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```referred to as the
zero-, ﬁrst-, second-, and third-mode edgewaves, is shown inFigure 5.8 and compared
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108 WAVES AND WAVE-INDUCED HYDRODYNAMICS
Figure 5.8 Free surface elevations corresponding to the ﬁrst four edge wave
modes and a fully reﬂected wave as a function of dimensionless distance offshore.
with the normally incident standing wave solution (Eq. (5.48)). Notice that the mode
number n corresponds to the number of zero crossings of the water surface elevation.
This wave motion must satisfy the following edge wave dispersion relationship:
σ 2 = gλ(2n + 1)m,
which relates the alongshore wave number λ to the wave frequency and the beach
slope. Ursell (1952) has shown that a more accurate representation of the dispersion
relationship is
σ 2 = gλ sin(2n + 1)m,
which is equivalent to the previous one for small beach slope m.
The waveform described by (Eq. (5.51)) is a standing edge wave, which will not
propagate along a beach. Propagating waveforms may be found by adding together
two standing waves as before. The second standing wave might be proportional to
sin λy sin σ t , in which case, we obtain
η(x, y, t) = Ae−λx Ln(2λx) cos(λy − σ t), (5.56)
a wave that propagates in the positive y direction. By subtracting, instead of adding,
we can obtain a wave propagating in the opposite direction.
Theedgewave solutionabove is valid only for planarbeaches; additionalmeans to
solve Eq. (5.46) are required for other beach proﬁles. For those that can be described
by an exponentially increasing depth, h = h0(1− e−r x), where h0 is the offshore
constant depth, Ball (1967) developed analytical solutions for the wave motion. For
arbitrary beach proﬁles, Holman and Bowen (1979) and Kirby, Dalrymple, and Liu
(1981) provide numerical methods.
The effects of a longshore current on edge waves has been investigated byHowd,
Bowen, and Holman (1992), who showed that the inﬂuence of the current is of the
same magnitude as a variable beach proﬁle. Furthermore, the equations govern-
ing edge waves on a uniform longshore current can be cast into the same govern-
ing equation as before through a transformation of the depth. After introducing a
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5.4 LOW-FREQUENCY MOTIONS AT THE SHORELINE 109
current-modiﬁed effective depth deﬁned as
h′(x) = h(x)(
1− V (x)C
)2 ,
Eq. (5.46) still can be used for the solution. The celerity C is the speed of the edge
wave C = σ/λ.
The generation of edge waves in nature has been a question of great interest,
particularly in conjunction with the study of nearshore circulation. Guza and Davis
(1974) developed a model showing that edge waves can be generated through a
nonlinear resonant mechanism with the incident wave train. An incident wave train
with a frequency σ can generate two edge waves with frequencies σ/2, which are
called subharmonic edge waves. This mechanism has been veriﬁed in the laboratory∗
(Guza and Inman 1975).
It should be pointed out that, from linear analyses, edge waves on a beach with
straight and parallel contours cannot be caused by an incident wave train from off-
shore. This can be proven with Snell’s law:
sin θ
C
= sin θ0
C0
If we consider a wave train generated in shallow water but directed obliquely off-
shore at angle θ to the beach normal, there are two possibilities: the wave propagates
offshore to deep water (so-called leaky modes), or, as the wave encounters deeper
water, thewave angle becomes larger until ﬁnally thewave propagates directly along-
shore. From Snell’s law, the wave angle θ becomes 90◦. Part of the crest will be in
shallowwater and part in deeper water, which will turn the wave back onshore. Then,
if a signiﬁcant portion of the wave is reﬂected back offshore, this process repeats.
This means that the wave is trapped against the shoreline and it is an edge wave. In
other words, if a wave generated in shallow water has a propagation angle between
0o and θc, the wave propagates offshore. For angles greater than θc, waves become
trapped as edge waves.
The only way edge waves can be generated by waves incident from offshore is
if the bathymetry is sufﬁciently irregular so that waves can approach the shallow
water with large angles of incidence. For man-made structures, two examples are the
trapping of waves by the ends of breakwaters (Dalrymple, Kirby, and Seli 1986) and
the reﬂection of waves from groins and jetties such that they have the correct wave
angle as they propagate onto the straight and parallel beach contours.
Gallagher (1971) proposed another mechanism for edge wave generation based
on the concept that the incident wave ﬁeld, which is composed of many waves and
many frequencies, can have wave groups occurring at a period that satisﬁes the edge
wave dispersion relationship. Bowen andGuza (1978) examined the limit case of two
∗ A very graphic laboratory experiment in a wave basin with reﬂecting sidewalls to illustrate this mecha-
nism is to generate normally incident waves onto a sloping beach. The wave period should be chosen so
that the width of the wave basin (length of the beach) is nπ/λ, where n ≥ 1. For n = 2, one edge wave
ﬁts along the beach. Within several minutes of starting the wave generator, the presence of the edge
waves will be obvious, for the run-up on the beach is distorted strongly by their presence.
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110 WAVES AND WAVE-INDUCED HYDRODYNAMICS
waves with slightly different frequencies in a laboratory which formed wave groups.
Edge waves were observed as predicted by this mechanism.
Lippmann, Holman, and Bowen (1997) provide a review of the mechanisms for
the generation of edge waves and show that edge wave generation by spatially and
temporally varying radiation stresses of the incident waves can account for most
of the low-frequency motion in the surf zone. They use the linear wave equation
(Eq. 5.46) with radiation stress driving terms.
In the ﬁeld, there are now numerous studies that show very large edge wave mo-
tions in the surf zone. In a major study, Huntley, Guza, and Thornton (1981) showed,
using a longshore array of current meters, that low-frequency energy in the surf zone
had wave lengths associated with the edge wave dispersion relationship given above.
Figure 5.9 shows the ﬁeld data and the associated dispersion relationships. This low-
frequency energy in some cases can have more energy content locally within the
shallow portions of the surf zone than the incident wave ﬁeld, which has important
ramiﬁcations (as yet not properly elucidated) for coastal processes.
Figure 5.9 Edge wave dispersion relationships and wave num-
bers forTorreyPinesBeach (Huntley,Guza, andThornton1981,
copyright by the American Geophysical Union).
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5.5 NEARSHORE CIRCULATION AND RIP CURRENTS 111
Amanifestation of the low-frequency wave motion is the slowly varying location
of the limit of wave uprush on the beach face. An observer standing at this limit of
wave uprush may soon ﬁnd it necessary to move to other locations up and down the
beach face as the surf zone slowly rises and falls owing to the edge wave or standing
wave motions within the surf zone over the course of minutes.
5.4.3 SHEAR WAVES
In 1986, a ﬁeld experiment atDuck,NorthCarolina, revealed a surprising behavior of
the longshore current (Oltman-Shay, Howd, and Birkemeier 1989). The very strong
longshore current that were caused by large waves began to oscillate with a low
frequency. The alongshore wavelengths of this motion, detected by current meters
distributedalong the shoreline,weremuch less thanpredictedbyedgewaveor gravity
wave theories. Bowen and Holman (1989) provided a theory for a wave motion that
depended on the cross-shore shear in the longshore```