COASTAL ENGINEERING 2004
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where ρ (ρs) = density of water (sand), Hb = breaking wave height, Cgb = wave
group velocity at the break point, K = non-dimensional empirical constant, and λ
= porosity of sand. If Eq. (3) is substituted into Eq. (2), the sand transport rate
can be written:
⎧
⎫
⎪ ⎡
⎛ ∂y ⎞⎤ ⎪
Q = Qo sin ⎨2 ⎢αo - arctan ⎜ ⎟⎥ ⎬
(5)
∂x ⎠⎦ ⎭
⎝
⎪ ⎣
⎪
⎩
If solved numerically, these equations may be applied to describe a variety
of situations and boundary conditions. To formulate an analytic solution,
however, we are restricted to more simplified and schematized situations. As a
first step towards an analytic approach, for beaches with mild slopes, it can
safely be assumed that the breaking wave angle relative to the shoreline and the
shoreline orientation, with respect to the chosen coordinate system, are small.
The consequences and validity of this assumption that linearizes Eq. (5) are
discussed further in Larson et al. (1987). Under the assumption of small angles,
to first order in a Taylor series:
∂y ⎞
⎛
Q = Qo ⎜ 2αo - 2 ⎟
(6)
∂x ⎠
⎝
If the amplitude of the longshore sand transport rate and the incident
breaking wave angle are constant (independent of x and t) the following
equation may be derived from Eqs. (1) and (6),
∂2 y
∂y
=ε 2
(7)
∂t
∂x
where,
2Qo
ε=
(8)
D
Eq. (7) is formally identical to the one-dimensional equation describing
conduction of heat in solids or the diffusion equation and was first derived in the
present context by Pelnard-Considre (1956). Thus, many analytical solutions
can be generated by applying the proper analogies between initial and boundary
conditions for shoreline evolution and the processes of heat conduction and
diffusion. Carslaw and Jaeger (1959) provide many solutions to the heat
conduction equation, and Crank (1975) gives solutions to the diffusion equation.
The coefficient ε, having the dimensions of length squared over time, is
interpreted as a diffusion coefficient expressing the time scale of shoreline
change following a disturbance (due to wave action in the present case).
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