Kinematic Wave Theory - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 52

2.1.1 Uniform media.
Here, we introduce the terminology used in KWT and the physical ideas
behind kinematic wave modeling. For one-dimensional, linear wave
propagation in uniform media (equations without variable coefficients),
elementary solutions take the sinusoidal form
cos( x- t) (2.1)

where the “dispersion relation,” or
= ( ) (2.2)
is real (i.e., not complex) in the absence of dissipation (e.g., see Chapter 1).
Equation 2.1 describes a cosine function traveling “to the right,” with an
amplitude , an arbitrary wavenumber , and a frequency ( ). The general

solution is given by Fourier superposition integrals of the form

(x,t) = F( ) cos( x - t) d (2.3)

0
where F( ) is chosen to satisfy initial and boundary conditions.
Geometrically, the wavenumber gives the number of wave crests per unit
distance, while the frequency describes the number of oscillations per unit time.
Any dependence of the phase velocity / on leads to the dispersion of wave
components having different wavelengths, that is, different waves traveling at
different speeds; hence, = ( ) is called the “dispersion relation.”

2.1.2 Example 2-1. Transverse beam vibrations.
Consider the lateral vibrations of a uniform beam without axial or torsional
loads. If is the mass density per unit volume, A the cross-sectional area, E the
Young' s modulus, and I the moment of inertia, the transverse displacement
v(x,t) satisfies 2 v/ t 2 + ( EI/ A) 4 v/ x 4 = 0. The uniform wave substitution
v(x,t) = e i(kx- t) leads to 2 = ( EI/ A) k 4 or = (EI/ A) k 2 . Thus, bending
waves are dispersive since the phase velocity /k = (EI/ A) k depends on k.
An initial disturbance containing different length components will disperse: it
will not be recognizable at large distances. Thus, use of bending waves for
MWD transmissions will not work.

2.1.3 Example 2-2. Simple longitudinal oscillations.
In contrast, the axial vibrations of a uniform bar follow 2 u/ t 2 - c 2 2 u/ x 2
= 0, where u(x,t) is the displacement and the sound speed is c = (E/ ). Then,
the substitution u(x,t) = e i(kx- t) leads to = (E/ ) k; since /k = (E/ ) is
independent of k, longitudinal waves are “nondispersive.” At least for ideal
uniform bars, axial waves can be used in downhole-to-surface acoustic MWD
communications; in reality, periodic joints in the drillstring may limit the