when g*(t) has been measured. Take, for example,the intensity g(w) as a function of frequency w ina spectrogram recorded with a spectrograph. Owingto the necessarily finite width of the slit in the spec-trograph, the actual intensity g(w) is smeared outinto a generally less-rapidly varying function g*(w).The operational solution of this problem loads tomany ways in which g can he expressed by a seriesin g*, suitable for numerical use.

Other subjects

In the foregoing we have discussed the mostimportant general subjects on which Van der Polwas engaged. Weshall now touch brieflyon variousether, more specialized problems on which hè workedfor shorter or longer periods - on some all his life -

1960/61, No. 2

"

BALTHASAR VAN DER POL 47

and which will be found in the bibliography of hisnumerous publications 20).

Amongst the earliest researches of Van der Polwere special subjects concerned with the theoryof electrical networks, including current distrihu-tions in an arbitrary number of coupled circuits.His interest in antenna theory was first aroused byinvestigations into the radiation and natural fre-quency of antennae possessing end capacitance,during which the inadequacy of the concept ofradiation resistance was discussed. In England hecarried out experimental research on the conduc-tivity of seawater, in connection with its bearingon the propagation of radio waves. When he joinedthe Philips laboratory his interest extended tovirtually all problems of radio technology, thenstill in its early stages. Partly in cooperation withK. Posthumus, Y. B. F. J. Groeneveld, T. J.Weijers,G. de Vries, C. J. Bakker, W. Nijenhuis, C. J. Bouw-kamp and others (mentioned elsewhere in thisarticle) he investigated the general properties oftriode characteristics, the distribution of the elec-trical field and electron paths inside a triode, thetheory of grid detection, general properties ofoscillators and filters (including the equivalent cir-cuit of a quartz oscillator, and the non-linear theoryof hysteresis effects in two coupled circuits, oneof which is part of a triode generator), noise in radiovalves, radiation patterns of beam antennae, andmany other subjects.Van der Pol had an important share in the pre-

paratory work leading to the introduetion of radiobroadcasting in the Netherlands. Together withR. Veldhuyzen, M. Ziegier and J. J. Zaalberg vanZelst he did extensive research into the field-strengthdistribution of broadcast waves over the Nether-lands. The results were plotted on charts, showingcontours of constant field strength. The deviationof these contours from a circular shape indicatedthe influence of the type of ground traversed be-tween transmitter and receiver. In the chart forthe Hilversum transmitter on 298.8 m wavelength,for example, the absorbent effect of the city ofAmsterdam and of the dry sandy ground of thechain of hills east of Utrecht could be clearly seen.This chart led Van der Pol to point out that thefield-strength distribution in the Northern provinceswould change after the damming of the Zuiderzee,since the conductivity of this large expanse ofwater would decrease as it became progressively

20) A complete bibliography of Van der Pol's publications,compiled by C. J. Bouwkamp, will appear shortly in PhilipsRes. Repts. . ...

less salty. This was confirmed by later measure-ments.

When "a new transmitter was planned to replacethe old one at Hilversum, the reciprocity theoremwas applied to determine its most favourablelocation: instead of starting with a transmitter in acentral (though not yet determined) point, meas-urements were made of the fields due to two auxil-iary'transmirters in two distant points in the coun-try, .where reception from a centrally-situatedtransmitter would have been weakest. The mostfavourable site for the new transmitter was foundby determining, in the central region ofthe country,at what point the received field was strongest onthe line connecting the points of intersection ofcorresponding contours charted for both auxiliarytransmitters (fig. lOa, b and fig. 11).

Long before there was any question of the prac-tical exploitation of frequency modulation, whichhas since become so important in broadcasting,Van der Pol had investigated the theory of thissystem of modulation. Fundamental insight hererequires much more mathematical knowledge thanthe simpler system of amplitude modulation, sincethe differential equations involved can no longer besolved by time functions proportional to exp jwt.In thè first place, Van der Pol gave a suitable defi-nition for the vague concept "instantaneous fre-quency", Winst. Writing the general frequency-modulated signal as:

t

A cos q;(t) = A cos [wo~+ mwo.! g(t)dt + cp], (12)o

this definition reads:

dWinst(t) = - q;(t) = Wo {I + mg(t)}. (13)

dt

Together with F. L. H. M. Stumpers he then inves-tigated the so-called quasi-stationary approximationwhereby the currents and voltages assume valuesat any given instant that conform to the thenexisting value of the instantaneous frequency asdefined by (13). Itwas found that the current pro-duced in an impedance as a result of a frequency-modulated voltage is then solely determined by thephase characteristic of the impedance 21). In the'forties Van der Pol also carried out experimentalresearch on frequency modulation, but lack of spacemust preclude its discussion here.

In the theoretical investigation of frequencymodulation an important part was played by

21) B. van der Pol, J. Instn, Electr, Engrs, 93 nr, 153, 1946.

48 PHILIPS TECHNICAL REVIEW VOLUME 22

a

Llselmeer

Noordzee

Noordzee

b

Fig. 10.Field-strength measurements carried out in 1934in thecentral region of the Netherlands for determining the mostfavourable site for the new broadcasting station, planned atthat time. In two remote parts of the country, where the poor-est reception might be expected from a central transmitter,two identical auxiliary transmitters were erected, one at theDollard estuary (wavelength 325 m), and the other at Maas-

tricht (wavelength 317 m).a) The field distribution of the Dollard transmitter repre-sented by contours of constant field strength. The long arrowpoints to the transmitter. 'b) The same for the Maastricht transmitter. (Note the shadoweffect due to Rotterdam.)

1960/61, No. 2· BALTHASAR VAN DER POL

Mathieu's differential equation. More generally,Van der Pol applied himself to the study of differen-tial and difference equations related to radioproblems. These studies again led to many andvarious investigations that had a less direct con-nection with physical or technical questions, orwere even of a purely methematical nature. Theyincluded theta functions, elliptic functions, thegamma function, the theory of numbers, the prop-

Noordzee

the distribution of Gaussian prime numbers. Inthe system of all complex numbers with integralreal and imaginary parts, these are the numbersthat cannot be obtained as the product of two ofthem. When all these two-dimensional prime num-bers are drawn in the complex plane, a curious pat-tern is produced (fig. 12). This was used as a table-cloth pattern which had particular success inAmerica.

IJselmeer

Apeldoorn()

,",,

Fig. 11. The bold lines connect the places where the two auxiliary transmitters are receivedwith equal strength (thick centralline, 1 : 1) or with an intensity ratio of 2 : 1 and 1 : 2.As can be seen, the highest field strength is found on the thick line at the point 18 nearthe village of Lopik (where in fact the station was subsequently built). The numbers atother places indicate the relative value of the fieldwhich, if the transmitter of the stationhad been sited there, would have been obtained in the more unfavourable of the two poor-reception areas (Dollard or Maastricht). (Fig. lOa, b and fig. 11 are taken from B. van derPol, Rapport van de veldmetingen van twee bij de Dollard en bij Maastricht opgesteldeproefzenders, T. Ned. Radiogen. 7, 173-195, 1935.)

erties of prime numbers, and so on. Nevertheless,these problems cannot be seen as entirely distinctfrom radio technology ..Theta functions, for exam-ple, are related to the theory of potential functions,and Van der Pol's interest in the theory of numberswas inspired by the part it plays, for instance, inthe above-mentioned synchronization effects inrelaxation oscillators. Incidentally, these abstractresearches also led to a quite unexpected "industrialby-product", as a result of Van der Pol's study of

The association of mathematics and radio technologysometimes worked in the opposite direction. We refer to aninvestigation whereby Van der Pol, in cooperation with C. C.J.Addink, used electrical techniques for studying a mathematicalfunction 22). The investigation concerned the determinationof the zero points of Riemann's C function, normally definedby the series:

22) B. van der Pol, Bull. Amer. Math. Soc. 53, 976,1947.

49

(14)

..--------------------------------------------~ - -.

50 PHILlPS TECHNICAL REVIEW VOLUME 22

,/I. • ... ,/J. ..... • ............, ...• ,/J.:-. • '•.?:_ .. ... • • ,I' ,/J ~.. • -:....• -:-.Y .• · ,/J • .._(.... •• Y. •

• ...... _,.,/J ._ • _,.• • »:Y. ,/J.Aa"-;.. ,/J. ~ -. • • .-. .- =-. /.Aa" -.- ....• • :- -.- •• .-. N' :-:.. • -.- _,. •

• • -.,;;. .._." • .Aa Y Y .....'! ._.. Z -.· ·,/J •• • -:-.- •• .- - -:- -:- • • -. -:- • •••,/J •• ' •• ,/J........ Y • • •• ,....:_......... :-- -:." ~,/J -: '!A....... ,I'JI! :- ;, --: ,.-.; .. ,/J .. • • ,/J ........ ;.~-!; -.. ~.

,1'. • •• ;,/J,/J. (;1 ;--;0..· :.···....... ......... ;=: .. ... .. -:; ~ .... • ,/J ... , .... ..}:- -:-..... -:- -:- ,/J,I' -:- -: ::... ...• • .. ._._-.y _,. ._-N _,..}IJ,/J .....• ...- .- ..... -. .- rI'rh -. ..... • -v:.'

... .::::- -:- ...~,/J0-:- -:-0(::'" -:--:{.. ...."Y. .. ,/J • y,/J •• ,/J • y-:' y .. ... .. Y)Y." ·.... ••• ..... s e .,/J" ••• • ...• ..:.: .. ,/J :; "':'..... rh.. '!;--;~......,/J~ :._:, -:..:-. •. ,I'.._.... .. -: ;--_,.......-.; :-,/J ,/J ~ _,. •..................... .-... • ,/J .... • • ,/J • -:-:-. ...... -=- .... -:- ..: -:-. ..__. · ,/J· . :-"-_j. ,/J. ... • L"_" •• -:-,/J ..... -: ._._ :-.~ .... -=- -. ...... .. ..-. • :-..-- •• • --..,/J •-:- ..... -:;. . -: : .... : :-. ..~..... -=- .... .... .. . .. ..- -.. .. .... .

.... :-.. .. .}IJ •• -: ....... • ...... • ..... • •• e » ....... .rI'1616

Fig. 12. The black squares represent in the complex plane (with origin in the centre)the complex prime numbers of Gauss, i.e. the numbers (m+ jn) that cannot be written asm + jn = (a + jb) (c+ jd), where m, n, a, b, c, d are integers. This pattern was woven intoa fabric marketed by an Eindhoven textile factory.

Although this series can be used only for Re z > 1, the functioncan nevertheless be defined for the whole complex plane bymeans of an analytic continuation. Riemann had conjecturedthat the function thus definedwould possess an infinite numberof zero points, all of which would lie on the line Re z = t.By means of a relation derived by Van der Pol, the functionalvalues along this line can be associated with the simple func-tion:

-x/2 [ 'l x/2e e-e

(the symbol [] has the samc meaning as in eq. (10}).In fact,when the function (IS) is cut off after a convenient large valueof x, and then periodically continued, the coefficients of theFourier series of the new function obtained in this way repre-sent approximations to a series of values of the functionl;(z}/z along the line Re z = t.The problem of determining thezero points conjectured by Riemarm was thus reduced to aFourier analysis of the said function derived from (IS). ThisFourier analysis was carried out in an elegant experiment inwhich the sawtooth pattern of one period of the function inquestion was cut along the edge of a disc (jig. 13). The discwas rotated at a highly constant speed and light was passedthrough a radial slit and caused to fall on the serrated edge and

(IS)

on a photocell placed behind it. The photo-current thus ob-tained, varying with time, was mixed with a sinusoidal currentof frequency f. The resultant signal, containing the differencetones between f and all harmonics of the periodic functionderived from (IS), was used to excite a mechanical resonatorofvery sharp resonance. When the frequencyfwas now slowlyraised from zero, eachsuccessive harmonic of the above func-tion excited the resonator in turn with its difference tone, andwith an intensity corresponding to the relevant Fourier coef-

ficicnt. By recording the excitation (as a function off) a spee-trum was obtained in which the required zero points of thel; function could immediately be seen as minima; see jig. 14.

Musical theory

Weshall now turn from Van der Pol's scientificand technical investigations and in conclusiontouch brieflyon his interest in music. Althoughdeeply interested in analysing the structure of amusical work, he realized that this could never bean approach to the creative aesthetic element -

1960/61, No. 2 BALTHASAR VAN DER POL

1617

Fig. 13. Disc with edge serrations representing part of thefunction (15), for experimentally determining the zeropoints of Riemann "g ~ function.

an element for which, as a competent mUSICIanhimself, he had a deep understanding. It was hispreoccupation with the theory of numbers thatled him to look for mathematicallaws in harmonic

intervals 23), or, for example, to find numericaldefinitions for concepts such as dissonance. Inparticular he discovered that the fractions presen-ted by the successive intervals in the diatonic scalecould be contained in a Farey series. In this case itis the series of all irreducible fractions whose denom-inators and numerators never exceed the number 5.Only two tones from the diatonic sequence (the band the d, with c as the fundamental) do not fitinto the Farey series, and the remarkable thing isthat it is precisely these tones about which thereis most uncertainty in tuning (if one is not boundto the equal-temperament scale).It will not be surprising after the foregoing that

Van del' Pol emphasised that the sub-harmonicsof a given note can be produced with the aid ofsynchronization effects on relaxation oscillations.In this way it was possible to confirm experimen-tally a postulate put forward by the music theoristRugo Riemann, namely that the minor triad canbe regarded as the combination of the 4th, 5th and6th sub-harmonics of a given tone, just as appliesto the major triad in respect of the 4th, 5th and 6thupper harmonics of another tone.23) B. van der Pol, Muziek en elementaire getallen theorie,

Arch. Mus. Teyler 9, 507-540, 1942 (in Dutch).

70, 100 'ip /40\

/60 170 I(JO 1!Xl 200 21()\ \ \ \ \

Fig. 14. Recording obtained with the aid of the disc in fig. 13, giving the Fourier coefficientsof the function cut into the disc. The minima in the recording are the zero points of the~ function on the line Re z = t. The first 29 zero points, denoted by dashes and Romanfigures, were already known. from calculations. (The imaginary part of z is set out alongthe top edge.)

51

52 PHILIPS TECHNICAL REVIEW VOLUME 22

Van der Pol possessed absolute pitch, and wasable, for example, to tune his violin to the rightpitch without a tuning fork. He enjoyed exercisingthis gift and it was typical of the man that he shoulddevelop from it yet another interesting investiga-tion. Together with Addink he devised an apparatususing an oscilloscope to check the tuning of anorchestra during a performance 24). From hundredsof observations, both on permanently tuned in-struments and on orchestras, they found that themiddle a varied from 430 to 447 cis, and also thatthere were quite distinct, more or less systematicvariations of this frequency during a performance.

Van der Pol's enquiring mind led him to applytheprinciples of science to music. For him, however,there existed a much profounder relation betweenthese fields. There is perhaps no more fitting wayto end this review of the many-sided work of thegreat man of science that Van der Pol undoubtedlywas, than to quote his own words, from a lecturegiven at the Teyler Foundation in Haarlem:

. 24) B. van der Pol and C. C. J. Addink, Philips tech. Rev. 4,217, 1939.

"Is there really any difference between the inspi-ration that leads to the conception' of a beauti-fully flowing melody, a rich theme or a brilliantmodulation, and that which leads to the concep-tion of a new, elegant, unexpected mathematicalrelation or postulate? Are not both born in thesame mysterious way and do they not both oftendemand laborious development? Art connotesskill, and skill too is the handmaid of science."

Summary. The review given deals prineipally with the workdone by the late Van der Pol in the years from 1922 to 1949,when he was with the Philips Research Laboratories at Eind-hoven. The main subjects discussed are the propagation ofradio waves, especially the influence of the ground conditionson ground-wave propagation; non-linear circait phenomena(in partienlar relaxation oscillations described by the "Van del'Pol equation", and their synchronization); transients andoperational calculus, which (following Heaviside, whom hegreatly admired) Van der Pol used as a fruitful heuristic andlater rigorously founded method. A bricf discussionis devotedto variousother subjects treated by Van der Pol in his numerouspublications, including mathematical investigations promp-ted by problems of radio technology, such as the applicationof Mathieu's equation to frequency-modulation problems,and also studies relating to subjects of pure mathematics, such.as elliptic functions and the theory of numbers (especially theproperties ofprime-numbers). Finally someexamples are given.ofVan der Pol 's interest in music and the theory ofmusic.