182 DYNAMICAL THEOEY OF SOUND 



MO, we get the case of a wave advancing into a region 

 previously at rest. The wave-velocity is given by 



V Q -V p-p po 



as first found by Stokes (1848), and afterwards independently 

 by Earnshaw, Riemann, and Rankine. A difficulty, first pointed 

 out by Lord Rayleigh, arises, however, as to the conservation of 

 energy. The rate at which work is being done on the portion 

 of air above considered is p u Q pu, whilst that at which the 

 kinetic energy is increasing is \ m (u? u<?). The difference is 



p u -pu-\m (u 2 - uf) = \m ( p + p) - v). . . .(37) 



If the two points (v, p), (v , p ) on the indicator diagram be 

 denoted by P, P , respectively, the expression (37) is m times 

 the area of the trapezium bounded by the straight line P P, 

 the axis of v, and the ordinates p 0) p. If the transition be 

 effected without gain or loss of heat, the points P , P will lie 

 on the same adiabatic, and the gain of intrinsic energy will be 

 represented by the area included between this curve, the axis 

 of v, and the same two ordinates. Since the adiabatics are con- 

 cave upwards, the latter area is (in absolute value) less than the 

 former. It appears on examination of the signs to be attributed 

 to the areas that if v > v the work done is more than is accounted 

 for by the increase of the kinetic and intrinsic energies, whilst if 

 VQ < v the work given out would be more than is equivalent to 

 the apparent loss of energy. 



It is evident that no complete theory of waves of discon- 

 tinuity can be attempted without some reference to viscosity 

 and to thermal conduction, since at the point of transition 

 the gradients of velocity and temperature are infinite. 



It does not appear probable that under ordinary conditions 

 the modifications due to finite amplitude are of serious im- 

 portance. In equation (30), for instance, the ratio of the 

 amplitude of the vibration of the second order to that of the 

 primary vibration is comparable with tfax/c 2 , or with n*a/g . x/H, 

 where H is the height of the homogeneous atmosphere. With 

 ordinary amplitudes a, and ordinary distances x, this ratio will 

 be very small. In three dimensions the effect must be very 



