6 Wave Transformation in Shallow Water
6.1 Spectrum Transformation
Due to Wave Refraction in Shallow Water
Wave spectrum refraction a in coastal area.
The wave description
given in Chap. 1 allows easy analysis of wind wave propagation in a coastal
area, 1 i.e. when relatively short sea waves propagate from deep to shallow
water, approaching a coastline. In this case refraction plays a special role
among the different factors affecting wave behavior. Due to depth variation
it results in the evolution of wave parameters, namely, propagation direction,
length, amplitude and wave profile. As noted in the Introduction, there are
many papers devoted to the problem of wave transformation in a coastal
area. An application of the spectral approach to this problem following from
the general formulation is considered in this chapter.
Sticking to physical terminology, wave refraction in shallow water can be
considered in the framework dispersive wave propagation in a spatially nonuniform isotropic medium. It follows from Sect. 1.6 that the propagation of
a wave packet is described with the help of the ray equations on a spherical
surface (1.86)-(1.89) and in a plane (5.2), with the frequency w being constant
along the ray:
w 2 = a 2 = f 2 (k, H (r)) = gkth (kH) = const.
(6.1)
This condition yields a useful ratio for determining the wave number lk I
and phase velocity c along the trajectory, depending on the slowly changing
depth H.
A second simple consequence of the general kinematic ratios can be obtained in the "cylindrical" case, i.e. with the depth H changing only in one
direction, for example H = H(x). It follows from (5.2) that a component of
the wave vector ky should be constant during wave packet propagation
ky = k sin ((3) = const ,
(6.2)
where n/2 - (3 is the angle between the direction of the wave vector k and
the isobath. The dependence between the angle (3 and the wave number k or
1 In this case the wave spectrum transformation apart from the surf zone is considered.
6.1 Spectrum Transformation
Due to Wave Refraction in Shallow Water
Wave spectrum refraction a in coastal area.
The wave description
given in Chap. 1 allows easy analysis of wind wave propagation in a coastal
area, 1 i.e. when relatively short sea waves propagate from deep to shallow
water, approaching a coastline. In this case refraction plays a special role
among the different factors affecting wave behavior. Due to depth variation
it results in the evolution of wave parameters, namely, propagation direction,
length, amplitude and wave profile. As noted in the Introduction, there are
many papers devoted to the problem of wave transformation in a coastal
area. An application of the spectral approach to this problem following from
the general formulation is considered in this chapter.
Sticking to physical terminology, wave refraction in shallow water can be
considered in the framework dispersive wave propagation in a spatially nonuniform isotropic medium. It follows from Sect. 1.6 that the propagation of
a wave packet is described with the help of the ray equations on a spherical
surface (1.86)-(1.89) and in a plane (5.2), with the frequency w being constant
along the ray:
w 2 = a 2 = f 2 (k, H (r)) = gkth (kH) = const.
(6.1)
This condition yields a useful ratio for determining the wave number lk I
and phase velocity c along the trajectory, depending on the slowly changing
depth H.
A second simple consequence of the general kinematic ratios can be obtained in the "cylindrical" case, i.e. with the depth H changing only in one
direction, for example H = H(x). It follows from (5.2) that a component of
the wave vector ky should be constant during wave packet propagation
ky = k sin ((3) = const ,
(6.2)
where n/2 - (3 is the angle between the direction of the wave vector k and
the isobath. The dependence between the angle (3 and the wave number k or
1 In this case the wave spectrum transformation apart from the surf zone is considered.