Amplitude and phase spectra are quite different in Case 2 where an 

 internal wave of many modes but of a single frequency, co^, is observed by 

 using transducers that are in motion. 



Case 2 : Because of the different phase velocity c^ of each mode we get the 

 periods 







U cos a 



(34) 



if a is the same for all c^- The process is recorded as 



F (z, t) = 



1 



77=1 



277 



A^W^ (z) cos^ — t 



^0,n 



(35) 



This record is much more complicated because a complete set of apparent 

 periods, ?" q, n for "= 1, 2, 3, . . . N, is recorded instead of one period that would 

 be recorded at a fixed point. Amplitude and phase specti-a are given by 



A (co ', z) 



/, z) 





, elsewhere 



A 



(36) 



for 



0, n 



not defined elsewhere 



with 77= 1,2, ...N. 



Figure 4 shows the same schematic amplitude spectrum as in figure 3 

 with /V = 4, using a towed sensor. Because of towing, the internal wave of the 

 period Jq is decomposed into the different modes which now occur at dif- 

 ferent apparent frequencies coq „• It may be mentioned that the integral over 

 the entire co range of equation 33 is different from the integral of the entire 

 o) ' range in equation 36. 



Case 3 : Several internal waves are considered with periods r^ which travel 

 in the directions a^, each one composed of a sum of different orders n. 

 Since the ship is moving, the apparent periods t^ ^ are given by 



(37) 



1 ( 



17 



