36
2 Mathematical Simulation of Wave Propagation at Global Distances
Fig. 2.1. Geometry of the formulation of the problem: {x,y} are local rectangular
coordinates; { r.p, €1} are spherical coordinates; Cg is the group velocity vector; f3 is the
angle between the wave vector direction and the x axis; Of, g~ are wave generation
area margins at r.p = r.po; /31, /32 are limiting angle values at which the source can
be "seen" at the observation point. The "petal" denotes the angular distribution
of the initial spectrum
It follows from the general formulation (1.86)-(1.90) of the problem in the
case of deep water and the absence of a current, that the wave number lkl
and the frequency a remain constant along the wave propagation trajectory.
The equation of the conservation of the wave action density (1.84) can be
expressed in terms of the frequency-angle energy spectrum S (a, (3). Using the
relation between the wave action density and the energy spectrum: N(k, (3) =
S(a,(3) · (aajak)/ka, the wave energy balance equation is:
as as . as . as .
at+ acp 'P + a'8 '8 + a(3(3 =G.
(2.1)
It should be noted that the source function G(a,(3,cp,'8,t) is of a local
character and it can be approximated by ratios used in the usual plane coordinate system, as mentioned in Chap. 1. Taking into account wave energy
advection, the left part of (2.1) is determined by the surface shape in which
the waves propagate.
In the case under consideration, the wave packets move along a geodesic
line, which is the shortest distance between the two nearest points in the
2 Mathematical Simulation of Wave Propagation at Global Distances
Fig. 2.1. Geometry of the formulation of the problem: {x,y} are local rectangular
coordinates; { r.p, €1} are spherical coordinates; Cg is the group velocity vector; f3 is the
angle between the wave vector direction and the x axis; Of, g~ are wave generation
area margins at r.p = r.po; /31, /32 are limiting angle values at which the source can
be "seen" at the observation point. The "petal" denotes the angular distribution
of the initial spectrum
It follows from the general formulation (1.86)-(1.90) of the problem in the
case of deep water and the absence of a current, that the wave number lkl
and the frequency a remain constant along the wave propagation trajectory.
The equation of the conservation of the wave action density (1.84) can be
expressed in terms of the frequency-angle energy spectrum S (a, (3). Using the
relation between the wave action density and the energy spectrum: N(k, (3) =
S(a,(3) · (aajak)/ka, the wave energy balance equation is:
as as . as . as .
at+ acp 'P + a'8 '8 + a(3(3 =G.
(2.1)
It should be noted that the source function G(a,(3,cp,'8,t) is of a local
character and it can be approximated by ratios used in the usual plane coordinate system, as mentioned in Chap. 1. Taking into account wave energy
advection, the left part of (2.1) is determined by the surface shape in which
the waves propagate.
In the case under consideration, the wave packets move along a geodesic
line, which is the shortest distance between the two nearest points in the