GENERAL THEORY OF SOUND WAVES 215 



since this makes 



If we differentiate the general equation of sound waves ((4) 

 of 70) with respect to x or y or z, we recognize that if <j> is a 

 solution so also is d<t>/dx, or d(j>/dy, or d<f>/dz. Thus from (1) we 

 derive the solution 



which satisfies the general differential equation except at the 

 singular point r = 0. The value of <f> thus obtained may be 

 interpreted as the velocity-potential of a " double source " due 

 to the juxtaposition of two simple sources which are always in 

 opposite phases. This will be explained more fully in 76, in 

 the particular case where the variation with time is simple- 

 harmonic. 



The problem of reflection of sound by a rigid infinite plane 

 is readily solved by the method of "images." If with every 

 source P of sound on the near side of the boundary we associate 

 a similar source at the geometrical image P' of P with respect 

 to the plane, it is obvious that the condition of zero normal 

 velocity over the plane would still be fulfilled if the boundary 

 were abolished. Hence, in the actual case, the motion on the 

 near side will be made up of that due to the given sources P 

 and of that due to the images P'. It may be mentioned that 

 the present case of a rigid plane boundary is the only one where 

 the physical " image " of a point-source is itself accurately a 

 point-source. 



The problem of reflection at the plane boundary of two 

 distinct fluid media has been discussed in 61, in the case of 

 direct incidence. The case of oblique reflection was solved by 

 Green (1847). The results are chiefly of interest for the sake 

 of the optical analogies, but one curious point, noticed by 

 Helmholtz, may be mentioned. Owing to the greater velocity 

 of sound in water, the conditions for total reflection may occur 

 when the waves are incident from air on water (in fact when- 

 ever the angle of incidence exceeds about 13), but not in the 

 converse case. This is of course the reverse of what holds with 

 regard to light. 



