A^ = k (yi - 72) sin (27r5/360) (A7) 



In Equation A7 , (y^ - 72) is known, Ai^ is measured with the two sensors 

 (as discussed below), and k can be found, using linear wave theory, from 

 knowledge of wave frequency (which is also measured with the two sensors) . 

 Linear wave theory relates wave number magnitude k to wave frequency f and 

 water depth d through the surface gravity wave dispersion relationship 



^TT^f^ = gk tanh kd (A8) 



where g is gravitational acceleration. Thus, the only unknown variable left 

 in Equation A7 is d , for which the solution is 



360 . - 

 sm 



In L Uyx - Yz) 



(A9) 



9. In combination, two sensors detect the amplitude and relative phase 

 of a plane wave that passes over them. With a little help from linear wave 

 theory, those data are used to estimate the propagation direction of the plane 

 wave. This is quite a simple result. Obviously, real ocean waves are not 

 that easy to analyze, or two sensors would always be used instead of larger 

 arrays . 



Requirements for More than Two Sensors 



10. In application of Equation A9 , it is desirable to be able to 

 resolve, for example, a wave incident at 20 deg from another at 30 deg. To do 

 this, it is necessary to resolve the phase of the 20-deg incident wave from 

 that of the 30-deg wave. The longshore separation of the sensors is critical 

 in this regard. Defining an ideal gage separation is called the resolution 

 problem. Unfortunately, there is no single ideal separation. This is because 

 the longshore component k^ of the wave number vector k is a factor in the 

 phase differences observed by the sensors (Equation A5) , and it varies with 

 propagation direction Equation A6) . 



11. To illustrate this problem. Figure A2a shows a vertical cross- 

 section (along the sensor array) of a wave incident at direction 6^ as it 



A5 



