1

(II-1-64)

2

1

(II-1-65)

2

(h) When the wave crests are not parallel to the bottom contours, some parts of the wave will be traveling

at different speeds and the wave will be refracted; in this case Equation II-1-65 does not apply (see Parts II-3

and II-4). The rate of energy transmission is important for coastal design, and it requires knowledge of Cg

to determine how fast waves move toward shore. The mean rate of energy transmission associated with

waves propagating into an area of calm water provides a different physical description of the concept of group

velocity.

(i) Equation II-1-65 establishes a relationship between the ratio of the wave height at some arbitrary

depth and the deepwater wave height. This ratio, known as the *shoaling coefficient *(see Part II-3 for detail

derivation), is dependent on the wave steepness. The variation of shoaling coefficient with wave steepness

as a function of relative water depth d/L0 is shown in Figure II-1-8. Wave shoaling and other related

nearshore processes are described in detail in Parts II-3 and II-4.

(10) Summary of linear wave theory.

(a) Equations describing water surface profile particle velocities, particle accelerations, and particle

dispk cements for linear (Airy) theory are summarized in Figure II-1-9. The Corps of Engineers'

a

microcomputer package of computer programs (ACES; Leenknecht et al. 1992) include several software

applications for calculating the linear wave theory and associated parameters. Detailed descriptions of the

ACES and CMS software to the linear wave theory may be found in the ACES and CMS documentation.

(b) Other wave phenomena can be explained using linear wave theory. For example, observed decreases

and increases in the mean water level, termed wave setdown and wave setup, are in essence nonlinear

quantities since they are proportional to wave height squared. These nonlinear quantities may be explained

using the concept of radiation stresses obtained from linear theory. Maximum wave setdown occurs

just seaward of the breaker line. Wave setup occurs between the breaker line and the shoreline and can

increase the mean water level significantly. Wave setdown and setup and their estimation are discussed in

P rt II-4.

(c) *Radiation stresses *are the forces per unit area that arise because of the excess momentum flux due

to the presence of waves. In simple terms, there is more momentum flow in the direction of wave advance

because the velocity *U *is in the direction of wave propagation under the wave crest when the instantaneous

water surface is high (wave crest) and in the opposite direction when the water surface is low (wave trough).

Also, the pressure stress acting under the wave crest is greater than the pressure stress under the wave trough

leading to a net stress over a wave period. Radiation stresses arise because of the finite amplitude (height)

of the waves. Interestingly, small-amplitude (linear) wave theory can be used to reasonably approximate

radiation stresses and explain effects such as wave set down, wave setup, and the generation of longshore

Water Wave Mechanics

II-1-29

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