In order to interpret these fluctuations as internal waves we assume W = 6li^; 

 then 



N 







(17) 



where 



K^n = A^W^ 



To find the amplitude A^ , we use the method of least squares. If the depths, 

 2 , are equally spaced, 



/ r 



/= 1 



7 0(2.)- 7 A W (z.) 



n = 1 



Minimum 



(18) 



must hold; this yields the system of equations 

 / 



u 



N 



au^)- ^A,^W,/2,) 

 « = 1 



M'J?,)=0, for r= 1,2,...N (19) 



from which the A^ are determined by using a computer. Equations for b{z-) 

 and B^ have a form similar to equations 17 and 19 and are therefore not 

 shown here. Equations 6 and 10 are used to find the phase velocities and 

 equation 19 is used to compute amplitude of the modes. The study of modes of 

 internal waves from anchor stations in shallow water is greatly facilitated when 

 the observations extend to the bottom. This basic insight into the internal wave 

 structure of an area is generally obtained by anchor stations because the re- 

 cords contain no doppler- shifted frequencies. 



ler Shifts of a Single Internal Wave Using a 

 Moving Sensor 



We assume that there is only one internal wave present with the period 

 T . If the ship is anchored at x = Xq , it can easily measure this period r. 



Case 1: Suppose the ship travels with the speed U in the +x 

 direction, and the wave moves with the velocity c ^ in the direction -x. 

 Then we have a doppler shift and observe the apparent period r', given by 



11 



