sometimes called the energy spectrum) of surface elevation can be obtained 

 from these coefficients. Relative phase information, which is used to 

 estimate wave propagation directions, can also be extracted. 



Estimating Propagation Direction of a Single Plane Wave with Two Sensors 



5. Propagation directional distributions are estimated frequency by 

 frequency, i.e., information at one frequency is assumed to be independent of 

 information at any other frequency. A frequency-direction spectrum is simply 

 the two-dimensional (2-D) presentation of each of these independent direction- 

 al estimates. Therefore, the following discussion will deal with directional 

 estimation at a single wind wave frequency. 



6. Phase difference between sensors is used to estimate wave propaga- 

 tion direction. How propagation direction is extracted from phase is best 

 shown in the simple case of a single plane wave passing through an array of 

 two pressure sensors (Figure Al) . The pressure sensors are in a line parallel 

 to the shoreline and are separated by some known distance. A plane wave 

 normally incident to the beach will have its crests and troughs arrive at both 

 sensors at the same time (Figure Ala) . Wave signals at the sensors will then 

 have equal phase; and their phase difference will be zero, independent of 

 their longshore separation. If the plane wave is incident at some angle off 

 the normal to the array, one sensor would see wave crests and troughs before 

 the other (Figure Alb) . The phase difference would then be nonzero and would 

 depend on the longshore separation of the two pressure sensors. 



7. To understand how phase differences can translate to directional 

 angle requires some trigonometry. The water surface r? of a progressive 

 plane wave is represented as a cosine function, 



r? = a cos(27rft - k-x) (Al) 



where a is amplitude, f is cyclic frequency (equal to 1/T ; where T is 

 wave period), t is time, k is wave number vector (having cross -shore and 

 longshore components k^ and ky , respectively) , and x is the coordinate 

 location vector (having components x and y as cross -shore and longshore 

 positions, respectively). The k-x term in the cosine argument of 

 Equation Al is constant if the plane wave is being observed from a fixed 



A2 



