EM 1110-2-1100 (Part II)
30 Apr 02
1
n0 '
(II-1-64)
2
1
E0 C0 ' EnC
(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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