10 SOUND WAVES 



The particle amplitude, in centimeters, is 



y1 



c 



From equations 1.20 and 1.24 the condensation is 



.= -^ 1.25 



bx 



1.5. Spherical Waves, — Many acoustic problems are concerned with 

 spherical diverging waves. In spherical co-ordinates x = r sin 6 cos 4/, 

 y = r sin d sin ^ and % = r cos Q where r is the distance from the center, 

 Q is the angle between r and the oz axis and ^ is the angle between the 

 projection of r on the xy plane and ox. Then V^0 becomes 



d^4> 2 d(j) 1 5 ^ . 50 1 (920 

 VV = — ^ + - — H ^ (sin 0) — + — ^ • -~ 1.26 



For spherical symmetry about the origin 



The general wave equation then becomes, 



The wave equation for symmetrical spherical waves can be derived in 

 another way. Consider the flux across the inner and outer surfaces of 

 the spherical shell having radii of r — Ar/2 and r + Ar/2, the difference is 



-4x- (^pV2-^ Ar 1.29 



br \ dtj 



The velocity is 



dr (90 

 dt dr 



where = velocity potential. 



The expression 1.29 employing equation 1.30 becomes 



1.30 



dr \ dr / 



Ar 1.31 



