EM 1110-2-1100 (Part II)
30 Apr 02
(n) When the relative water depth becomes shallow, i.e., 2πd/L < 1/4 or d/L < 1/25, Equation II-1-8 can
be simplified to
C ' gd
(o) Waves sufficiently long such that Equation II-1-18 may be applied are termed long waves. This
relation is attributed to Lagrange. Thus, when a wave travels in shallow water, wave celerity depends only
on water depth.
(p) In summary, as a wind wave passes from deep water to the beach its speed and length are first only
a function of its period (or frequency); then as the depth becomes shallower relative to its length, the length
and speed are dependent upon both depth and period; and finally the wave reaches a point where its length
and speed are dependent only on depth (and not frequency).
(3) The sinusoidal wave profile. The equation describing the free surface as a function of time t and
horizontal distance x for a simple sinusoidal wave can be shown to be
' a cos θ
η ' a cos kx & ωt '
where η is the elevation of the water surface relative to the SWL, and H/2 is one-half the wave height equal
to the wave amplitude a. This expression represents a periodic, sinusoidal, progressive wave traveling in the
positive x-direction. For a wave moving in the negative x-direction, the minus sign before 2πt/T is replaced
with a plus sign. When θ = (2πx/L - 2πt/T) equals 0, π/2, π, 3π/2, the corresponding values of η are H/2, 0,
-H/2, and 0, respectively (Figure II-1-1).
(4) Some useful functions.
(a) Dividing Equation II-1-9 by Equation II-1-13, and Equation II-1-10 by Equation II-1-15 yields,
' tanh kd
(b) If both sides of Equation II-1-20 are multiplied by d/L, it becomes
(c) The terms d/Lo and d/L and other useful functions such as kd = 2πd/L and tanh (kd) have been
tabulated by Wiegel (1954) as a function of d/Lo (see also SPM 1984, Appendix C, Tables C-1 and C-2).
These functions simplify the solution of wave problems described by the linear theory and are summarized
in Figure II-1-5. An example problem illustrating the use of linear wave theory equations and the figures and
tables mentioned follows.
Water Wave Mechanics