CETN IV-17
March 1999
Tr = angular frequency relative to the current
(Other source terms, such as atmospheric input and bottom friction, are neglected here because
propagation distances are relatively short, on the order of a few miles or less.)
Consider the situation where waves are traveling into a tidal inlet (in the +x direction) and are
opposed by an ebb-tidal current flowing out through the inlet (in the -x direction). As the waves
propagate from the ebb shoal into the inlet channel, the ebb-current speed increases (where the
flow is confined in the inlet opening), and the wave-current interaction reduces the relative group
velocity and increases the relative angular frequency. Thus, the term (Cgr + U)/Tr decreases in
size. To balance this decrease, the energy E increases through shoaling, and/or the excess energy
is dissipated through wave breaking. The depth within the inlet channel may be as great or
greater than the ebb shoal, but the wavelength decreases because of the interaction with the
opposing current (see CETN IV-9). If the wave dissipation is neglected or underestimated, the
wave energy predicted by Equation 1 grows unrealistically large in the inlet. Thus, a method for
estimating wave breaking on an opposing current is required.
The discussion of the one-dimensional wave-action equation illustrates the process of wave
shoaling on a current, but for general inlet wave-transformation problems, solution of the two-
dimensional wave-action equation with refraction is required (see, e.g., Smith, Militello, and
Smith 1998 and Holthuijsen, Ris, and Booij 1998). Details on calculating the relative group
celerity and relative angular frequency to solve the one-dimensional problem (Equation 1) are
given in CETN-IV-9 (Smith 1997).
BREAKING CRITERIA: Miche (1951) specified the maximum monochromatic wave height as
a function of wavelength and water depth:
Hmax = 0.142 L tanh kd
(2)
where
H = wave height
L = wavelength
= wave number (k = 2B/L)
k
d
= water depth
In deep water, Equation 2 reduces to a maximum wave steepness Hmax/L = 0.142, and in shallow
water, it reduces to a maximum height-to-depth ratio Hmax/d = 0.88. This criterion is powerful
because it includes both the impacts of depth- and steepness-limited breaking. Equation 2 is
implemented as the monochromatic breaking criterion in the one-dimensional wave-current
interaction program presented in CETN IV-9.
2