544 Prof. H. Lamb on Wave-Patterns 



and put yjr = 0-\-(o, where 6 is a root o£ 



/W=0, (25) 



we have 



M)=f(0)+l*>f"m+ (26> 



Owing to the oscillations in sign of the periodic function, 

 the most important part of the integral is that for which 

 the argument is nearly stationary, i. e. for which &> is small. 

 The variation of ty in the remaining factors of the integrand 

 is therefore neglected, and the range of co may at the same 

 time be extended to +co without serious error, since only 

 mutually destructive elements are on the whole thus 

 introduced. 



Hence in the case of* (19) for example we obtain 



6 ~ V (2tt) cos - U) V|/"(<9)| ' ° u ' ' [Z n 



according as x cos + y sin Q'^J). This is in virtue of the 

 formula 



£.**-«.=, ifi^g,) . . . (28) 



The sign to be prefixed to \iir in (27) must be that of /"(#). 

 The diffusion-factor e~ Kb may be supposed included, if 

 necessary, in the value of (j)(/c). 



The locus of points where f is in any assigned phase is 

 therefore given by an equation of the type 



k(x cos d+y sin 0) = const., . . . (29) 



provided be determined as a function of #, y by the relation 



•^/e(tf?cos0 + ?/sin0)=O, .... (30) 



k being known as a function of 6 from the fact that 27t/k is 

 the wave-length corresponding to the wave-velocity ccos#. 



In other words, the locus in question is the envelope of 

 the straight line 



# cos0+#sin# = p, (31) 



where 



p=n\ . (32) 



\ being the wave-length just defined, and n a numerical 

 coefficient which changes by unity in passing from one curve 

 to the next of the same phase. This is in agreement with 

 the results of the more synthetic investigation "*. 

 * Lamb, ' Hydrodynamics,' 4th ed., Art. 256. 



