4.2 Long Surface Gravity Waves
69
Fig. 4.3 Configuratio of the one-dimensional shallow-water model. Undisturbed water depth
is h o
4.2.4 The Governing Equations
With the above simplifications the equations governing the dynamics of long surface waves can be written as:
∂u
∂t
= −g
∂η
∂ x
(4.12)
∂η
∂t
= −
∂ (u h)
∂ x
(4.13)
where u is speed in the x-direction, t is time, g is acceleration due to gravity, η is
sea-level elevation, and h is total water depth.
The f rst equation is an expression of Newton’s laws of motions and states that a
slope in the sea surface operates to change the lateral velocity. The second equation –
the vertically integrated form of the continuity equation – relates temporal changes
in sea level to convergence/divergence of the depth-integrated lateral f ow.
4.2.5 Analytical Wave Solution
Total water depth h can be approximated as constant for a fla seafloo together
with wave amplitudes small compared with total water depth. In this case, the wave
solution of the above equations is:
η(t, x) = η o sin (2π x/λ − 2π t/T )
(4.14)
u(t, x) = u o sin (2π x/λ − 2π t/T )
(4.15)
where η o is wave amplitude, λ is wavelength, T is wave period, and the magnitude
of u is given by:
u o = η o
g
h