EM 1110-2-1100 (Part II)
30 Apr 02
(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):
H gT cosh [ 2π(z%d)/L ]
H gT sinh [ 2π(z % 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
θ = π/2, 5π/2, etc., and the maximum vertical velocity in the negative direction occurs when θ = 3π/2, 7π/2,
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,
gπH cosh [ 2π(z % d)/L ]
sin θ '
gπH sinh [ 2π(z%d)/L ]
αz ' &
cos θ '
(d) Positive and negative values of the horizontal and vertical fluid accelerations for various values of
θ are shown in Figure II-1-2.
(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.
Water Wave Mechanics