Because the right-hand side of (64) is positive, the condition is fulfilled if 

 (a) H and m are real ( waves with finite crests ) ; ( b ) j? = or m = 0, the 

 conditions for waves with infinite crests; or (c) m = im*,lt real and (£/L)^ > im/E 

 which are Kelvin waves which travel in the x direction. The supposition for 

 these cases is that the tide- generating force is expandable in terms of L and B. 

 (2) The case where angular frequency of internal waves is less than Coriolis 

 parameter, i.e., 



Because of the negative sign on the right-hand side, equation 64 can be fulfilled 

 only with m=im* and(j2/L)^ < im/B)'^ These are Kelvin waves. Under the sup- 

 position that the tide - generating force can be expanded into a series of Kelvin 

 waves, internal tidal waves of the Kelvin type with infinitely high amplitudes 

 are possible in all areas where wq^ < f^ . They are really no longer of the 

 Kelvin type since, because of the resonance factor, the amplitude is infinite 

 everywhere in the frictionless case. They are, therefore, identical with 

 internal waves with infinitely long crests. 



The condition in equation 64 is identical with the chai'acteristic 

 equation of free internal waves for the odd modes, which are always in phase 

 with the corresponding harmonics of the tide - generating force (harmonics in 

 regard to the ocean basin). To the extent that the surface tides can be consider- 

 ed to be forced waves, the direction in which the internal tides travel should 

 correspond with that of the surface tides. 



Considering the Pacific Ocean, we should expect semidiurnal waves in 

 the entire ocean corresponding to the expansion of the force and, therefore, to 

 the surface Mg-tide. The diurnal internal wave is influenced by q!, = 28° 

 or 0= 30° , where Wq = f for the K^ — or 0^ — tide. North of these latitudes, the 

 internal wave should have infinitely long crests and travel parallel to the coast. 

 South of the latitude coq = f , the internal tidal waves should travel in about the 

 same direction as the surface waves. 



Free Diurnal Internal Tidal Waves at 30° N 



To determine whether diurnal internal waves in the ocean are mainly 

 forced or free waves, we may carry out observations near 30°N. In contrast to 

 the forced internal waves mentioned in the preceding section, free internal tidal 

 waves of the diurnal period can occur only in the area co^yf , that is, south of 

 30°N for the K^ -tide and south of 28°N for the 0^ -tide. The behavior of free 

 waves in the entire area where w ^f in the case of a variable Coriolis parameter 

 is given below. 



Suppose f=f(y) and waves if/(x,y,z,t)=i//{x,y,z) exp (icDt) . Then the 

 following equations describe free internal waves: 



. o o dP 



icou + fv + — =0 (65) 



dx 



-ip 

 icov -fu + — = (66) 



dy 



25 



