Geology Reference
In-Depth Information
with a cosine dependence. The free surface position is also an unknown, and a
functional assumption for it must be made. Since Equation 3.67 requires that
/ y =
/ t, we choose
(x,t) = - a sin (kx -
t)
(3.77)
with a sine dependence, so that the trigonometric functions cancel. In Equation
3.77, “a” represents the wave elevation or amplitude . Substitution of Equations
3.76 and 3.77 in Equation 3.67 shows that C = a
/k, so that
/k) e +ky cos (kx-
(x,y,t) = (a
t)
(3.78)
thus completing the solution.
3.3.4 Energy considerations.
In order to demonstrate some elementary properties of waves, we drop
surface tension for simplicity, so that the dispersion relation in Equation 3.75
reduces to 2 = g k or
= (gk) 1/2 (3.79)
In this discussion, we assume the positive square root, and ignore the “ ” for
brevity. This result for deep water gravity waves has the “phase velocity”
c p =
/k = (g/k)
(3.80)
and the “group velocity”
c g =
/ k = (1/2) (g/k)
(3.81)
so that
c g = (1/2) c p
(3.82)
If we refer to a mass element (
dx) and consider its potential energy increase
for a wave elevation
, simple center of gravity considerations lead to
/2
2 dx = 1/2 g
2 dx
PE = g
(3.83)
0 0
where = 2 /k is the wavelength. Substituting Equations 3.77 and 3.78 into
Equation 3.83, we find that the potential energy per unit wavelength is
PE = 1/4 g a 2
(3.84)
Now, the kinetic energy per unit wavelength is
0
KE = 1/2
/ x) 2 + (
/ y) 2 } dx dy
{(
(3.85)
-
0
Search WWH ::




Custom Search