242 Prof. Challis on the Modification of Sounds 



instance of the hydrodynamical problem of the propagation 

 of sound in plane-waves, Mr. Stokes has violated the above 

 rule by accepting the analytical solution for certain values of 

 the co-ordinates and the time, and rejecting it for other values. 

 I proceed to make this assertion good. 



The following is the enunciation of the problem. Assuming 

 that a medium in which the pressure p and density p are 

 related to each other by the equation p = a^f), is capable of 

 propagating waves in such a manner that the velocity and 

 density of the particles of the medium are functions of the 

 distance from a given plane, it is required to determine the 

 motion. 



Let V be the velocity of any particle at the distance z from 

 the given plane, and at the time t reckoned from a given 

 epoch. Then by a known process, which it is unnecessary 

 to go through, we have for the case of propagation in a single 

 direction, the general solution, 



t; = a . Nap. log p ==/(^2; — (ct + 1;) / + c^, 



and giving to the arbitrary function a form corresponding to 

 a particular series of waves, we have the definite solution, 



i:;=a.Nap. \ogp = m sin — (z—{a + 'o)t-\-c). 



By these equations the velocity and density are given at all 

 distances from the given plane at any instant, and the solution 

 is in all respects complete. If, to ascertain the rate at which 

 a given velocity and density are propagated, we differentiate 

 the equations supposing v and p to be constant, it will be 



found that the velocity v and corresponding density c*"" are 

 propagated with the velocity a-\-v. The effect of the diffei'- 

 ence of rate of propagation of different parts of the wave pre- 

 sents itself as a continuous modification of the form of the 

 wave. Mr. Stokes has exhibited this modification by a geo- 

 metrical curve (see Phil. Mag. vol. xxxiii. p. 359), and in so 

 doing has accepted the indications of analysis during an in- 

 terval equal to — — , but has rejected all its subsequent indica- 

 ^ 27rW2 -^ ^ 



tions. To take a numerical instance, let « = 916 feet in a 

 second, m = 1 foot in a second, and A = 10 feet. Then it 

 would be admitted in this instance, that during an interval of 

 one second and six tenths, in which the wave travels over 

 14.59 feet, the analysis indicates truly, but not for any longer 

 period. This procedure Mr. Airy has signified his approval 

 of by saying, that "Mr. Stokes has shown that a distinct 



