LINEAR PLANE WAVES
65
Figure 3.5 Space-time représentation for a progressive plane wave.
The foregoing discussion includes the fondamental concepts and most of the
définitions needed to describe classical wave théories. It is noted that the Symbol
H, often used for wave height, has yet to be defined. The reason for this will
soon become apparent.
3.2 LINEAR PLANE WAVES
The theory of linear waves is alternatively known as Airy’s theory, small amplitude theory, and first-order theory. Developed primarily by Airy (1845) and
by Laplace (1816), it is the most important of the classical théories because it
is both easy use and it forms the basis for the probabilistic spectral description of waves. For a full development and discussion of the theory, the reader
should consult the intellectually entertaining work by Kinsman (1965). We now
summarize the assumptions, the governing équations, and the solutions for the
wave velocity and pressure profiles usefol for predicting wave-induced forces on
offshore structures.
Suppose that the simple harmonie plane wave defined in Figure 3.6 is propagating in the positive x direction in water of density p. The vertical coordinate
is z, directed positive upward; and the origin is located at the still water line
(SWL) or the mean surface level. The nine assumptions inhérent in linear theory
are as follows:
1. The amplitude A of the surface disturbance is very small relative to the
wave length A and the water depth d.
2. The velocity head (u2 + w2)/2g is small compared with the hydrostatic
pressure head pgz. Here u and w are the horizontal and vertical water particle
velocities, respectively.
65
Figure 3.5 Space-time représentation for a progressive plane wave.
The foregoing discussion includes the fondamental concepts and most of the
définitions needed to describe classical wave théories. It is noted that the Symbol
H, often used for wave height, has yet to be defined. The reason for this will
soon become apparent.
3.2 LINEAR PLANE WAVES
The theory of linear waves is alternatively known as Airy’s theory, small amplitude theory, and first-order theory. Developed primarily by Airy (1845) and
by Laplace (1816), it is the most important of the classical théories because it
is both easy use and it forms the basis for the probabilistic spectral description of waves. For a full development and discussion of the theory, the reader
should consult the intellectually entertaining work by Kinsman (1965). We now
summarize the assumptions, the governing équations, and the solutions for the
wave velocity and pressure profiles usefol for predicting wave-induced forces on
offshore structures.
Suppose that the simple harmonie plane wave defined in Figure 3.6 is propagating in the positive x direction in water of density p. The vertical coordinate
is z, directed positive upward; and the origin is located at the still water line
(SWL) or the mean surface level. The nine assumptions inhérent in linear theory
are as follows:
1. The amplitude A of the surface disturbance is very small relative to the
wave length A and the water depth d.
2. The velocity head (u2 + w2)/2g is small compared with the hydrostatic
pressure head pgz. Here u and w are the horizontal and vertical water particle
velocities, respectively.