following current produces an increase in *L*, and an opposing current a

decrease in *L *in comparison to the case of no current (all other factors held

constant).

0.75

Blocking current

Opposing current

E

0.25

A

No current

D

C

B

Following current

-0.25

-0.75

0.0

0.1

0.2

0.3

0.4

0.5

Relative Water Depth, d/L

Figure 2. Examples of solutions to dispersion equation with current present

(after Jonsson 1990)

In the presence of a current, the wave energy will not be conserved along

the wave orthogonals, instead the energy is conserved along the wave rays

that have the absolute group speed *C*ga as a tangent at all locations. The wave

ray direction β depends of the relative wave group speed and the current

considerations (Jonsson 1990; also, see Figure 1) to yield:

1/ 2

2

(8)

* U *sin(δ - α)

β = α + arctan

(9)

*U *cos(δ - α) + *C*gr

where *C*gr is the relative group speed and α the direction of the wave

orthogonal. The relative group speed is determined from linear wave theory

according to:

1

2*kd *

(10)

2 sinh 2*kd *

From Equation 9, it may be concluded that if *U *cos(δ-α) = -*C*gr, the

denominator is zero, and the wave rays form a 90-deg angle with respect to

14

Chapter 3 Wave Model

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