ACOUSTICS the wheel makes 8 N +1 revolutions in N clock beats, and the fork makes 32 N + 4 vibrations in the same time. If the clock is going exactly right, this gives a frequency for 4 the fork of 32 If the fork has rather less frequency than 32 then the flashes appear to move forward and the 4 frequency will be 32 In Rayleigh’s experiment the 32 fork was made to drive electrically one of frequency about 128, and somewhat as with the phonic wheel, the frequency was controlled so as to be exactly 4 times that of the 32 fork. A standard 128 fork could then be compared either optically or by beats with the electrically driven 128, and the frequency of the standard determined. A very noticeable illustration of the alteration of pitch by motion occurs when a wdiistling locomotive moves rapidly past an observer. As it passes, the pitch of the principle whistle falls quite appreciably. The explanation is simple. The engine follows up any wave that it has sent forward, and so crowds up the succeeding waves into a less distance than if it remained at rest. It draws off from any wave it has sent backward and so spreads the succeeding waves over a longer distance than if it had remained at rest. Hence the forward waves are shorter and the backward waves are longer. Since U = wA where IT is the velocity of sound, A the wave length, and n the frequency, it follows that the forward frequency is greater than the backward frequency. The more general case of motion of source, medium, and receiver, may be treated very easily if the motions are all in the line joining source and receiver. Let S (Fig. 5) be the source at a given instant, and let its frequency of vibration, or the number of waves it sends out per second, be ?i. Let S' be its position, one second later, its velocity being u. Let R be the receiver at a given instant, R' its position a second later, its velocity being v. Let the velocity of the air from S to R be w, and let U be the velocity of sound in still air. If all were still, the n waves emitted by S in one second would spread over a length U. But through the wind velocity the first wave is carried to a distance S «
s'
+ K u' w Fig. 5. U + w from S, while through the motion of the source the last wave is a distance u from S. Then the n waves occupy a space U+ Now turning to the receiver, let us consider what length is occupied by the waves which pass him in one second. If he were at rest, it would be the waves in length U + w, for the wave passing him at the beginning of a second would be so far distant at the end of the second. But through his motion v in the second, he receives only the waves in distance ] + w-v. Since there are n waves in distance U + w-w the number he actually receives is n J] + w-v TT, ) + w-u If the velocities of source and receiver are equal then the frequency is not affected by their motion or by the wind. But if their velocities are different, the frequency of the waves received is affected both by these velocities and by that of the wind. The change in pitch through motion of the source may be illustrated by putting a pitch-pipe in one end of a few feet of rubber tubing and blowing through the other end, while the tubing is whirled round the head. An observer in the plane of the motion can easily hear a change in the pitch as the pitch-pipe moves to and from him. A musical note has a definite pitch or frequency, that is, it is a disturbance of definite periodicity. Yet notes of the same pitch, emitted by different instruua y ‘ ments, have quite different timbre or quality. The three characteristics of a longitudinal periodic disturbance are its amplitude, the length after which it repeats itself, and its form which may be represented by the shape of the displacement curve. Now the amplitude evidently corresponds to the loudness, and the length of
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period corresponds to the pitch or frequency. Hence we must put down the quality or timbre as depending on the form. The simplest form of wave, so far as our sensation goes, that is, the one giving rise to a pure tone, is, we have every reason to suppose, one in which the displacement is represented by a harmonic curve or a curve of sines, y—a sin m{x - «). If we put this in the form • 2,L 'i y=as —{x-e), A A c + 2A we see that y = 0, for x = c, e + -, ~2> e + 3 anc^ 50 on’ that i/ is + from x=e to e + ^, - from e + ^ to e + and so on, and that it alternates between the values + a and — a. The form of the curve is evidently as represented in Fig. 6, and it may easily be drawn to exact scale from a table of sines.
In this curve ABCD are nodes. OA=e is termed the epoch, being the distance from 0 of the first ascending node. AC is the shortest distance after which the curve begins to repeat itself; this length X is termed the wave length. The maximum height of the curve HM = a is the amplitude. If we transfer 0 to A, e = 0, and the curve may be represented by . 2tt y—asm -r-x. A If now the curve moves along unchanged in form in the direction ABC with uniform velocity U, the epoch e = OA at any time t will be Mt, so that the value of y may be represented as 27T 2/ = asin—(as-IR) ... (1) The velocity perpendicular to the axis of any point on the curve at a fixed distance x from 0 is ‘2irla 27T dy —-—cos T (a;-IB) (2) dt The acceleration perpendicular to the axis is d?y 4x2lT2<t. 8m . 2x (a! u<) TTjx T X2 4x2U2 - - -yf-ywhich is an equation characteristic of simple harmonic motion. The chief experimental basis for supposing that a train of longitudinal waves with displacement curve of this kind arouses the sensation of a pure tone, is that the more nearly a source is made to vibrate with a single simple harmonic motion, and therefore, presumably, the more nearly it sends out such a harmonic train, the more nearly does the note heard approximate to a single pure tone. The average energy per cc. of a harmonic train of waves may easily be calculated. If p is the density, the kinetic energy in a length X parallel to ABC, of cross section 1 sq. cm., is UCtJ dx K 2 2 2 4x U a 2 2x —^2—cos --(as- Jt)dx
. from (2)
px‘U2rt2 X ' The potential energy in volume cross section 1 length dx, manifested by the strain, may be calculated since it is equal to average pressure x strain produced, P vdx. or 2 where p is the pressure excess and v the volume change per unit volume.
