40
2 Mathematical Simulation of Wave Propagation at Global Distances
{
(
sin(rp)
)
S(w, {3, rp, {), t) = S0 w, {30 , {)-arctan Jcos 2 (rp)/a 2 - 1
(2.14)
+arctan (
sin(rpo)
)
Jcos 2 (rpo)/a 2 - 1 '
2Rw [
( sin(rp) )
( sin(rp 0 ) )] }
t- - 9 - arcsin ~ -arcsin ~
The solution (2.14) for the stationary case 8Sj8t = 0 will be found. The
initial wave energy spectral density is assumed to be independent of time t
and is given as:
Sa(w, f3o, {), t)
= { ~o(w) 3 ~ sin 4 (f3o[H({J- {)~)- H({)- {)2)])
where H({)) is the Heaviside function.
(2.15)
with 7t < {30 < 2n,
with 7t < {30 < 7t ,
This problem formulation corresponding to the spectral density source of
wave energy is given at the parallel rp = rpo between two meridians {)~ < {) <
{)2 (see Fig. 2.1). The maximum of the angular energy distribution is emitted
along a meridian to the south. It is possible to assume that this problem
formulation describes the swell evolution propagating in the Pacific Ocean.
This could be a result of persistent north winds blowing in the Bering Sea
(see Fig. 2.1).
Substituting (2.15) into (2.14), the angle {3 can be limited by the condition:
. 2
( cos(rp)
)
2
sm {30 = 1 -
( ) cos(f3)
;::: 0 .
cos rpo
(2.16)
In fact, this means the change in the angular distribution width of the
spectral energy density due to the surface sphericity in which the waves propagate. If, at some surface point, the initial wave energy is distributed in the
interval of the angles {31 ~ {3 ~ {32, corresponding to two arcs of the great
circle 0 1 and 0 2 , then the energy would not scatter over the entire spherical
surface, but remain between the arcs 0 1 and 0 2 , beyond which "a shadow
zone" is observed.
Besides the condition (2.16), the energy range of the change of the angle {3
at the calculated point is limited by the finite linear width of the initial
spectral energy density source (2.15), i.e. {32 ~ {3 ~ {31. The values {31 and {32
at a given point { rp, {)} can be found by solving the transcendental equation
(2.7). The spectral zero moment is presented as:
2 Mathematical Simulation of Wave Propagation at Global Distances
{
(
sin(rp)
)
S(w, {3, rp, {), t) = S0 w, {30 , {)-arctan Jcos 2 (rp)/a 2 - 1
(2.14)
+arctan (
sin(rpo)
)
Jcos 2 (rpo)/a 2 - 1 '
2Rw [
( sin(rp) )
( sin(rp 0 ) )] }
t- - 9 - arcsin ~ -arcsin ~
The solution (2.14) for the stationary case 8Sj8t = 0 will be found. The
initial wave energy spectral density is assumed to be independent of time t
and is given as:
Sa(w, f3o, {), t)
= { ~o(w) 3 ~ sin 4 (f3o[H({J- {)~)- H({)- {)2)])
where H({)) is the Heaviside function.
(2.15)
with 7t < {30 < 2n,
with 7t < {30 < 7t ,
This problem formulation corresponding to the spectral density source of
wave energy is given at the parallel rp = rpo between two meridians {)~ < {) <
{)2 (see Fig. 2.1). The maximum of the angular energy distribution is emitted
along a meridian to the south. It is possible to assume that this problem
formulation describes the swell evolution propagating in the Pacific Ocean.
This could be a result of persistent north winds blowing in the Bering Sea
(see Fig. 2.1).
Substituting (2.15) into (2.14), the angle {3 can be limited by the condition:
. 2
( cos(rp)
)
2
sm {30 = 1 -
( ) cos(f3)
;::: 0 .
cos rpo
(2.16)
In fact, this means the change in the angular distribution width of the
spectral energy density due to the surface sphericity in which the waves propagate. If, at some surface point, the initial wave energy is distributed in the
interval of the angles {31 ~ {3 ~ {32, corresponding to two arcs of the great
circle 0 1 and 0 2 , then the energy would not scatter over the entire spherical
surface, but remain between the arcs 0 1 and 0 2 , beyond which "a shadow
zone" is observed.
Besides the condition (2.16), the energy range of the change of the angle {3
at the calculated point is limited by the finite linear width of the initial
spectral energy density source (2.15), i.e. {32 ~ {3 ~ {31. The values {31 and {32
at a given point { rp, {)} can be found by solving the transcendental equation
(2.7). The spectral zero moment is presented as: