Overview and Fundamental Ideas - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 21

displacements, H T , where T is the constant tension acting along the string.
Thus, the vertical force V acting at one side of the control element equals
T u/ x or T u/ x.

du

dx

Equilibrium position

Figure 1.2. Transverse vibrations of a string.

At the opposite side, the force is T u/ x + (T u/ x) x dx, so that the
incremental force equals (T u/ x) x dx. Since the weight is - l g dx, we obtain

l

2 u/ t 2 - T 2 u/ x 2 = -

l g which, upon division, yields 2 u/ t 2 - c 2

2 u/ x 2 = -g.

In the free “fall limit” with T = 0, the familiar u -1/2 gt 2 result reappears.

1.5.2 Example 1-9. String falling under its own weight.
We solve Equation 1.50 subject to the initial elevation u(x,0) = 0 and the
static condition u(x,0)/ t = 0. The string is rigidly attached to a wall x = 0, that
is, u(0,t) = 0; at infinity, its geometric slope is zero, with u/ x( ,t) = 0. The
reason for our recipe will be apparent shortly. We multiply each side of
Equation 1.50 by e -st (with s > 0), and integrate with respect to time from t = 0
to . Integration by parts shows that e -st 2 u/ t 2 dt = s 2 e -st u(x,t) dt -
su(x,0) - u(x,0)/ t, where we have omitted the limits (0, ) for brevity.
The definite integral on the right side is known as the “Laplace transform”
of the function u(x,t), and is denoted by
L {u(x,t)} = U(x,s) =

e -st u(x,t) dt (1.51)

0
The time integration has eliminated the appearance of t, replacing it by the
positive parameter s (this choice ensures that U(x,s) exists). The key advantage
with Laplace transforms is a practical one: once U(x,s) is available, where x is
regarded as a constant parameter, the solution u(x,t) can be obtained by “table
look-up.” Sophisticated “Bromwich contour integration” using “complex
variables” provides another powerful option (e.g., see Carrier, Krook and
Pearson, 1966). The former is preferable, since there is no shortage of tables. It
goes without saying that the opposite process, finding transforms for given
polynomial, trigonometric, and transcendental functions, follows similar lines.
Advances in software now permit real-time integration using “symbolic