160
5 Wave Evolution in Non-uniform Currents in Deep Water
the one-dimensional case, for a= sin(,80 ). The divergence between the points
A and B becomes greater with increasing parameter a.
It should be noted that straight waves roll back and downstream ( Cgx < 0)
in the trajectory segment from the blocking point A to point B. They become
reversed only later at the point B with the current velocity being equal to
VB = -w ( 4gJ1- (a/4)2) -l. The waves are taken downstream after the
point B. The inequality Cgx < 0 is always satisfied. The angle ,8 is decreased
to zero with wave propagation to the area with small current velocities.
As has been noted, the group velocity component Cgx of straight waves
can be both positive and negative. The condition Cgx = 0 can be written
in the form V cos 3 (,8) - 2 cos(,8) = 0. The curve III, corresponding to this
relation, is also plotted in Figs. 5.1a,b. It divides the plane { V, ,8} into two
domains with different values of the group velocity component Cgx, with
Cgx > 0 being to the left of the curve III for straight waves. There are waves
with a > 1, having positive values Cgx in the segment of the trajectory II
between two blocking points A' and A". The maximum of the function l1f I is
reached at ,8 =arccos( y'2/3) being equal to 4/3y'2/3. There are no existing
waves with Cgx > 0, for non-dimensional velocities V greater than 4/3y'2/3,
although straight waves can exist for V > 1.
At the point B the ambiguity is removed in the relations (5.9), (5.10) and
(5.12). In these expressions the ( +) sign corresponds to the straight waves,
and the (-) sign to the reversed waves. The wave number k at the point B
is equal to 4w 2 j g, i.e. depending neither on the current velocity, nor on the
angle ,8.
In the wave spectrum expression (5.9) a singularity occurs at 1 + V cos(,8)
= 0. The value of the frequency-angular spectrum tends to infinity at the
point B. This singularity appears as a result of variable substitution in the
expression (5.4). The Jacobian (5.4) becomes infinite at 1 + V cos(,8) = 0. In
the case (kG g) = 0, i.e the projection of the group packet velocity onto the
chosen direction, determined by the angle ,8, is equal to zero. In this direction
the measured packet time of the frequency w = -gj4V cos(,8) is increased
indefinitely. The wave packet is presented as a regular, monochromatic wave.
Its spectrum is approximated by the delta functionS rv 8(w + gj4V cos(,8)),
having a singularity at w = -gj4V cos(,8). Thus, the value of the time spectrum s±(w, ,8) is arbitrarily large for this component. Further it is shown that
if the spatial spectrum were used instead of the temporal one this singularity
would not occur. The singularity of the spectrum (5.9) is integrable at the
point B.
The spectral value of the reversed waves s- (w, ,8) is increased with decreasing current velocity. A non-integrable singularity occurs at V ---+ 0, indicating the unlimited increase of gravity wave amplitudes. The lengths of
these waves are decreased with decreasing current velocity according to (5.9).
5 Wave Evolution in Non-uniform Currents in Deep Water
the one-dimensional case, for a= sin(,80 ). The divergence between the points
A and B becomes greater with increasing parameter a.
It should be noted that straight waves roll back and downstream ( Cgx < 0)
in the trajectory segment from the blocking point A to point B. They become
reversed only later at the point B with the current velocity being equal to
VB = -w ( 4gJ1- (a/4)2) -l. The waves are taken downstream after the
point B. The inequality Cgx < 0 is always satisfied. The angle ,8 is decreased
to zero with wave propagation to the area with small current velocities.
As has been noted, the group velocity component Cgx of straight waves
can be both positive and negative. The condition Cgx = 0 can be written
in the form V cos 3 (,8) - 2 cos(,8) = 0. The curve III, corresponding to this
relation, is also plotted in Figs. 5.1a,b. It divides the plane { V, ,8} into two
domains with different values of the group velocity component Cgx, with
Cgx > 0 being to the left of the curve III for straight waves. There are waves
with a > 1, having positive values Cgx in the segment of the trajectory II
between two blocking points A' and A". The maximum of the function l1f I is
reached at ,8 =arccos( y'2/3) being equal to 4/3y'2/3. There are no existing
waves with Cgx > 0, for non-dimensional velocities V greater than 4/3y'2/3,
although straight waves can exist for V > 1.
At the point B the ambiguity is removed in the relations (5.9), (5.10) and
(5.12). In these expressions the ( +) sign corresponds to the straight waves,
and the (-) sign to the reversed waves. The wave number k at the point B
is equal to 4w 2 j g, i.e. depending neither on the current velocity, nor on the
angle ,8.
In the wave spectrum expression (5.9) a singularity occurs at 1 + V cos(,8)
= 0. The value of the frequency-angular spectrum tends to infinity at the
point B. This singularity appears as a result of variable substitution in the
expression (5.4). The Jacobian (5.4) becomes infinite at 1 + V cos(,8) = 0. In
the case (kG g) = 0, i.e the projection of the group packet velocity onto the
chosen direction, determined by the angle ,8, is equal to zero. In this direction
the measured packet time of the frequency w = -gj4V cos(,8) is increased
indefinitely. The wave packet is presented as a regular, monochromatic wave.
Its spectrum is approximated by the delta functionS rv 8(w + gj4V cos(,8)),
having a singularity at w = -gj4V cos(,8). Thus, the value of the time spectrum s±(w, ,8) is arbitrarily large for this component. Further it is shown that
if the spatial spectrum were used instead of the temporal one this singularity
would not occur. The singularity of the spectrum (5.9) is integrable at the
point B.
The spectral value of the reversed waves s- (w, ,8) is increased with decreasing current velocity. A non-integrable singularity occurs at V ---+ 0, indicating the unlimited increase of gravity wave amplitudes. The lengths of
these waves are decreased with decreasing current velocity according to (5.9).