Examples from Classical Mechanics - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 98

p = - T 2 / x 2

(3.68)

The complete unsteady Bernoulli equation (Lamb, 1945) is

/ t - gy - (u 2 + v 2 )/2 + F(t) p/ = - (3.69)

where g is the acceleration due to gravity. If p is constant at the free surface,
e.g., equal to atmospheric pressure, then F(t) = 0. Since a mean flow does not
exist, the quadratic terms in Equation 3.69 are smaller than the linear ones, and
we have the approximation

p/ = - / t - gy (3.70)

Equations 3.66, 3.68 and 3.70 imply the combined kinematical-dynamical free
surface condition

2

/ t 2 = ( T/

2 / x 2 -g)

/ y (3.71)

3.3.3 Problem solution.
We will solve Equation 3.65 for (x,y,t), subject to Equation 3.71 at the
unknown free surface y = (x,t) and the condition that all motions cease at
infinity. As usual, we attempt separable solutions,

(x,y,t) = (y) e i(kx-

t)

(3.72)

where k is a wavenumber and is the frequency (waveguide motions are
assumed in the x direction only). The function (y) provides the “cross-
structure” for the “modal” boundary value problem. Substitution of Equation
3.72 into 2 / x 2 + 2 / y 2 = 0 l e ads to "(y) - k 2 (y) = 0. The general
solution takes the form (y) = Ae -ky + B e +ky where A and B are constants, but
only the contribution (y) = e +ky is meaningful since it gives decaying motions
at y = - . For this solution, we obtain '(y) = ke +ky (0) = 1, and '(0) = k.
Note that this modal problem, unlike our previous modal problem for
acoustical waveguides, is not a discrete eigenvalue problem. In fact, substitution
of Equation 3.72 in Equation 3.71 shows that

) 2 (y) = T/ (ik) 2 (-i '(y) - g '(y) (3.73)

) 2 (0) = T/ (ik) 2

(-i '(0) - g '(0) (3.74)

Consequently, we have the dispersion relation
2 = Tk 3 / + gk

(3.75)

for any wave component with wavenumber k. Now, since (y) = e +ky is real, we
consider only the real part of Equation 3.72 and write

(x,y,t) = C e +ky cos (kx -

t) (3.76)