(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

(II-1-39)

cos θ & ρ*gz *% *p*a

2π*d*

2 cosh

where *p*N is the total or absolute pressure, *p*a 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

cos θ & ρ *g z*

(II-1-40)

2π*d*

2 cosh

(b) Equation II-1-40 can be written as

2π(*z *% *d*)

cosh

(II-1-41)

& ρ*gz*

2π*d*

cosh

since

2π*x*

2π*t*

η'

(II-1-42)

cos

cos θ

&

'

2

2

Water Wave Mechanics

II-1-21

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