DOMAINS OF VALIDITY FOR WAVE THEORIES
75
3. Also on the free surface at z = r] the pressure is uniform. This implies
that Bernoulli’s équation for unsteady flow must be satisfied, or
1/9
O.
1 dd>
»? +
+ wi------ — = constant
(3.38)
2p
g at
'
'
This is known as the dynamic free surface boundary condition.
If the wave propagates without change in form, then a uniform velocity
field of magnitude c can be imposed on the field of motion (Dean, 1967). The
boundary conditions of équations (3.37) and (3.38) are thus reduced to
With this last resuit, Bernoulli’s équation for steady flow becomes
T) + — [(u — c)2 + w2] = constant
The algorithm used by Dean (1967) to solve for tj, u, and v is outlined as
follows. Assume a form of tp, in terms of undetermined coefficients, which
satisfies Laplace’s équation (3.34) and also satisfies the boundary conditions of
équations (3.36) and (3.37). The form of the solution is such that the coefficients,
the wave number, and the free surface value of the stream function are computed
based on a least squares fit to the dynamic free surface boundary condition,
équation (3.38). This free stream function theory can also be used to compute
the characteristics of nonsymmetrical waves for which the surface profiles are
specified. Published tables (Dean, 1974) aid in applications. However, such
results can be computed directly using numerical methods.
The use of nonlinear wave theory is summarized. Trochoidal wave descriptions, which include fluid rotation, were historically used by naval architects,
but are not generally used by offshore structural engineers. Stokes approximate
theory is practical for describing short waves of finite height; but this theory becomes cumbersome and impractical for long waves of finite height. Fortunately,
alternative théories (cnoidal, solitary, numerical) hâve been developed for this
latter case. The numerical theory based on free stream fonctions is appropriate
for deepwater waves of finite height, for nonsymmetrical waves, and for shallow
water waves for which the application of linear theory is often inappropriate.
3.4
DOMAINS OF VALIDITY FOR WAVE THEORIES
A question that frequently arises concerns the sélection of a wave theory for
a given situation. Unfortunately, there are several bases for evaluating these
varions théories. No consensus has yet emerged as to a common basis. Dean
(1974) and LeMehaute (1976) hâve studied the problem and provided Figures
3.9 and 3.10, respectively, to aid in selecting an appropriate theory. Here H is
the wave height (twice the ampitude A of Figure 3.2); Un is the value of H
when the wave breaks; and d is the water depth.
75
3. Also on the free surface at z = r] the pressure is uniform. This implies
that Bernoulli’s équation for unsteady flow must be satisfied, or
1/9
O.
1 dd>
»? +
+ wi------ — = constant
(3.38)
2p
g at
'
'
This is known as the dynamic free surface boundary condition.
If the wave propagates without change in form, then a uniform velocity
field of magnitude c can be imposed on the field of motion (Dean, 1967). The
boundary conditions of équations (3.37) and (3.38) are thus reduced to
With this last resuit, Bernoulli’s équation for steady flow becomes
T) + — [(u — c)2 + w2] = constant
The algorithm used by Dean (1967) to solve for tj, u, and v is outlined as
follows. Assume a form of tp, in terms of undetermined coefficients, which
satisfies Laplace’s équation (3.34) and also satisfies the boundary conditions of
équations (3.36) and (3.37). The form of the solution is such that the coefficients,
the wave number, and the free surface value of the stream function are computed
based on a least squares fit to the dynamic free surface boundary condition,
équation (3.38). This free stream function theory can also be used to compute
the characteristics of nonsymmetrical waves for which the surface profiles are
specified. Published tables (Dean, 1974) aid in applications. However, such
results can be computed directly using numerical methods.
The use of nonlinear wave theory is summarized. Trochoidal wave descriptions, which include fluid rotation, were historically used by naval architects,
but are not generally used by offshore structural engineers. Stokes approximate
theory is practical for describing short waves of finite height; but this theory becomes cumbersome and impractical for long waves of finite height. Fortunately,
alternative théories (cnoidal, solitary, numerical) hâve been developed for this
latter case. The numerical theory based on free stream fonctions is appropriate
for deepwater waves of finite height, for nonsymmetrical waves, and for shallow
water waves for which the application of linear theory is often inappropriate.
3.4
DOMAINS OF VALIDITY FOR WAVE THEORIES
A question that frequently arises concerns the sélection of a wave theory for
a given situation. Unfortunately, there are several bases for evaluating these
varions théories. No consensus has yet emerged as to a common basis. Dean
(1974) and LeMehaute (1976) hâve studied the problem and provided Figures
3.9 and 3.10, respectively, to aid in selecting an appropriate theory. Here H is
the wave height (twice the ampitude A of Figure 3.2); Un is the value of H
when the wave breaks; and d is the water depth.