68
DETERMINISTIC DESCRIPTIONS OF OFFSHORE WAVES
TABLE 3.1 Results for Linear Small Amplitude Wave Theory
Parameter
Formula
Surface wave profile
r] = A cos (kx — ut)
Horizontal particle velocity
u =
Vertical particle velocity
Horizontal particle accélération
Vertical particle accélération
— 4tt2A sinh k(z+d)
/1
_ .
W~
1*
sinh kd COS
W'
Hydrostatic pressure
ps = -pgz
Dynamic pressure
P-P9AC ^^c0S{kx-ut)
Wave celerity
c =
tanh fcd)1,/2
Wave group velocity
r — Ç fl _i___ 2kd \
^9
2
sinh ‘ ZkdJ
’t)
Consider a few comments about the results for linear wave theory that are
summarized in Table 3.1. First, note that the water particle velocities, the
wave celerity, and the wave group velocity (derived by Sarpkaya and Isaacson,
1981) are ail different in form and hâve different physical meanings. The water
particle velocities and particle accélérations are those used in Morison’s équation
to compute the drag and inertial forces of these waves on offshore structures.
Second, the origin x = 0 of the wave is arbitrary, which implies that a constant,
arbitrary phase angle can be added to the term (kx — ut) in the formulas of
Table 3.1. To the casual reader, this may be a source of confusion since many
référencés locate the origin at the trough of the wave rather than at its crest.
Third, some classical référencés define the coordinate z as positive downward,
which has the effect of reversing signs for those tenus containing z. Further,
the vertical coordinate is sometimes labeled y in place of the more common z.
3.3
NONLINEAR WAVES
Two distinguishing features of a small amplitude wave based on linear theory
are its sinusoidal surface profile and its circulai fluid particle orbit. These two
features do not coexist in a finite amplitude wave based on nonlinear theory.
Summarized now are the most important features of several nonlinear wave
théories, the trochoidal, cnoidal, Stokes, solitary, and numerical théories. For
a complété historical background and detailed description of these and other
nonlinear wave théories, see Sarpkaya and Isaacson (1981). In the summary
that follows, the term nonlinear wave implies a wave of finite amplitude.
DETERMINISTIC DESCRIPTIONS OF OFFSHORE WAVES
TABLE 3.1 Results for Linear Small Amplitude Wave Theory
Parameter
Formula
Surface wave profile
r] = A cos (kx — ut)
Horizontal particle velocity
u =
Vertical particle velocity
Horizontal particle accélération
Vertical particle accélération
— 4tt2A sinh k(z+d)
/1
_ .
W~
1*
sinh kd COS
W'
Hydrostatic pressure
ps = -pgz
Dynamic pressure
P-P9AC ^^c0S{kx-ut)
Wave celerity
c =
tanh fcd)1,/2
Wave group velocity
r — Ç fl _i___ 2kd \
^9
2
sinh ‘ ZkdJ
’t)
Consider a few comments about the results for linear wave theory that are
summarized in Table 3.1. First, note that the water particle velocities, the
wave celerity, and the wave group velocity (derived by Sarpkaya and Isaacson,
1981) are ail different in form and hâve different physical meanings. The water
particle velocities and particle accélérations are those used in Morison’s équation
to compute the drag and inertial forces of these waves on offshore structures.
Second, the origin x = 0 of the wave is arbitrary, which implies that a constant,
arbitrary phase angle can be added to the term (kx — ut) in the formulas of
Table 3.1. To the casual reader, this may be a source of confusion since many
référencés locate the origin at the trough of the wave rather than at its crest.
Third, some classical référencés define the coordinate z as positive downward,
which has the effect of reversing signs for those tenus containing z. Further,
the vertical coordinate is sometimes labeled y in place of the more common z.
3.3
NONLINEAR WAVES
Two distinguishing features of a small amplitude wave based on linear theory
are its sinusoidal surface profile and its circulai fluid particle orbit. These two
features do not coexist in a finite amplitude wave based on nonlinear theory.
Summarized now are the most important features of several nonlinear wave
théories, the trochoidal, cnoidal, Stokes, solitary, and numerical théories. For
a complété historical background and detailed description of these and other
nonlinear wave théories, see Sarpkaya and Isaacson (1981). In the summary
that follows, the term nonlinear wave implies a wave of finite amplitude.