308 'Prof. K. Pearson, On plane waves of the [May 25, 



12. Let us suppose this form of solution retained when v is 

 not considered zero and endeavour to discover what changes are 

 introduced. "We take 



A cos n (z + Jv{3 2 + k 2 t) = <f> (/3) - j3# (/3), 



.-. A cos n <f>' (j3) = (j> OS) - p$ Q3) 



is an equation to discover the form of </>. 



PutX = </>'(/3), 



/. AcosnX=fXdj3-{3X, 



-nA&mnX=X^-/3-X %. 

 dX dX 



.'. f3 = nA sin nX. 



This gives /3 in terms of <f>' (/?)*. 



Hence we may write 



u = A cos n (z + Jv/3* + k 2 t) + x (/3) • t, 



where /3 = -nA sin n (z + Jvfi 2 + te 2 t). 



The complete value of the term ^ (/3.) is 



^j- v log(V^ + 7^T7 2 )-^V^M 2 ] + C 



= - q— + etc., 



by a proper choice of C. 



Now let us see what happens to /3 when t is indefinitely in- 

 creased. Obviously sin n (z + Jv(3 2 + k 2 t) will not increase in- 

 definitely. Hence as t grows large /3 remains finite. 



v/3H 

 Thus it follows that % (/3) £ = — ^ — does become infinite with t. 



OK 



Or: 



The terms tuhich give rise to the anomalous wave grow larger 

 and larger as the time increases indefinitely and the motion departs 

 more and more from the simple wave form 



u — A cos n (z + /ct). 

 The velocity of wave propagation k is given by 



k 2 = k 2 + v/3 2 = k 2 + vn 2 A 2 sin 2 n (z + Jv& + * 2 t). 

 * More generally we might have taken 



We should have found 



p =-F' (X) 



and u = F(X) + x((3)t, where X—z + Jv^ + K?t and xi§) has the value in text. 



