Wu 



For the mode aperiodic in z, given by Eq. (45b), the phase velocity c is 

 horizontal, and its value can be obtained much the same as before. However, 

 calculation of Cg should depend on the physical circumstance of the problem. 

 For instance, boundary conditions and nature of disturbances give rise to a rela- 

 tionship (often an eigenvalue problem) between w, k^., and a- (other than that in 

 Eq. (45b)); consequently a is no longer arbitrary. Even new modes may arise, 

 such as the irrotational wave modes in a stratified deep ocean. Some classical 

 examples can be found in Lamb (13) and Yih (1); another will be discussed later. 



The above results of elementary waves can be used to construct qualitative 

 solutions of more complicated form. The following are two typical examples. 



RADIATION OF INTERNAL WAVES DUE TO AN 

 OSCILLATING SINGULARITY 



Consider the flow due to a point singularity oscillating up and down with a 

 fixed (circular) frequency w in a uniformly stratified fluid (/3 = const.) which is 

 unbounded and otherwise at rest. First we consider the case w < n. Since w is 

 fixed, all the steady- state waves that are oscillating in z must have their wave 

 vector k ending at the hyperbola (see Eq. (45a)) 



k^ + fi^ = M^k, 



1/2 



(51a) 



or 



/3V 



Evidently, 9 is bounded in the range 



e^<e<'n-e^ {6^= cot" ^m), 



and (51b) gives k for each 6 lying in this range, as illustrated in Fig. 4. The 

 group velocity of these waves therefore has the components (by substituting (51) 

 in (49)) 



(51b) 



(52) 



Fig. 4 - Radiation of internal waves due to 

 an oscillating singularity 



562 



