104 90 



P=P^+p^c^s, s=^ — [l80a, b] 



where V is the volume of unit mass and p^, V^ denote values when p = p^; 

 s represents the strain and PqCj^ the elasticity, since c^ = (elasticity/ 

 density) 2. More generally, p can be written as a series in powers of s: 



Since V = 1/p, V, = ^/p, 



-^, ds = —f dp, -— = — -— 

 P P^ ds Po dp 



s = 1 - -^, ds = -^ dp, 

 Hence from [^16] 



^ = '^oV^j(-77)^<^s = CoVp7J(PoCo^ + 262S • • O^ ds 

 



or, after expanding in powers of s and integrating, 



^ = PoCo^ + Y^'a^^ • • • 

 Subtraction of [l8l] from this equation gives 



P = P - Po - ^^2^^ • ■ • 



Thus, if Hooke's law holds so that 62 ^^'^ ^^^ higher coefficients 

 vanish, as In [l80a], n = p - p^, and [179] gives for the pressure on the 

 wall due to waves of any amplitude, p - p^ = 2{p. - pg), as for small waves. 



If only terms through s^ are to be kept, s^ may conveniently be re- 

 placed by its value as found from the first three terms of [181]; then, as 

 far as terms in s^, 



^2 (_ _ ,2 



P = P - Po ~ o„2, 4 (p - Po) [""S^] 



At the wall, [179] then gives, with [l82], 



'0 "0 



o„2 4 (p - Po) = 2(p, - p,) - —277 (P, - Po) 



or, since in the small quadratic term it is sufficiently accurate to write 

 P - Po = 2(p, - pj, 



P - Po = 2(p, - Po) + TT7x(p, - Po)^ [183] 



''0 "-o 



