(5) Local fluid velocities and accelerations.

(a) In wave force studies, the local fluid velocities and accelerations for various values of *z *and *t *during

the passage of a wave must often be found. The horizontal component *u *and the vertical component *w *of the

local fluid velocity are given by the following equations (with *θ*, *x*, and *t *as defined in Figure II-1-1):

cos θ

(II-1-22)

2* L*

cosh (2π*d*/*L*)

(II-1-23)

sin θ

2* L*

cosh (2π*d*/*L*)

(b) These equations express the local fluid velocity components any distance (*z + d*) above the bottom.

The velocities are periodic in both *x *and *t*. For a given value of the phase angle *θ = (2πx/L -2πt/T)*, the

hyperbolic functions *cosh *and *sinh*, as functions of *z *result in an approximate exponential decay of the

magnitude of velocity components with increasing distance below the free surface. The maximum positive

horizontal velocity occurs when *θ *= 0, 2π, etc., while the maximum horizontal velocity in the negative

direction occurs when *θ *= π, 3π, etc. On the other hand, the maximum positive vertical velocity occurs when

etc. Fluid particle velocities under a wave train are shown in Figure II-1-2.

(c) The local fluid particle accelerations are obtained from Equations II-1-22 and II-1-23 by

differentiating each equation with respect to *t*. Thus,

M*u*

sin θ '

αx '

(II-1-24)

M*t*

cosh (2π*d*/*L*)

M*w*

αz ' &

(II-1-25)

cos θ '

M*t*

cosh (2π*d*/*L*)

(d) Positive and negative values of the horizontal and vertical fluid accelerations for various values of

(e) Figure II-1-2, a sketch of the local fluid motion, indicates that the fluid under the crest moves in the

direction of wave propagation and returns during passage of the trough. Linear theory does not predict any

net mass transport; hence, the sketch shows only an oscillatory fluid motion. Figure II-1-3 depicts profiles

of the surface elevation, particle velocities, and accelerations by the linear wave theory. The following

problem illustrates the computations required to determine local fluid velocities and accelerations resulting

from wave motions.

II-1-12

Water Wave Mechanics

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