where

η = wave setup/setdown)

Determination of the kinematic wave properties in the presence of a

current involves solving Equations 2-5 simultaneously to obtain *L*, *C*a, *C*r,

and *T*r. Thus, it is assumed that the absolute wave period is known, as well as

the current magnitude and direction (with respect to the waves, that is, δ-α).

By employing Equations 2-5, a dispersion relationship including a current

may be derived according to (Jonsson, Skovgaard, and Wang 1970),

* U *cos(δ - α)*T*a d

tanh *kd *=

1 -

(6)

where

(7)

2π

which is the deepwater wavelength for the case of no current. For a specific

absolute wave period, water depth, and current, *L *may be calculated from

Equation 6 through some iterative procedure. After *L *is known *C*a is

determined from Equation 4, *C*r from Equation 5, and finally *T*r from

Equation 3.

Solving Equation 6 is not straightforward because the equation may have

several solutions or no solution at all, depending on the properties of the

current with respect to the wave properties. Jonsson (1990) comprehensively

discusses this topic and only a few main characteristics are pointed out here.

For the case of a following current, that is, *U *cos(δ-α) > 0, there are always

two solutions to Equation 6, whereas the equation can have two, one, or no

solutions for an opposing current (note that the left-hand side of Equation 6

can attain both positive and negative values for the general case).

Figure 2 illustrates typical solutions to Equation 6 for various types of

currents, plotting the right-hand side of the equation as straight lines

beginning in (*d*/*L*o)1/2 (point A in the figure) and the left-hand side as the two

curves with decreasing gradients with increasing *d*/*L *(dashed lines in

Figure 2). For the case of waves travelling from deeper water on to a current,

it is always the solution corresponding to the lower wave number (longer

wavelength) that is physically reasonable (note that the solution

corresponding to larger *d*/*L *is not shown in Figure 2; it is located beyond the

chosen axis range). By solving Equation 6 (employing the plus sign on the

left-hand side), only the desired solution is obtained for a following current

(point B). In the situation of an opposing current, the wrong solution is

always for wave numbers larger than the wave number corresponding to

wave blocking (point E), which is discussed in the following paragraphs.

Point D indicates a solution for an opposing current not strong enough to

yield blocking. Thus, by comparing the solutions corresponding to points B,

C, and D the modification of wavelength by a current is clearly seen: a

13

Chapter 3 Wave Model