EM 1110-2-1100 (Part II)
30 Apr 02
(h) Thus, in deep water, the water particle orbits are circular as indicated by Equation II-1-36 (see Fig-
ure II-1-4). Equations II-1-37 and II-1-38 show that in transitional and shallow water, the orbits are elliptical.
The more shallow the water, the flatter the ellipse. The amplitude of the water particle displacement
decreases exponentially with depth and in deepwater regions becomes small relative to the wave height at a
(i) Water particle displacements and orbits based on linear theory are illustrated in Figure II-1-4. For
shallow regions, horizontal particle displacement near the bottom can be large. In fact, this is apparent in
offshore regions seaward of the breaker zone where wave action and turbulence lift bottom sediments into
suspension. The vertical displacement of water particles varies from a minimum of zero at the bottom to a
maximum equal to one-half the wave height at the surface.
(7) Subsurface pressure.
(a) Subsurface pressure under a wave is the sum of two contributing components, dynamic and static
pressures, and is given by
2π(z % d)
ρ g H cosh
L
p )'
(II-1-39)
cos θ & ρgz % pa
2πd
2 cosh
L
where pN is the total or absolute pressure, pa is the atmospheric pressure, and ρ is the mass density of water
(for salt water, ρ = 1,025 kg/m3 or 2.0 slugs/ft3, for fresh water, ρ = 1,000 kg/m3 or 1.94 slugs/ft3). The first
term of Equation II-1-39 represents a dynamic component due to acceleration, while the second term is the
static component of pressure. For convenience, the pressure is usually taken as the gauge pressure defined
as
2π (z % d)
ρ g H cosh
L
p ' p ) & pa '
cos θ & ρ g z
(II-1-40)
2πd
2 cosh
L
(b) Equation II-1-40 can be written as
2π(z % d)
cosh
L
p ' ρgη
(II-1-41)
& ρgz
2πd
cosh
L
since
H
2πx
2πt
H
η'
(II-1-42)
cos
cos θ
&
'
2
L
T
2
Water Wave Mechanics
II-1-21
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