THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA VOLUME 22, NUMBER 1 JANUARY, 1950

Letters to the Editor

Application of Vector Analysis to the Wave Equation W. J. CUNNINGHAM

Yale University, New Haven, Connecticut Received October 11, 1949

HERE is in most books on acoustics a section in which is derived the differential equation for the propagation of sound waves. The derivation of this equation almost invariably is done separately for waves in rectangular, spherical, or other coordinate systems. Such duplication of effort is hardly needed since a single derivation in vector notation will apply to any system. The mathematics for electromagnetic waves usually is treated in vector form; the following is an application of a similar analysis to the acoustical problem.

FIG. 1.

continuity is Ot,/Otq-div(t,q) =0. (2)

The condensation, s, is defined so that o=o0(lq-s). If only small signals are allowed, so that s<<l, then Eq. (2) may be rewritten as

Os/Ot+divq=O. (2a)

The equation of elasticity is p=ks, where k is a constant de- pendent upon the nature of the fluid. The velocity potential, 4•, is defined by the equation, q=--grad4•. All these equations may be combined to give the wave equation in its usual form,

o% / ot•- dv% = o.

The velocity of propagation is c= (k/t,o)L These equations are perfectly general and are valid in any

coordinate system. The vector operators may be written in special forms, if desired, when any particular system is being used.

* These forms for the gradient and for the divergence may be converted readily to familiar forms in rectangular or spherical coordinate systems by choosing an elementary volume in the system and applying the indicated operation to it. See, for example, A. P. Wills, Vector Analysis (Prentice- Hall, Inc., New York, 1931), p. 83, or R. W. P. King, Electromagnetic gineering (McGraw-Hill Book Company, Inc., New York, 1945), Vol. I, pp. 26, 158.

In the figure is shown a small element of the fluid through which a disturbance is passing. The element has a volume, /Xr, and a total surface area, 2•. At one point on the surface is shown an element of area, &r, which carries a normal vector, n, of unit length and directed outwardly. The particle velocity at &r is represented by the vector, q. The instantaneous sound pressure at any point is p. The static density of the fluid is p0, and the instantaneous density at any point is p.

The forces acting on the element will be considered first. The total incremental force normal to the surface and directed out-

wardly is ,/'znpdv. The mass of fluid in the volume is o0/Xr, pro- vided only small signals are al19wed. The acceleration of the fluid is /Xq//Xt. Newton's second law of motion applies here, with the result that

- fznp&r = t, oArAq/At. The negative sign is needed to direct the force toward the element itself. Limits may be taken as both the element and the time in- terval become very small, so that

--t, oOq/Ot= lim (1/Ar) fznp&r. Ar-}O

The quantity on the right side of the equation is a standard form for the gradient* of the scalar p. Therefore the equation of mo- tion is

t, oOq/Ot q- gradp = 0. (1)

In a similar manner the flow of fluid through the element will be considered. The total mass of fluid leaving the volume per unit time is fz(n.t,q)&r. The change of mass in the volume per unit time is --/XrAt,/At. Conservation of mass applies here, with the result that

g(n.aq)&r= -- Araa/At. Limits may be taken as both the element and the time interval become very small, so that

--Oa/Ot= lim (1/Ar)L(n.pq)&r. A r.-}O

The quantity on the right side of the equation is a standard form for the divergence* of the vector (•q). Therefore the equation of

61

Subjective Effects in Binaural Hearing W. KOENIG

Bell Telephone Laboratories, Inc., New York City. New York October 17, 1949

HIS is a brief qualitative report of some experiments with a completely binaural telephone system--that is, a system with a separate microphone, amplifier, and receiver for each ear. The microphones and receivers were both moving coil instruments, Western Electric No. 630A and 711A respectively. An alternate system was provided in which two receivers were fed from a common pick-up. A switching system enabled the listener to alter- nate between these two conditions without changing receivers.

1. Room E•ects.--A remarkable property of the binaural sys- tem is its ability to "squelch" reverberation and background noises, as compared to the system with a common pick-up. The following experiments illustrate different manifestations of this property:

a. With the binaural system, no unnatural or objectionable re- verberation was noticed either from speech or from incidental noises, even when the gain was set as much as 12 db above unity. (Unity gives the same level in the ears as direct listening.) With a single microphone feeding one or two receivers, room reverbera- tion was surprisingly apparent even at unity setting, and became worse with increasing gain.

b. When listening with one pick-up and two receivers, the back- ground noises in the room--street noises, fans, people walking nearby, etc.--were very noticeable; when the system was sud- denly switched binaural, the background noises continued to be heard at first, but then they seemed to fade away in a matter of seconds to a very low level. Any single noise could still be heard if desired, but when not listened to consciously, tended to dis- appear.

c. If several conversations were going on simultaneously at the transmitting end, a hopeless jumble resulted when listening with only one pick-up. With the binaural system, each party could be distinguished separately, and it was fairly easy to understand everything that was said.

d. With the binaural system, it was possible to understand speech under conditions of extremely high room noise, even with negative signal-to-noise ratios.

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62 LETTERS TO THE EDITOR

e. The simplest and most reproducible demonstration of the squelching effect is a sharp rap on a hard surface such as a table. The echoes of this sound die down (subjectively) much faster with the binaural system than with the single pick-up. It is not necessary that the two microphones be located in a manner analogous to the two ears. They can be located anywhere in the room, provided only that they are not too close together.

In all the above experiments, the phasing of the receivers seemed to make no difference whatever. Nor did moderate dif-

ferences in level in the two receivers destroy the virtues of the binaural systpm. A network was also built to produce adjustable amounts of phase shift, with a frequency characteristic approxi- mating what might be expected acoustically from two pick-ups separated by 6 inches or so in space. This network, when in- serted ahead of one of the receivers, did not produce any squelch- ing effect in the system with one pick-up, nor did it impair the binaural system in this respect.

Finally, to test the argument that any two pick-ups, such as the two ears, form a direcfive array, and thus might discriminate against sounds from any but a preferred direction, a pair of similar microphones was tried, with their outputs merged and fed to two receivers. This system showed no squelching effect, no matter where the microphones were placed.

In binaural listening, the two ears receive acoustic signals which differ somewhat in phase and in magnitude, and the ears very likely differ in their sensitivity and frequency response characteristics. These two signals are combined, either in the nerves or in the brain, in a manner for which no explanation has been found, either in physiological terms or in terms of an elec- trical circuit analogy.

2. Directional Perception.--When the talker walked around the room, the listener with the binaural system perceived an im- pression of changes of location or direction. However, just as in previous experiments of this type, the talker seemed to be con- fined to the azimuth semicircle behind the listener; he could never be imagined in front of the listener. A mechanical system was therefore arranged so that when the listener turned his head, the pick-up microphones, attached to an artificial head, did likewise, in the same amount. With this system, it was found possible to locate the position of the talker and make the dummy face him almost exactly, regardless of his initial position. The talker then seemed to be directly in front of the listener. Ap- parently our directional perception depends partly on our ability to draw conclusions from the manner in which binaurally re- ceived sounds are affected by movements of the head.

Erratum: Acoustics in Comfort and Safety LEO L. BERANEK

Acoustics Laboratory, Massachusetts Institute of Technology, Cambridge 39, Massachusetts

[J. Acous. Soc. Am. 21, 302 (1949)]

WO paragraphs of this paper were inserted incorrectly in the published article. These two paragraphs are the ones in the first column of page 304, beginning, "The noises that are produced-.-" and ending, "..-industrial and residential areas." These paragraphs should follow the 23rd line of the second column on page 303.

Rise in Pitch of Pure Tone on Introduction

of Thermal Noise

N. A. WAXSON AND T. V. FRAZIER

Department of Physics, University of California, Los Angeles, California October 25, 1949

N the abstract of Paper 19 of the November, 1949, Meeting of the Acoustical Society of America, E. D. Schubert reports a rise in the pitch of a pure tone when thermal noise is introduced into the ear simultaneously with the pure tone by air conduction.

Two months ago, one of the writers (T.V.F.), without knowl- edge of the work being done at Michigan, observed a rise in the pitch of a pure tone introduced by air conduction in the presence of a loud pure tone of lower frequency introduced by bone con- duction. A pitch rise in the airborne pure tone was also observed when thermal noise was impressed by bone conduction. Finally, the rise in pitch of an airborne pure tone was observed upon the introduction of thermal noise by air conduction, a result which tends to confirm that observed at Michigan.

After reading the abstract of Paper 19, the writers extended the experiment further and found that if the pure tone is intro- duced by bone conduction and the thermal noise by air conduction from loudspeakers in the ear boxes of the Air and Bone Conduc- tion Audio Testing Assembly, x the effect is present. With the fre- quency of the pure tone at approximately 2000 c.p.s. and with the maximum range of thermal noise passed by the loudspeakers, the pitch rise in the pure tone was approximately a half tone. The effect was a function of the levels of the pure tone and the thermal noise.

• N. A. Watson, J. Acous. Soc. Am. 16,7194 (1945).

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