Overview and Fundamental Ideas - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 42

At the risk of being repetitious, we also give several relations which are
derivable from the foregoing equations, but which are of significance in
themselves in Fourier transform analysis.

e i

x

(x) dx = 1 (1.155)

e i

x

'(x) dx = - i (1.156)

e -i

x d

= 2 (x) (1.157)

1.8 Point Force and Dipole Wave Excitation
We have discussed the physical meaning and analytical modeling of point
forces and dipoles in the context of the wave equation; we have also given
Laplace transform solutions, and introduced and '. We now combine these
ideas to solve for the transverse oscillations of a string excited by a point load at
x = x* with a varying acceleration f(t). That is, we consider

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

(1.158)

1.8.1 Example 1-15. Finite string excited by a time-varying
concentrated point force.
We solve Equation 1.158 subject to the initial flat elevation u(x,0) = 0 and
the static condition u(x,0)/ t = 0 in the interval 0 < x < L. The string is rigidly
pinned at its end points with u(0,t) = 0 and u(L,t) = 0. We multiply each side of
Equation 1.158 by e -st and time integrate the result from t = 0 to , allowing the
solution process to proceed exactly as before. 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, but we do encounter a
new f(t) (x-x*) term. It is clear that e -st f(t) (x-x*) dt equals F(s) (x-x*),
where the function of x moves across the integral, and F(s) represents the
transform of f(t) with respect to time. These steps reduce Equation 1.158 to the
ordinary differential equation s 2 U( x ,s ) - c 2 U" ( x ,s ) = F( s ) (x-x*), where
intermediate results were simplified using initial conditions. This new equation
contains (x-x*), responsible for a slope discontinuity in the transverse string.
In elementary courses, this necessitates a break-up of the problem into two parts,
and then, the matching of two solutions assuming continuity of displacement,
plus jump conditions based on momentum considerations.