(II-1-7)

(b) An expression relating wave celerity to wavelength and water depth is given by

2π*d*

(II-1-8)

tanh

2π

(c) Equation II-1-8 is termed the *dispersion relation *since it indicates that waves with different periods

travel at different speeds. For a situation where more than one wave is present, the longer period wave will

travel faster. From Equation II-1-7, it is seen that Equation II-1-8 can be written as

2π*d*

tanh

(II-1-9)

2π

(d) The values *2π/L *and *2π/T *are called the *wave number k *and the *wave angular frequency ω*,

respectively. From Equation II-1-7 and II-1-9, an expression for wavelength as a function of depth and wave

period may be obtained as

2π*d*

tanh

tanh *kd*

(II-1-10)

'

2π

ω

(e) Use of Equation II-1-10 involves some difficulty since the unknown *L *appears on both sides of the

equation. Tabulated values of d/L and d/L0 (SPM 1984) where *L*0 is the deepwater wavelength may be used

to simplify the solution of Equation II-1-10. Eckart (1952) gives an approximate expression for Equa-

tion II-1-10, which is correct to within about 10 percent. This expression is given by

4π2 d

tanh

(II-1-11)

2π

(f) Equation II-1-11 explicitly gives *L *in terms of wave period *T *and is sufficiently accurate for many

engineering calculations. The maximum error 10 percent occurs when d/L . 1/2. There are several other

approximations for solving Equation II-1-10 (Hunt 1979; Venezian and Demirbilek 1979; Wu and Thornton

1986; Fenton and McKee 1990).

(g) Gravity waves may also be classified by the water depth in which they travel. The following

classifications are made according to the magnitude of *d/L *and the resulting limiting values taken by the

function *tanh (2πd/L)*. Note that as the argument of the hyperbolic tangent kd = 2πd/L gets large, the tanh (kd)

approaches 1, and for small values of kd, tanh (kd) . kd.

(h) Water waves are classified in Table II-1-1 based on the relative depth criterion *d/L*.

Water Wave Mechanics

II-1-7

Integrated Publishing, Inc. |