5.2 Frequency-Angular Spectrum Evolution in a Current
157
Wave propagation from the area without the current (V = 0) to the
area with the current velocity directed along the Ox axis V = {V(x, y); 0}
is considered. The initial wave spectrum (i.e. at V = 0) is assumed to be
uniform and stationary So = So(w, (3). The spectrum in the current can be
written in accordance with the relation (5.4) as follows:
s±(w, (3, V) =
16So(w, f3o)
'
( 5 _ 9 )
V1 + V cos((3) ( 1 ± V1 + V cos((3))
where V = 4 V w j g is the non-dimensional current velocity. The ( ±) sign in
this expression indicates the ambiguity of determining the wave spectrum
in the current depending on the frequency w, angle (3 and velocity V. An
analogous ambiguity also occurs in determining the wave number k = lkl in
a current:
(5.10)
In order to determine the spectral value in (5.9), it is necessary to define (30 . This can easily be done when the velocity V depends only on one of
two coordinates. The coordinate xis cyclic at V = V(y). According to (5.2),
the component kx is constant with wave packet propagation and (30 can be
defined as:
fJo ~ a
In the other case, for V = V(x), ky = const, (30 can be written as:
(5.12)
The first case (5.11) corresponds to the situation of wave propagation in
shear horizontal non-uniform flow. It is considered below.
The second of the aforementioned cases is studied in detail. Variations of
the angle (3 are considered as then dependend on the non-dimensional velocity V for wave propagation in the increasing current V ( x). Proceeding from
the preservation conditions of the frequency w and the wave vector component ky along the trajectory, the wave motion integral in the variables V
and (3 can be written in the form:
157
Wave propagation from the area without the current (V = 0) to the
area with the current velocity directed along the Ox axis V = {V(x, y); 0}
is considered. The initial wave spectrum (i.e. at V = 0) is assumed to be
uniform and stationary So = So(w, (3). The spectrum in the current can be
written in accordance with the relation (5.4) as follows:
s±(w, (3, V) =
16So(w, f3o)
'
( 5 _ 9 )
V1 + V cos((3) ( 1 ± V1 + V cos((3))
where V = 4 V w j g is the non-dimensional current velocity. The ( ±) sign in
this expression indicates the ambiguity of determining the wave spectrum
in the current depending on the frequency w, angle (3 and velocity V. An
analogous ambiguity also occurs in determining the wave number k = lkl in
a current:
(5.10)
In order to determine the spectral value in (5.9), it is necessary to define (30 . This can easily be done when the velocity V depends only on one of
two coordinates. The coordinate xis cyclic at V = V(y). According to (5.2),
the component kx is constant with wave packet propagation and (30 can be
defined as:
fJo ~ a
In the other case, for V = V(x), ky = const, (30 can be written as:
(5.12)
The first case (5.11) corresponds to the situation of wave propagation in
shear horizontal non-uniform flow. It is considered below.
The second of the aforementioned cases is studied in detail. Variations of
the angle (3 are considered as then dependend on the non-dimensional velocity V for wave propagation in the increasing current V ( x). Proceeding from
the preservation conditions of the frequency w and the wave vector component ky along the trajectory, the wave motion integral in the variables V
and (3 can be written in the form: