158
5 Wave Evolution in Non-uniform Currents in Deep Water
a V cos(,6) = 4 ( sin(,6) - J a sin(,6)) ,
(5.13)
where a = gky/w 2 is a non-dimensional parameter, which is constant along
the trajectory. The value of the parameter a is less than one and equal to
sin(,6) for wave packets going out of the area with no current (Vo = 0). The
value a can be greater than one for waves first generated in the current.
The relation (5.13) can be considered as the trajectory of wave packet
propagation in the plane { V, ,6 }· This trajectory with a < 1.0 is shown as
the curve na (or curve lib for a> 1) in Fig. 5.1a. The dependence ,6 = ,6(V)
does not have a single meaning, i.e. two values of the angle ,6 correspond
to the same value V, with a < 1. When ,6 > 1.0, there are three values ,6
corresponding to the one value V. If the wave packet propagates from the area
with no current (Vo = 0) to the increasing countercurrent V, then the motion
goes to the right upper part along the left curve II up to the point A (the
curve na in Fig. 5.1a or in Fig. 5.1b). The projection of the group velocity
on the axis Ox is positive, i.e.:
1/g
Cgx = "2V k cos(,6) + V 2: 0 at V < 0.
The greatest current velocity is reached at the point A within the trajectory, whereas (5.13), as the function V(,6), is maximal. The point A is the
turning point, after passing which the value Cgx becomes negative. Then the
motion along the lower part of the curve II starts from the point A to the
left (Fig. 5.1a, curve Ilb), i.e. to the area of decreasing current velocities. The
ambiguity of determining the angle ,6 disappears at the turning point A. The
angle can be found by solving the proper algebraic equation of the fourth
degree (5.13). The real part of the solution is the following expression:
] 2}
1
3 ~
1
{11 - v + {11 + v
'
(5.14)
where v = J1-16/27a 2 . It is necessary to use (-) with a < 1 in this
expression.
There are two turning points: A' and A" (see Fig. 5.1a) for waves with
a > 1. The wave blocking current velocity depends on the parameter a (for
a < 1, VA < -1.038g/4w, !3A < 0.096n). The turning point disappears
completely for a> 3J3/4.
5 Wave Evolution in Non-uniform Currents in Deep Water
a V cos(,6) = 4 ( sin(,6) - J a sin(,6)) ,
(5.13)
where a = gky/w 2 is a non-dimensional parameter, which is constant along
the trajectory. The value of the parameter a is less than one and equal to
sin(,6) for wave packets going out of the area with no current (Vo = 0). The
value a can be greater than one for waves first generated in the current.
The relation (5.13) can be considered as the trajectory of wave packet
propagation in the plane { V, ,6 }· This trajectory with a < 1.0 is shown as
the curve na (or curve lib for a> 1) in Fig. 5.1a. The dependence ,6 = ,6(V)
does not have a single meaning, i.e. two values of the angle ,6 correspond
to the same value V, with a < 1. When ,6 > 1.0, there are three values ,6
corresponding to the one value V. If the wave packet propagates from the area
with no current (Vo = 0) to the increasing countercurrent V, then the motion
goes to the right upper part along the left curve II up to the point A (the
curve na in Fig. 5.1a or in Fig. 5.1b). The projection of the group velocity
on the axis Ox is positive, i.e.:
1/g
Cgx = "2V k cos(,6) + V 2: 0 at V < 0.
The greatest current velocity is reached at the point A within the trajectory, whereas (5.13), as the function V(,6), is maximal. The point A is the
turning point, after passing which the value Cgx becomes negative. Then the
motion along the lower part of the curve II starts from the point A to the
left (Fig. 5.1a, curve Ilb), i.e. to the area of decreasing current velocities. The
ambiguity of determining the angle ,6 disappears at the turning point A. The
angle can be found by solving the proper algebraic equation of the fourth
degree (5.13). The real part of the solution is the following expression:
] 2}
1
3 ~
1
{11 - v + {11 + v
'
(5.14)
where v = J1-16/27a 2 . It is necessary to use (-) with a < 1 in this
expression.
There are two turning points: A' and A" (see Fig. 5.1a) for waves with
a > 1. The wave blocking current velocity depends on the parameter a (for
a < 1, VA < -1.038g/4w, !3A < 0.096n). The turning point disappears
completely for a> 3J3/4.