43
where 3? designates the real part. This is an important consideration especially when ob
taining the radiation stresses.
It is important that in obtaining Eq. (3.69) that squares of the products of the current
components be retained until the last step. The reason for this is seen in that terms such
as kUV and kU2 are combined with terms of ojV and uU using the dispersion relationship
to give terms of aU and &V.
Equation (3.71) has been successfully used by Kirby in the form given and with modifi
cations. The modifications are the addition of a non-linear term which makes the equation
applicable for weakly non-linear waves. However for present purposes Eq. (3.71) presents a
certain problem which will shortly be corrected. The problem is that for waves propagating
at an angle to the x axis it is not possible to obtain a correct form of the conservation
of wave action. The case of no current on a beach with straight and parallel contours is
given as an illustration. The equation for the conservation of wave action reduces to the
conservation of wave energy flux. In mathematical terms this is
^Cscos0a2j =0 (3.73)
where a is the wave amplitude. For waves approaching the beach in a direction normal to
the beach Eq. (3.71) reduces to
+(C,M' = 0 (3.74)
Multiplying Eq. (3.74) by the complex conjugate of A', and add to it the complex conjugate
equation of (3.74) multiplied by A', yields
[ct\A\2]x = 0 (3.75)
which is Eq. (3.73) for the special case of cos# = 1. However it is not possible to obtain
Eq. (3.73) from Eq. (3.71) for the general case of waves at angle 6 on a beach with straight
and parallel contours.
The importance of this inability to give the correct form of Eq. (3.73) is that if energy
flux outside the breaker line is not conserved then gradients of the Sxv radiation stress