In applying assumption (g) to waves in water of varying depth encountered when waves approach a beach the local depth is usually used. This
can be rigorously justified, but not without difficulty, for most practical
cases in which the bottom slope is flatter than about 1 on 10. A progressive wave moving into shallow water will change its shape significantly.
Effects due to viscosity and vertical velocity on a permeable bottom may
be measurable in some situations, but these effects can be neglected in
most engineering problems.
2.23 ELEMENTARY PROGRESSIVE WAVE THEORY (Small-Amplitude Wave Theory)
The most fundamental description of a simple sinusoidal oscillatory
wave is by its length L (the horizontal distance between corresponding
points on two successive waves); height H (the vertical distance to its
crest from the preceding trough); period T (the time for two successive
crests to pass a given point) ; and depth d (the distance from the bed
to the stillwater level). (See Appendix B for a list of common symbols.)
Figure 2-2 shows a two-dimensional simple progressive wave propagating
in the positive x-direction. The symbols used here are presented in the
figure. The Symbol n dénotés the displacement of the water surface
relative to the stillwater level (SWL) and is a fonction of x and time.
At the wave crest, n is equal to the amplitude of the wave a, or onehalf of the wave height.
Small-amplitude wave theory and some finite-amplitude wave théories
can be developed by introduction of a velocity potential(x, z, t). Horizontal and vertical components of the water particle velocities are defined
at a point (x, z) in the fluid as u = 3/3x and w = 34>/9z. The velocity
potential, Laplace’s équation, and Bemoulli’s dynamic équation together
with the appropriate boundary conditions provide the necessary information
needed in deriving the small-amplitude wave formulas. Such a development
has been shown bÿ Lamb (1932), Eagleson and Dean (See Ippen 1966b), and
others.
2.231 Wave Celerity, Length and Period. The speed at which a wave form
propagates is termed the phase velocity or wave celerity, C. Since the
distance traveled by a wave during one wave period is equal to one wavelength, the wave celerity can be related to the wave period and length by
(2-1)
An expression relating the wave celerity to the wavelength and water depth
is given by
I gL
,
C = / — tanh
2îrd
L
(2-2)
2-7
can be rigorously justified, but not without difficulty, for most practical
cases in which the bottom slope is flatter than about 1 on 10. A progressive wave moving into shallow water will change its shape significantly.
Effects due to viscosity and vertical velocity on a permeable bottom may
be measurable in some situations, but these effects can be neglected in
most engineering problems.
2.23 ELEMENTARY PROGRESSIVE WAVE THEORY (Small-Amplitude Wave Theory)
The most fundamental description of a simple sinusoidal oscillatory
wave is by its length L (the horizontal distance between corresponding
points on two successive waves); height H (the vertical distance to its
crest from the preceding trough); period T (the time for two successive
crests to pass a given point) ; and depth d (the distance from the bed
to the stillwater level). (See Appendix B for a list of common symbols.)
Figure 2-2 shows a two-dimensional simple progressive wave propagating
in the positive x-direction. The symbols used here are presented in the
figure. The Symbol n dénotés the displacement of the water surface
relative to the stillwater level (SWL) and is a fonction of x and time.
At the wave crest, n is equal to the amplitude of the wave a, or onehalf of the wave height.
Small-amplitude wave theory and some finite-amplitude wave théories
can be developed by introduction of a velocity potential
at a point (x, z) in the fluid as u = 3
potential, Laplace’s équation, and Bemoulli’s dynamic équation together
with the appropriate boundary conditions provide the necessary information
needed in deriving the small-amplitude wave formulas. Such a development
has been shown bÿ Lamb (1932), Eagleson and Dean (See Ippen 1966b), and
others.
2.231 Wave Celerity, Length and Period. The speed at which a wave form
propagates is termed the phase velocity or wave celerity, C. Since the
distance traveled by a wave during one wave period is equal to one wavelength, the wave celerity can be related to the wave period and length by
(2-1)
An expression relating the wave celerity to the wavelength and water depth
is given by
I gL
,
C = / — tanh
2îrd
L
(2-2)
2-7