The first and simplest change to be noted is a change in wa 

 period, or frequency. To a shipboard observer in a wind-generated set 

 the wave frequency varies with direction of the ship. If the ship i 

 sailing against the waves, more wave crests are met within a given lengtl 

 of time; hence the frequency seems larger. However, if the ship is 

 sailing with the waves, fewer wave crests are met within the same perioc 

 of time; hence the frequency seems smaller. At the extreme, a reference 

 frame moving with a wave's phase velocity makes the wave appear 

 stationary, with infinite period or zero frequency. Note that this 

 change in reference frame does not change the wave geometry. All 

 wavelengths and other length scales are unchanged. 



The general case is described by the Doppler shift, i.e., 



Co = a + u • k i 



where u = current velocity 



ic = wave number vector (magnitude k = 2TT/L, direction perpen- 

 dicular to wave crests and troughs, i.e., in their direction 

 of propagation) 



'jj = waves' radian frequency in the frame of reference in which u 

 is the current velocity 



o = waves' radian frequency relative to the water moving with 

 the current u 



A physical interpretation of this Doppler equation (3), after 

 dividing through by k, is that the phase speed equals the relative phase 

 speed plus the component of current velocity in the direction orthogonal 

 to the wave crests. 



The distinction between OJ and is important. The reference frame 

 of ojis that in which the current u is defined. Examples of such a 

 reference frame are a fixed bed u^der the sea or a fixed measuring 

 instrument immersed in the sea. The reference frame of is that in 

 which the current is zero. For brevity a is referred to as frequency 

 relative to the current. These symbols are used consistently in this 

 sense . 



As indicated by the distinction between oj and a , when analyzing 

 the interaction of waves and currents, it is necessary to precisely 

 define the motion of both waves and currents. To do this it is 

 necessary to establish clearly the reference frame in which the motion 

 is considered, and it is often useful to relate this primary reference 

 frame to a second reference frame in which only wave motion is observed. 

 Typically, the primary reference frame is fixed to the earth or a 

 structure imbedded in the earth, but it may be the reference frame of a 

 measuring instrument, or an observer, moving relative to the earth. 



Only if the current is perfectly uniform is the second reference 

 frame easy to define. Then it corresponds to that of an observer moving 

 with the current and is the reference frame in which the wave frequency 



14 



