Page:1902 Encyclopædia Britannica - Volume 25 - A-AUS.pdf/68

 48

ACOUSTICS

volume proportional to TS in volume proportional to MN, or if v is the decrease of volume per original volume Y, v TS Y-MN' But if E is the elasticity of the air, and if is the excess of pressure producing the decrease of volume, by definition we have T . mn S. Pm —Ey—E. E from (2) (3) — Yj.Wm This relation shows that wherever the particle velocity is forward Pm is in excess, and that wherever it is backward pm is in defect of the undisturbed or normal pressure. We may evidently put pat = - E. dy dx Forming a similar expression for p^ and subtracting E — Pm — " uai), and substituting in (1) U2 Px - Pm a_ (4) E NM Turning now to the forces acting on the layer originally between M and N and 1 sq. cm. in cross section, we have P +pn at N and P +pn at M, where P is the normal pressure. There is also the applied force. If this is X per mass 1, and if the normal density is p, then the total force on mass pMN is Pn-Pm + X/jMN, and the acceleration is, Pn-Pm + X (5) pNM Equating (4) and (5) we obtain E SlM X (6) P Pn-f-Pm / If the disturbance is so small that the change in volume may be taken as exactly proportional to the change in pressure, then E is constant. If then X = 0, or there is no external applied force, U2=E/p (the value found in 0. A. § 15). But if the disturbance is great we can no longer assume that E is the same for all changes of volume. We may, however, suppose that the change of pressure consists of two terms : (1) a term exactly proportional to the change in volume ; (2) a term, in general very small, expressing the divergence from proportionality. If we like, we may regard the second term as an applied force superposed on the force expressed in the first term, and then trace the effect in equation (6). The relation between pressure and volume in air for sound disturbance is the adiabatic relation PV^ = constant, where 7 = ratio of specific heats. If P change to P-fp and V to y -v, (P+p) (V-r)v=PVY, P (-i1^ l+£= Y Expanding the right hand and neglecting powers above the second, p _ v (7 + 1) F22 7y + 7 V If the second term on the right did not exist, we should have P = 7Py, and the elasticity would have the constant value E = 7P. We assume now that the elasticity does have this value, and that the second term is an applied force superposed on the pressures deduced from a constant elasticity which were used in obtaining equation (6). In that equation we must take the applied force on element cross section l'2, length NM, as p. XM. X = excess of pressure denoted by second term at X, “ >> »> ,, at M. =^p'{(U(v):} iminator of the fraction in (6), pn-Pm ^ Xow in the denominator represents
 * d from a constant elasticity, so that,

pressures deduced

Substituting these values in (6) u’=EPIV"7v} ^since E = 7P and is in the limit equal to Jt(l+y±lP p 7 p =7(1+<7+1)c)

from (3)

/E 7 + 1 (7) V ~P+ 2 ' This result implies that the different parts of a wave move on at different rates, so that its form must change. As we obtained the result on the supposition of unchanged form, we can of course only apply it for such short lengths and such short times that the part dealt with does not appreciably alter. We see at once that, where u — 0, the velocity has its “normal” value, while where u is positive the velocity is in excess, and where u is negative the Fis 2. velocity is in defect of the normal value. If, then, a (Fig. 2) represent the displacement curve of a train of waves, b wall represent the pressure excess and particle velocity, and from (7) we see that while the nodal conditions of b, with = 0 and m = 0, travel with velocity the crests exceed 7 + 1 that velocity by 2 u, and the hollows fall short of it by 7 + 1 u, with the result that the fronts of the pressure 2 waves become steeper and steeper, and the train b changes into something like c. If the steepness get very great our investigation ceases to apply, and neither experiment nor theory has yet shown what happens. Probably there is a breakdown of the wave somewhat like the breaking of a water-wave when the crest gains on the next trough. In ordinary sound-waves the effect of the particle velocity in affecting the velocity of transmission must be very small. Experiments, referred to later, have been made to find the amplitude of swing of the air particles in organ pipes. Thus Mach found an amplitude 0’2 cm. when the issuing waves were 250 cm. long. The amplitude in the pipe was probably much greater than in the issuing waves. Let us take it as OT mm. in the waves—a very extreme value. The maximum particle velocity is 2wna (where n is the frequency and a the amplitude), or 27raU/. This gives maximum u = about 80 ora7sec> which would not seriously change the form of the wave in a few wave lengths. Meanwhile the waves are spreading out and the value of u is falling in inverse proportion to the distance from the source, so that very soon its effect must become negligible. But in loud sudden sounds, such as a peal of thunder or the report of a gun, the effect may be more considerable, and there is no doubt that with such sounds the normal velocity is quite considerably exceeded. Thus there is the old observation of Parry (§ 23, O. A.) that from a distance =