With a large number of waves (a large n), energy decreases with
increasing m, and the leading wave will eventually lose its identity.
At the group center, energy increases and decreases rapidly - to nearly
maximum and to nearly zéro. Conséquently, an energy front is located
at the center wave group for deepwater conditions. If waves had been
examined for shallow rather than deep water, the energy front would
hâve been found at the leading edge of the group. For any depth, the
ratio of group to phase velocity (C^/C) generally defines the energy
front. Also, wave energy is transported in the direction of phase propagation, but moves with the group velocity rather than phase velocity.
2.239 Summary - Linear Wave Theory. Equations describing water surface
profile particle velocities, particle accélérations, and particle displacements for linear (Airy) theory are summarized in Figure 2-6.
2.24
HIGHER ORDER WAVE THEORIES
Solution of the hydroynamic équations for gravity-wave phenomena can
be improved. Each extension of the théories usually produces better
agreement between theoretical and observed wave behavior. The extended
théories can explain phenomena such as mass transport that cannot be
explained by linear theory. If amplitude and period are known precisely,
the extended théories can provide more accurate estimâtes of such derived
quantifies as the velocity and pressure fields due to waves than can
linear theory. In shallow water, the maximum wave height is determined
by depth, and can be estimated without wave records.
When concem is primarily with the oscillating character of waves,
estimâtes of amplitude and period must be determined from empirical data.
In such problems, the uncertainty about the accurate wave height and
period leads to a greater uncertainty about the ultimate answer than does
neglecting the effect of nonlinear processes. Thus it is unlikely that
the extra work involved in using nonlinear théories is justified.
The engineer must define régions where varions wave théories are
valid. Since investigators differ on the limiting conditions for the
several théories, some overlap must be permitted in defining the régions.
Le Mehaute (1969) presented Figure 2-7 to illustrate approximate limits
of validity for several wave théories. Théories discussed here are indicated as are Stokes’ third- and fourth-order théories. Dean (1973),
after considering three analytic théories, présents a slightly different
analysis. Dean (1973) and Le Mehaute (1969) agréé in recommending cnoidal
theory for shallow-water waves of low steepness, and Stokes’ higher order
théories for steep waves in deep water, but differ in régions assigned
to Airy theory. Dean indicates that tabulated stream function theory is
most intemally consistent over most of the domain considered. For the
limit of low steepness waves in transitional and deep water, the différence between stream function theory and Airy theory is small. Additional
wave théories not presented in Figure 2-7 may also be useful in studying
2-33
increasing m, and the leading wave will eventually lose its identity.
At the group center, energy increases and decreases rapidly - to nearly
maximum and to nearly zéro. Conséquently, an energy front is located
at the center wave group for deepwater conditions. If waves had been
examined for shallow rather than deep water, the energy front would
hâve been found at the leading edge of the group. For any depth, the
ratio of group to phase velocity (C^/C) generally defines the energy
front. Also, wave energy is transported in the direction of phase propagation, but moves with the group velocity rather than phase velocity.
2.239 Summary - Linear Wave Theory. Equations describing water surface
profile particle velocities, particle accélérations, and particle displacements for linear (Airy) theory are summarized in Figure 2-6.
2.24
HIGHER ORDER WAVE THEORIES
Solution of the hydroynamic équations for gravity-wave phenomena can
be improved. Each extension of the théories usually produces better
agreement between theoretical and observed wave behavior. The extended
théories can explain phenomena such as mass transport that cannot be
explained by linear theory. If amplitude and period are known precisely,
the extended théories can provide more accurate estimâtes of such derived
quantifies as the velocity and pressure fields due to waves than can
linear theory. In shallow water, the maximum wave height is determined
by depth, and can be estimated without wave records.
When concem is primarily with the oscillating character of waves,
estimâtes of amplitude and period must be determined from empirical data.
In such problems, the uncertainty about the accurate wave height and
period leads to a greater uncertainty about the ultimate answer than does
neglecting the effect of nonlinear processes. Thus it is unlikely that
the extra work involved in using nonlinear théories is justified.
The engineer must define régions where varions wave théories are
valid. Since investigators differ on the limiting conditions for the
several théories, some overlap must be permitted in defining the régions.
Le Mehaute (1969) presented Figure 2-7 to illustrate approximate limits
of validity for several wave théories. Théories discussed here are indicated as are Stokes’ third- and fourth-order théories. Dean (1973),
after considering three analytic théories, présents a slightly different
analysis. Dean (1973) and Le Mehaute (1969) agréé in recommending cnoidal
theory for shallow-water waves of low steepness, and Stokes’ higher order
théories for steep waves in deep water, but differ in régions assigned
to Airy theory. Dean indicates that tabulated stream function theory is
most intemally consistent over most of the domain considered. For the
limit of low steepness waves in transitional and deep water, the différence between stream function theory and Airy theory is small. Additional
wave théories not presented in Figure 2-7 may also be useful in studying
2-33