solution exists (for a following current, blocking cannot occur). Inspecting

Equation 6, the right side is a linear function of the wave number (*k *= 2π/*L*),

whereas the left side is a more complicated function of *k*. There is only one

solution if the linear function constitutes the tangent to the function on the

left-hand side in the point of solution (Jonsson 1990), which is equivalent to

the two functions having the same gradient in *k*. To clarify the derivation of

the blocking condition, Equation 6 is rewritten in terms of *k *to yield:

2π

- *kU *cos(δ - α)

(17)

Differentiating with respect to *k *gives,

(

)

(18)

where *U*s denotes the current speed at blocking. The left side of Equation 18

corresponds to the relative group speed, giving the following criterion for

wave blocking:

(19)

This criterion implies that wave blocking occurs if the current projected on

the wave orthogonals has an opposing speed corresponding to the relative

group speed, producing an angle between the resulting direction for *C*ga and

the wave orthogonals of 90 deg.

At the point of blocking, the wavelength attains a minimum value, which

may be estimated by substituting Equation 19 into Equation 6 to yield,

1

tanh *kd *=

(20)

1 - *n L*o

where

1

2*kd *

(21)

2 sinh 2*kd *

The required blocking speed associated with Equation 20 may be obtained

from Equation 19, once the wavelength *L *at blocking has been determined for

a specific *L*o and *d*. This criterion may be written in nondimensional form as:

=-

(22)

tanh *kd*

2π

Thus, for a specific ratio *d*/*L*o, the required blocking speed can be determined

from Equations 20 and 22. Figure 3 displays the nondimensional blocking

speed as a function of *d*/*L*o.

18

Chapter 3 Wave Model

Integrated Publishing, Inc. |