Overview and Fundamental Ideas - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 39

1.7.2.6 On the subtle meaning of impulse.

When we state that the dynamical response of a physical system to any
excitation is determined completely by its impulse response, we must
understand that this refers to specific differential equations and excitations. For

example, if we are considering
l 2 u/ t 2 + u/ t - T 2 u/ x 2 + l g = F(t) (x-x s ) (1.135)
as the model for an entity u(x,t), the actual excitation F must experimentally
create a jump discontinuity in u/ x in order for us to use its u(x,t) response as
the impulse response. Those u p (x,t)' s so obtained can be used to produce more
general solutions for u(x,t) using the convolution integral in Equation 1.134.
For the transverse string modeled by Equation 1.135, the impulsive strike of a
guitar pick or a violin bow, as suggested in Figure 1.4, might do.

1.7.2.7 Example 1-14: Incorrect use of impulse response.
Suppose Joe Engineer uses a small clamp-on excitation source that actually
implements the dipole load in Figure 1.5. Because the “sharp source” acts over
a short distance, he (mistakenly) believes that he has excited the system with a
point impulse. His measurements for u p (x,t), and their application to Equation
1.134 for u(x,t), would be useless: the convolution result is based upon the
assumption of a true (x-x s ) source that creates jumps in u/ x, but this is
clearly not the case from Figure 1.5 (the dipole here, rather than being (x-x s ), is
actually '(x-x s ), which does not lead to Equation 1.134).
If Joe Engineer insists on using the loading of Figure 1.5, then in what
sense are his measurements meaningful? Since the [ u/ t] or [ u] suggested by
Figure 1.5 really represent jumps in the spatial derivative

/ x (see Equation

1.107), his measurements really pertain to solutions of

2 / t 2 +

/ t - T 2 / x 2 = ... (x-x s ) (1.136)

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Any attempt to use Equation 1.134 for u(x,t) would be incorrect; Joe Engineer's
convolution is restricted to the variable (x,t) only ... and no other .

1.7.2.8 Additional models.
Equations 1.99 and 1.136 for u(x,t) and (x,t) are not the only formulations
available for transverse strings. Alternative formulations are easily created to
treat new applications. Suppose Joe Engineer invents a hypothetical bow that
creates internal discontinuities in 2 u/ x 2 . How would such excitations be
modeled? In this case, one can differentiate l 2 u/ t 2 + u/ t - T 2 u/ x 2 +
l g = 0 with respect to x, and add G(t) (x-x s ) to the right side to obtain

2

u/ x)/ t 2 + u/ x)/ t - T 2 u/ x)/ x 2 = G(t) (x-x s ) (1.137)

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