Mud Acoustics in Modern Drilling - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 264

5.1.7.1 Closed-closed ends.
The condition X(0) = 0 leads to B = 0, while X(L) = A sin p L = 0 requires
that p L = n , where n = 1, 2, 3, ... Thus, the displacement mode shapes are
X(x) = A sin n x/L

(5.21)

with T(t) = C sin n ct/L + D cos n ct/L, or
T(t) = C sin

t + D cos t (5.22)

where

= n c/L (5.23a)
f = /2 = nc/2L (5.23b)
with the frequency f being measured in Hertz.

5.1.7.2 Open-open ends.
At open ends in the long wave limit, the acoustic pressure is approximately
zero (Morse and Ingard, 1968; Pierce, 1981). Thus, we consider the derivative
X'(x) = A p cos p x - B p sin p x. The condition X'(0) = 0 implies A = 0, while
X'(L) = -B p sin p L = 0 requires p L = n , n = 0, 1, 2, 3, ... and so on. Hence, the
mode shapes are
X(x) = B cos n x/L (5.24)
with T(t) = C sin n ct/L + D cos n ct/L, or
T(t) = C sin

t + D cos t (5.25)

where

= n c/L (5.26a)
f = /2 = nc/2L (5.26b)
We observe that the eigenfunctions for closed-closed and open-open problems
obviously differ, but that the eigenvalues or natural frequencies are identical.

5.1.7.3 Closed-open ends.
Here, X(0) = 0 requires B = 0, so that the function X(x) = A sin p x. Then,
X'(L) = A p cos p L = 0 requires that, p L = (2n-1) /2, with n = 1, 2, 3, ... or p =
(2n-1) /2L. The mode shapes are
X(x) = A sin (2n-1) x/2L

(5.27)

with T(t) = C sin (2n-1) ct/2L + D cos (2n-1) ct/2L, or
T(t) = C sin t + D cos t (5.28)

where

= (2n-1) c/2L (5.29a)
f = /2 = (2n-1)c/4L (5.29b)