Overview and Fundamental Ideas - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 14

solution as the sum of four terms, in particular, u(x,t) 1/2 C 3 sin (x/c + t) +
1/2 C 4 cos (x/c + t) + 1/2 C 3 sin (x/c - t) + 1/2 C 4 cos (x/c - t). Now, the sin
(x/c + t) and cos (x/c + t) terms represent left-going waves, and the physical
solution is right-going. But if we set the left-going coefficients C 3 and C 4 to
zero, the right-going terms also vanish. Something is logically inconsistent, and
we will see why using a reformulated approach.

1.3.5.2 Example 1-7b. Correct approach.
Let us replace the “obvious” choice assumed in Equation 1.37 by the
separable product
u(x,t) e i t U(x) (1.39)
where both u(x,t) and U(x) may be complex functions. If we substitute into
Equation 1.36, we again obtain Equation 1.38. Now, we take its solution in the
form U(x) = D 1 e i x/c + D 2 e -i x/c so that u(x,t) D 1 e i x/c+t + D 2 e i -x/c+t .
Since the D 1 term is a left-going wave, we discard it, setting D 1 = 0; this leaves
u(x,t) D 2 e i -x/c+t , which satisfies u(0,t) D 2 e i t . Because we have
prescribed u(0,t) = A cos t, we select D 2 = A, so that u(x,t) Ae i t-x/c .
Next we use the substitution u = u r + i u i , and the complex identity e i =
cos + i s i n , where i = (-1). Since A was assumed to be real, the real part of
our complex solution yields u r (x,t) A cos (t - x/c), while the imaginary part
leads to u i (x,t) A sin (t - x/c). The real (right-going) part u r (x,t) A cos
(t - x/c) clearly satisfies the boundary condition u r (0,t) A cos t, and solves
the boundary value problem. But our procedure gives an additional ' free”
solution, u i (x,t) A sin (t - x/c), which satisfies u i (0,t) = A sin t. Thus,

complex variables produce simplifications when properly employed.

1.3.5.3 Example 1-7c. Faster approach.
We could have written the solution immediately from first principles.
Since the waves must be right-going, u(x,t) must be a function of “x-ct,” that is,
the dimensionless quantity (t - x/c). The equation u(0,t) A cos t motivates

us to replace t, which does not contain x, by (t - x/c), which does. This yields

u(x,t) A cos (t -x/c).

1.3.6 Example 1-8. Dissipative wave solution.
Solutions to Equation 1.35 are more complicated, and analysis shows that
the damping u/ t affects both amplitude and “phase” (that is, the (t - x/c)
and (t + x/c) in the above solutions). Equations 1.35 and 1.39 yield the
complex differential equation