

SIMPLE -HARMONIC WAVES. DIFFRACTION 233 



If, following a procedure explained in 11, we equate the rate 

 of decay of this energy to W, we find 



and therefore A = A e- T , ..................... (20) 



8 o 

 provided T= /y ...- ................... (21) 



p 



The ratio (nr/^Tr) of the modulus of decay to the period is 

 therefore usually very great. 



78. Effect of a Local Periodic Force. 



Corresponding results can, with the help of more or less 

 intuitive considerations, be obtained for other forms of vibrating 

 solid, but the work is much simplified by a preliminary theorem, 

 which has also an independent interest. This relates to the 

 effect of a periodic extraneous force concentrated about a point 

 in a gaseous medium. 



An elementary proof can be derived at once from the pre- 

 ceding investigation. The result will obviously be the same if 

 the force be imagined to act on an infinitely small sphere 

 having the same density as the surrounding fluid. The effect 

 is therefore that of a double source ; and if we now denote the 

 concentrated force, supposed acting parallel to x, by Pe int , we 

 find, putting M = f Trpa 3 in 77 (11), 



P=2i7rpkca 3 A, ..................... (1; 



and therefore, by 77 (15), for large values of kr, 



p p -ikr 



<t> = T~ -cos<9 ................... (2) 



4f7TpC T 



The following investigation is of a more formal character; 

 but it involves mathematical processes more intricate than 

 those which are employed in other parts of this book. The 

 work depends on the solution of the equation 



V0+#* = <l>, ..................... (3) 



where < is a given function of x, y, z which vanishes outside a 

 certain finite region R. In the theories of attraction, and of 

 thermal and electric conduction, we meet with the equation 



*, ........................ (4) 



