358 Mr. W. G. Fraser on 



satisfies the ordinary equation of continuity, and we must 

 find the variation of the density p by means of the equation 



or # oy ot 



^VB-z" 3?// w By B// "*" d* 



Thus 



i^- H- p{ — c* s «-^ cos fc(a? + c£) -f mc&e-** cos * (a? + ct) } 



. ;Bp", /dp A 

 0# Bj/ 



To find the variation of density in time at a fixed point we 

 omit the last two terms, which are in any case small quantities 



of the second order, since u\ t^^r • ^T" are all supposed small. 



Thus we have 



I ^° + ( m _ xs'c&er** cos * (a? + d) = 0. 



At the barrier, where x = 0, y = 0, this becomes 



1 1£ + ( m _ 1)^.2 cos to _ 0? 

 p ot 

 so that, if p be the original density, 



p .— p Al- m)lc sin Tcct ^ 



But, at the origin, 77= —sin ket, 

 .-. p=p/ m - 1)ky > 



— p e {m—\)2iTYtlK 

 = p e 0n-l)gr}/c*. 



where X is the wave-length, and c the velocity of propagation. 

 Thus if rj/X or rj/c 2 do not exceed a certain amount, the 

 cohesive forces may suffice to prevent rupture, but if this 

 amount be exceeded, the tendency to discontinuity will pre- 

 vail, and the wave will break. 



Waves in Shallow Water. 



When the water is of depth h the velocity potential for a 

 train of waves 



r) = smk(w — ct) 



is 



6 = c . 1 /7 — ^cos Mx — ct), 



