Examples from Classical Mechanics - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 85

r (x) = sinh x - sin x
- (sinh L + sin L)(cosh x - cos x)/(cosh L + cos L) (3.14)

The general solution takes the form

v(x,t) = (E sin 2 (EI/ A) t + F cos 2 (EI/ A) t) r (x) (3.15)

where the summation is taken over the eigenvalue . Unlike Example 3-1a, the
coefficients E and F are not easily obtained in terms of the initial transverse
displacement and speed v(x,0) and v t (x,0). The difficulty arises from the fact
that these initial conditions must be expanded not in sine or cosine series, but in
terms of the eigenfunctions in Equation 3.14. In principle, the simple boundary
conditions used here do show that the above eigenfunctions are “complete,” so
that procedures similar to those used in deriving formulas for Fourier
coefficients will lead to solutions. If the initial conditions vary rapidly with
space, then more terms in must be summed and Equation 3.12 must be
correspondingly solved.
For simple boundary condition combinations, e.g., clamped-free, pinned-
pinned and deflection-torsional spring, it is possible to establish the
“orthogonality” of the resulting eigenfunctions, so that representation using
series such as that obtained in Equation 3.15 is guaranteed. However, time-
dependent boundary conditions resulting from mass or dashpot contributions
lead to difficulties, since orthogonality is not achieved and alternative solutions
are required. Of course, even for the present example, the “exact” solution to
Equation 3.15 may be less accurate and more cumbersome than a direct
numerical integration of Equation 3.1. For these reasons, finite difference
solutions are developed in Chapter 4.

3.2 Example 3-2. Acoustic Waves in Waveguides
In Chapter 1, we studied the classical equation 2 u/ t 2 - c 2 2 u/ x 2 = 0 ,
for which the nondispersive speed for all disturbances is a constant c. For the
simple beam, we learned that the fourth-order equation A 2 v/ t 2 + EI 4 v/ x 4
= 0 also possesses traveling wave solutions. However, the waves are dispersive,
with different component wavelengths propagating at different speeds.

3.2.1 Simple waveguides.
Here, we will demonstrate how geometrical constraints cause some
disturbances to propagate, while damping out others. We will show how 2 p/ t 2
- c 2 ( 2 p/ x 2 + 2 p/ y 2 + 2 p/ z 2 ) = 0 (a natural extension to 2 u/ t 2 - c 2
2 u/ x 2 = 0) admits wave motions with speeds other than c, and also, how group
velocity concepts arise for three-dimensional waveguide motions. A
“waveguide” is just a medium that transmits wave motion. A vibrating string,