Van Wijngaarden 



Regarding simple-harmonic solutions in the form 



exp [i(aX - i^T)] , 



we obtain from (2) the dispersion relation 



[l+Ca^A^)] 



1/2 



(4) 



(cf . (2.17) in the paper). This result shows that the phase velocity v/a and 

 group velocity dvMa. are a maximum for a - o, i.e., for very long waves; and we 

 note that this is a property in common with infinitesimal surface waves on water 

 of uniform undisturbed depth. According to a well-known principle which has 

 often been used in water-wave theory, the front of any wave propagating into the 

 undisturbed medium will eventually be composed only from Fourier components 

 with group velocities near the maximum. Hence a good asymptotic approxima- 

 tion to the properties of the wave front may be obtained by taking a first approx- 

 imation to (4) for small a^, thus 



1 



2\2 



(5) 



and using this in a Fourier -integral representation of the wave (see H. Jeffreys 

 and B. S. Jeffreys, "Methods of Mathematical Physics," Cambridge University 

 Press, 1946, §17.09). 



For example, consider the solution p(X,T) of (2) that is initially a step func- 

 tion, such that 



This solution is 



p(X,0) 



1, X < , 



= 0, X > . 



P(X 



277i J a 



(6) 



with the path of integration indented under the origin. Substituting (5) for v and 

 then writing 



-'^^^"' 



3 \i'3 



we obtain as the asymptotic approximation 



CO p , ,-| CO 



(7) 



130 



