Overview and Fundamental Ideas - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 36

1.7.2.3 Multiple forces and higher order moments.
Higher-order forces and couples may be constructed to model different
kinds of external excitations. In Figure 1.6, we show two equal forces pointing
upwards, and one force with the combined magnitude of the two opposing loads
pointing downwards.

-2F

F

F

Figure 1.6. An unconventional force arrangement.
This force configuration introduces a slope discontinuity similar to that of
a single force acting alone; however, the net force here vanishes, while a single
force is always nonzero. The use of this or other load models will be dictated by
physical requirements. Other load models are easily conceptualized.

1.7.2.4 Symmetries and anti-symmetries.
Force loadings are associated with additional properties. To fix ideas, refer
back to Equation 1.99 and consider an infinite string. A single-force or
monopole will create a jump [ u/ x] , but the jump is such that the two u/ x's at
its sides are equal and opposite; also, as proven earlier, they will propagate in
opposite directions. A double-force dipole load will create a jump [u], but the
jump is such that the two u' s at its sides are equal and opposite; also, they will
propagate in opposite directions. Modeling decisions will also depend on
required symmetries or anti-symmetries (in the dependent variable or its
derivative) desired at the source point.

1.7.2.5 Impulse response.
Engineers talk freely about “impulse responses.” We hear statements that
the system response to an impulse excitation completely characterizes the
dynamics of the system. Often, explanations are given in terms of Green' s
functions and similar abstractions, but it is possible to develop these ideas
without higher math. What is an impulse? To develop the ideas, we return to
the single-force model for u(x,t), with a general time-dependent F(t),

2 u/ t 2 +

u/ t - T 2 u/ x 2 + l g = F(t) (x-x s ) (1.123)

l