Chapter III
Numerical Modeling
32
Cartesian coordinates
The equation of wave action conservation can be written as follows in horizontal Cartesian
coordinates
Equation III-8
Table III-2 Symbols and Significations in Wave Action Conservation Equation.
Symbol
Signification
N (í µí² ⃗ ⃗ , í µí½, í µí¼½, í µí²)
Wave action density.
t
Time.
í µí² ⃗ ⃗ = (í µí², í µí²)
Cartesian coordinates.
í µí½
Wave action density.
S
Source term for the energy balance equation.
í µí² ⃗ ⃗ (í µí² í µí² ,í µí² í µí² ,í µí² í µí½ ,í µí² í µí¼½ ) Velocity of propagation of a wave group (in the four-dimensional phase
space)
í µí¼µ
Four-dimensional differential operator in the space : í µí², í µí², í µí½, í µí¼½.
The four characteristic propagation velocities are specified by the following formulas:
Equation III-9
Equation III-10
Equation III-11
Equation III-12
Table III-3 Symbols and Significations for Characteristic Propagation Velocities.
Symbol
Signification
S
Spatial coordinate in the direction of wave propagation θ.
m
Coordinate perpendicular to S.
í µí¼µí µí² ⃗⃗⃗⃗⃗
Two-dimensional differential operator in the space í µí±¥
⃗⃗ .

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