(1) A progressive wave may be represented by the variables *x *(spatial) and *t *(temporal) or by their

combination (phase), defined as *θ = kx - ωt*, where *k *and *ω *are described in the following paragraphs. The

values of *θ *vary between 0 and 2π. Since the *θ*-representation is a simple and compact notation, it will be

used in this chapter. Figure II-1-1 depicts parameters that define a simple, progressive wave as it passes a

fixed point in the ocean. A simple, periodic wave of permanent form propagating over a horizontal bottom

may be completely characterized by the wave height *H *wavelength *L *and water depth *d*.

(2) As shown in Figure II-1-1, the highest point of the wave is the *crest *and the lowest point is the

the distance of the trough below the SWL are each equal to the wave amplitude *a*. Therefore *a = H/2*, where

given point is the *wave period T*. The *wavelength L *is the horizontal distance between two identical points

on two successive wave crests or two successive wave troughs.

(3) Other wave parameters include ω = 2π/T the *angular *or *radian frequency*, the *wave number k = 2π/L*,

the *phase velocity *or *wave celerity C *= *L/T *= *ω/k*, the *wave steepness ε = H/L*, the *relative depth d/L*, and the

motion can be defined in terms of dimensionless parameters *H/L, H/d*, and *d/L*; these are often used in

practice. The dimensionless parameters *ka *and *kd*, preferred in research works, can be substituted for *H/L*

and *d/L*, respectively, since these differ only by a constant factor 2π from those preferred by engineers.

II-1-4

Water Wave Mechanics

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