the wave orthogonals. This condition, in fact, corresponds to wave blocking,
discussed in the following paragraphs.
As can be seen from Equation 6, the wave angle α must be known before
the wave properties can be calculated. Here, Snell's law is employed to
determine wave refraction and how α varies across the profile. Taken
between two locations with different depth and current characteristics,
denoted with index 1 and 2, Snell's law may be expressed as (Jonsson and
Skovgaard 1978):
sin α1 sin α2
=
(11)
L1
L2
In calculating the wave properties at location 2, assuming all quantities are
known at location 1, Equations 6 and 11 are solved simultaneously because
both α and L are unknown at the new location.
Wave Breaking and Energy Dissipation
In Equation 1, the wave energy dissipation must be estimated before the
wave transformation can be calculated. As previously stated, only the
dissipation due to breaking PD is considered here, because in the surf zone it
boundary layer Pf. Wave breaking occurs because the wave form is not
stable for the existing hydrodynamic and topographic conditions. In shallow
water, the topography typically induces the breaking (called depth-limited
breaking). However, in the presence of a current, the hydrodynamic
conditions may cause the waves to break because the wave steepness exceeds
a critical limit (steepness-limited breaking). Typically, for depth-limited
breaking a criterion on the maximum wave height to water depth is employed
(e.g., H/d = 0.78, where H is the wave height), whereas for steepness-limited
breaking the maximum wave steepness is used (e.g., H/L = 1/7).
The Miche criterion (Miche 1951), as modified by Battjes and Janssen
(1978), provides a reliable estimate of the maximum wave height before
breaking, including both hydrodynamic and topographic controls on the
waves (i.e., includes both steepness- and depth-limited breaking). This
maximum wave height is given by,
Hb = 0.88 / k tanh(γb kd / 0.88)
(12)
where γb is the maximum ratio between wave height and water depth in
shallow water (depth-limited breaking), typically taken to be 0.78 (done here
also). Thus, the asymptotes of Equation 12 for shallow and deep water are
Hb/d = γb and Hb/L = 0.14, respectively. The breaker index γb is known to
depend on wave steepness (e.g., Kaminsky and Kraus 1994), but introduction
of such a dependence would require yet another iteration between waves and
currents. Kaminsky and Kraus (1994) found an average value of γb of 0.78
for a database comprising more than 400 measurements from a variety of
15
Chapter 3 Wave Model