Overview and Fundamental Ideas - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 10

applying initial conditions . The substitutions u(x,0) = (x) = u n (x,0) = A n

sin ( nx/L), together with Equations 1-11 and 1-12, lead to
A n = { 2 (x) sin n x/L dx}/L (1.25)
Similarly, u t (x,0) = (x) = u n (x,0)/ t = ( nc/L) B n sin ( nx/L) yields
B n = { 2 (x) sin n x/L dx}/( nc) (1.26)
Since A n and B n are known, the solution for u(x,t) is completely determined.
This formal construction does not necessarily produce a solution, e.g., if the
derived series diverges or if it is not differentiable. Here we have introduced the
use of eigenvalues and their corresponding eigenfunctions.

1.3.2 Example 1-4. String with distributed forces.
Now let us consider the inhomogeneous wave equation, with a non-zero
right side, namely,

2 u/ t 2 - c 2 2 u/ x 2 = f(x,t)

(1.27)

where the right side may represent the acceleration of a concentrated force, a
distributed force, or both. Again, we consider the boundary conditions u(0,t) = 0
and u(L,t) = 0, and the initial conditions u(x,0) = (x) and u t (x,0) = (x).
Separation of variables again shows that sin nx/L is the appropriate function
satisfying boundary conditions. But since f(x,t) is nonzero and arbitrary, we
take a modified approach and write
u(x,t) = u n (t) sin nx/L (1.28)
To be consistent, we also expand the functions (x), (x) and f(x,t) in
similar series so that, for example, we obtain the Fourier series pair

f(x,t) = f n (t) sin nx/L (1.29a)

f n (t) = {2 f(x,t) sin n x/L dx}/L (1.29b)

where t is a parameter. Likewise, we obtain,
(x) =

n sin nx/L

(1.30a)

n = { 2

(x) sin n x/L dx}/L (1.30b)

(x) =

n sin nx/L

(1.31a)

n = { 2 (x) sin n x/L dx}/L (1.31b)
Substitution into Equation 1.27 leads to (sin nx/L){-c 2 ( n/L) 2 u n (t) -
u n "(t) + f n (t)} = 0. This is satisfied if the coefficients within the { } brackets
vanish, that is, if the functions u n (t) obey the ordinary differential equation