Liquids in Tubes.—As in the case of gas-filled tubes stationary waves may be set up in tubes filled with liquid. On account of the small compressibility of a liquid compared with that of gas, however, a correction is required due to the yielding of the walls of the containing tube. This yielding produces an apparent lowering of the wave velocity—the lowering being smaller the thicker the tube. If be the theoretical velocity of sound in the liquid, and c the actual velocity in the liquid in a tube of small thickness h then c +2tca/hE) (23) where a is the internal radius, K the bulk modulus for the liquid, and E the value of Young's modulus for the material of the tube (see Lamb's Sound, p. 174). In the other extreme, when the walls are very thick [(K-F tz.)/i.1] where 1.2, is the rigidity of the material of the tube. Kundt and Lehmann succeeded in obtaining "dust" (fine iron filings) figures in liquids, as in gases, and thus measured the velocity of sound in tubes of different diameters and thickness. Application of Lamb's formula to their experimental values gives a mean value 1436 m/sec. at 19°C for the velocity of sound in open water. At frequencies within the audible range, the resonance of a liquid column can be observed by ear. The vibration is readily excited for example, by means of a steel diaphragm electro-magnetically maintained by current of variable frequency from a valve oscillator.
When the wave-length of the sound in the liquid is sufficiently small compared with the diameter of the tube the "correction" disappears. Hubbard and Loomis (Nature, Aug. 6, 1927 and Phil. Mag., June 1928) using a quartz oscillator zoo mm. diameter emitting plane-waves at frequencies between 200,000 and 400,000, have determined the velocity of sound in various liquids in a tube, with an accuracy of r in 3,000. The results for fresh and s2!t water at different temperatures agree well with the values obtained by other observers for these liquids in bulk. In Solids.—The velocity of sound in a solid rod of length 1 is very simply determined by observing the frequency N of its longitudinal vibration, the rod being clamped at the midpoint. The velocity is equal to NX or 2N1. The rod may be excited mechanically by striking or rubbing, or electro-magnetically. The velocity in steel or glass, for example, is about 5 X r o'cms./sec. as compared with 0-33 X cm./sec. in air and 1•5 X cm./sec. in water. The values of the velocity determined experimentally in this way agree very closely with the calculated velocities where E is Young's modulus of elasticity. The converse of the method is therefore convenient as a means of determining E approximately. Lang (loc. cit.) has determined the velocity in short steel rods (5 cms.) vibrating at a frequency of 50,000 using Kundt's tube as a means of estimating the frequency.
The values of velocity at supersonic frequencies are found to be the same as at audible frequencies. Apart from seismic obser vations there is little or no experimental data relating to the velocity of sound-waves in solids in bulk.
Velocity of Waves of Large-Amplitude.—Explosion Waves.—Hitherto it has been assumed that the displacement amplitude and condensation s are always small, and that the wave travels through the medium with velocity c=1,1(K/p) without change of type. If, however, the condensation becomes large, as in an explosion-wave, the velocity may be considerably modified for the curve connecting pressure p and density p is not a straight line. The bulk modulus, K= poaP/ap, increases as the density p is increased by compression, and diminishes as the density is reduced. Consequently the com pression wave travels faster and the rarefaction slower than a "small-amplitude" sound-wave. The result is a change of wave form as a large-amplitude wave travels through the medium. Lamb (Sound, p. 177) shows that the velocity of propagation is c(i s) relative to the undisturbed medium (or c[r in the adiabatic case) indicating increase of velocity with increase of condensation s. The parts of the wave of greater density therefore gain continuously on those where it is less—i.e., the crests tend to overtake the troughs, as indicated graphically in figs. ro a and b. The wave becomes steeper in front and more gradual behind. A continuation of such a process would even tually lead to a discontinuity when the wave-front becomes vertical—a condition which is physically impossible. As Rayleigh has remarked, such a tendency is held in check by the divergence of the wave and the influence of viscosity tending to diminish amplitudes and therefore reduce the velocity to its "normal" value. As an illustration, change of type in a progressive wave may be observed when sea waves approach a shelving beach. Here the crests gain on the troughs and the wave "fronts" be come steeper and steeper until they curve over and break. A large amplitude explosion wave has initially a velocity con siderably greater than ordinary sound-waves, gradually approach ing this latter value as the distance from the origin increases. Regnault found that explosion-sounds increased in velocity with increase in the intensity of the explosion. Foley (Nat. Acad. Soc. Proc., June, 192o) showed that the velocity of sound from an intense electric spark varied from 66o m./sec. at a dis tance of 3.2 mm. from the spark to 38o m./sec. at a distance of 18 metres. The shadow of the high pressure region near the wave-front of the sound-pulse was photographed at known short time intervals after the instant of production of the spark, the velocities at various distances being deduced therefrom.