wave phenomena. For given values of H, d and T, Figure 2 7 may be used
as a guide in selecting an appropriate theory. The magnitude of the
Ursell parameter U^ shown in the figure may be used to establish the
boundaries of régions where a particular wave theory should be used.
The Ursell parameter is defined by
L2 H
un
UR = — .
(2-45)
K
J3
For linear theory to predict accurately the wave characteristics, both
wave steepness, H/gT^, and the Ursell parameter must be small as shown in
Figure 2-7.
2.25
STORES’ PROGRESSIVE, SECOND-ORDER WAVE THEORY
Wave formulas presented in the preceding sections on linear wave
theory are based on the assumption that the motions are so small that the
free surface can be described to the first order of approximation by
Equation 2-10:
H
?27tx
2rrt\
H
n = — cos I---- — ---- I = — cos 0 or a cos 0 .
2
\ L
T /
2
More specifically, it is assumed that wave amplitude is small, and the
contribution made to the solution by higher order terms is negligible. A
more general expression would be:
tj = acos(0) + a2 B2 (L, d) cos(20)
(2-46)
+ a3 B3 (L, d) cos (30) + • • • anBn (L, d) cos(n0) ,
where a = H/2, for first- and second-orders, but a < H/2 for orders
higher than the second, and B£, Bj etc. are specified fonctions of the
wavelength L, and depth d.
Linear theory considers only the first term on the right side of
Equation 2-46. To consider additional terms represents a higher order of
approximation of the free surface profile. The order of the approximation is determined by the highest order term of the sériés considered.
Thus, the ordinate of the free surface to the third order is defined by
the first three terms in Equation 2-46.
hhen the use of a higher order theory is warranted, wave tables, such
“Jhose PreP*red by Skjelbreia (1959), and Skjelbreia and Hendrickson
(1962), should be used to reduce the possibility of numerical errors made
in using the équations. Although Stokes (1880) first developed équations
u°\fl,JioL?"PlltUde waves> the équations presented here are those of
Miche (1944).
2-36
as a guide in selecting an appropriate theory. The magnitude of the
Ursell parameter U^ shown in the figure may be used to establish the
boundaries of régions where a particular wave theory should be used.
The Ursell parameter is defined by
L2 H
un
UR = — .
(2-45)
K
J3
For linear theory to predict accurately the wave characteristics, both
wave steepness, H/gT^, and the Ursell parameter must be small as shown in
Figure 2-7.
2.25
STORES’ PROGRESSIVE, SECOND-ORDER WAVE THEORY
Wave formulas presented in the preceding sections on linear wave
theory are based on the assumption that the motions are so small that the
free surface can be described to the first order of approximation by
Equation 2-10:
H
?27tx
2rrt\
H
n = — cos I---- — ---- I = — cos 0 or a cos 0 .
2
\ L
T /
2
More specifically, it is assumed that wave amplitude is small, and the
contribution made to the solution by higher order terms is negligible. A
more general expression would be:
tj = acos(0) + a2 B2 (L, d) cos(20)
(2-46)
+ a3 B3 (L, d) cos (30) + • • • anBn (L, d) cos(n0) ,
where a = H/2, for first- and second-orders, but a < H/2 for orders
higher than the second, and B£, Bj etc. are specified fonctions of the
wavelength L, and depth d.
Linear theory considers only the first term on the right side of
Equation 2-46. To consider additional terms represents a higher order of
approximation of the free surface profile. The order of the approximation is determined by the highest order term of the sériés considered.
Thus, the ordinate of the free surface to the third order is defined by
the first three terms in Equation 2-46.
hhen the use of a higher order theory is warranted, wave tables, such
“Jhose PreP*red by Skjelbreia (1959), and Skjelbreia and Hendrickson
(1962), should be used to reduce the possibility of numerical errors made
in using the équations. Although Stokes (1880) first developed équations
u°\fl,JioL?"PlltUde waves> the équations presented here are those of
Miche (1944).
2-36