2.254 Subsurface Pressure. The pressure at any distance below the fluid
surface is given by
H cosh [2n(z + d)/L]
P
2
cosh (2nd/L)
3
ïtH2 tanh (27rd/L) cosh [4tt(z + d)/L]
1
8 Pg L sinh2 (27rd/L)
sinh2 (27rd/L)
3 cos
(2-56)
1
îtH2 tanh (27rd/L)
8 Pg
sinh2 (27rd/L)
4n(z 4- d)
cosh -----—----- — 1
2.255 Maximum Steepness of Progressive Waves. A progressive gravity wave
is physically limited in height by depth and wavelength. The upper limit,
or breaking wave height in deep water is a function of the wavelength, and
in shallow and transitional water is a function of both depth and wavelength
Stokes (1880) predicted theoretically that a wave would remain stable
only if the water particle velocity at the crest was less than the wave
celerity or phase velocity. If the wave height were to become so large
that the water particle velocity at the crest exceeded the wave celerity,
the wave would become unstable and break. Stokes found that a wave having
a crest angle less than 120 degrees would break (angle between two lines
tangent to the surface profile at the wave crest). The possibility of
existence of a wave having a crest angle equal to 120 degrees was shown by
Wilton (1914). Michell (1893) found that in deep water the theoretical
limit for wave steepness was
(2-57)
Havelock (1918) confirmed Michell’s findings.
Miche (1944) gives the limiting steepness for waves traveling in
depths less than L^/2 without a change in form as
0.142 tanh
(2-58)
Laboratory measurements by Danel (1952) indicate that the above
équation is in close agreement with an envelope curve to laboratory
observations. Additional discussion of breaking waves in deep and
shoaling water is présented in Section 2.6, BREAKING WAVES.
2.256 Comparison of the First- and Second-Order Théories. A comparison
of first- and second-order théories is useful to obtain insight about the
2-39
surface is given by
H cosh [2n(z + d)/L]
P
2
cosh (2nd/L)
3
ïtH2 tanh (27rd/L) cosh [4tt(z + d)/L]
1
8 Pg L sinh2 (27rd/L)
sinh2 (27rd/L)
3 cos
(2-56)
1
îtH2 tanh (27rd/L)
8 Pg
sinh2 (27rd/L)
4n(z 4- d)
cosh -----—----- — 1
2.255 Maximum Steepness of Progressive Waves. A progressive gravity wave
is physically limited in height by depth and wavelength. The upper limit,
or breaking wave height in deep water is a function of the wavelength, and
in shallow and transitional water is a function of both depth and wavelength
Stokes (1880) predicted theoretically that a wave would remain stable
only if the water particle velocity at the crest was less than the wave
celerity or phase velocity. If the wave height were to become so large
that the water particle velocity at the crest exceeded the wave celerity,
the wave would become unstable and break. Stokes found that a wave having
a crest angle less than 120 degrees would break (angle between two lines
tangent to the surface profile at the wave crest). The possibility of
existence of a wave having a crest angle equal to 120 degrees was shown by
Wilton (1914). Michell (1893) found that in deep water the theoretical
limit for wave steepness was
(2-57)
Havelock (1918) confirmed Michell’s findings.
Miche (1944) gives the limiting steepness for waves traveling in
depths less than L^/2 without a change in form as
0.142 tanh
(2-58)
Laboratory measurements by Danel (1952) indicate that the above
équation is in close agreement with an envelope curve to laboratory
observations. Additional discussion of breaking waves in deep and
shoaling water is présented in Section 2.6, BREAKING WAVES.
2.256 Comparison of the First- and Second-Order Théories. A comparison
of first- and second-order théories is useful to obtain insight about the
2-39