1.2

Bottom Velocity (α = -1/2)

1.1

Pade (2,2) (α = -2/5)

1

Modified Pade

(α = -0.392)

Depth-Averaged Velocity

(α = -1/3)

0.9

Velocity at Free Surface

(α = 0)

0.8

0.1

0.2

0.3

0.4

0.5

Comparison of normalized phase speeds for different values of α

Figure 1

wave train consisting of two small amplitude periodic waves with amplitudes, *a*1

and *a*2, frequencies, ω1 and ω2, wave numbers, *k*1 and *k*2, and propagating in

directions θ1 and θ2 respectively. The water-surface elevation can be written as:

η(1) ( *x*, *t*) = *a*1 cos(*k*1 ⋅ *x *- ω1t) + *a*2 cos(*k*2 ⋅ *x *- ω2t)

(11)

where *k *= (*k*cosθ, *k*sinθ). The second-order wave will consist of a subharmonic

quencies 2ω1, 2ω2 and ω+ = ω1 + ω2. This can be written as:

η ( *x*, *t *) =

(2)

2

2

+

(12)

2

+ *a*1a2G (ω1, ω2 , θ1 , θ2 ) cos(2*k* ⋅ *x *- 2ωt)

where *k* = *k*1 *k*2, and *G*(ω1, ω2, θ1,θ2) is a bidirectional quadratic transfer func-

tion that relates the amplitude of the second-order waves to the amplitude of the

first-order waves. Dean and Sharma (1981) derived expressions for the bidirec-

tional quadratic transfer function based on second-order Stokes theory. Nwogu

7

Chapter 2 Theoretical Background

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