Mud Acoustics in Modern Drilling - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 270

u (x,t) = u m (x,t) + u' (x,t) (5.35)
p(x,t) = p m (x,t) + p'(x,t) (5.36)
We now assume, in an absolute value sense, that the (primed) disturbances
represent small deviations from mean conditions, that is,
m (x,t) >> '(x,t) (5.37)
u m ( x , t) >> u' (x,t) (5.38)
p m (x,t) >> p'(x,t) (5.39)
Substitution of Equations 5.37 to 5.39 in Equations 5.31 and 5.33, and
subsequent algebraic expansion, lead to a system of equations amenable to
simple limit processes.
Limit No. 1. If the primed disturbances vanish identically, and m is
constant, the mean pressure p m and velocity u m satisfy Equation 5.32, that is,
Bernoulli's equation.
Limit No. 2. If the mean density m is constant, and the mean speed u m
vanishes, our acoustic disturbances satisfy
m u' t = - p' x

(5.40)

and

' t + m u ' x = 0 (5.41)
to leading order. To obtain Equations 5.40 and 5.41, formally small quadratic
quantities in the disturbances have been dropped. Let us now introduce the
“compressibility” defined by
' =

m p'

(5.42)

We differentiate Equation 5.40 with respect to x, and Equation 5.41 with respect
to t, to find that

m u' tx = - p' xx and ' tt + m u' xt = 0. Then, using Equation 5.42,

we have the pressure equation
2 p'/ t 2 - (B/ m ) 2 p'/ x 2 = 0 (5.43)
where the “bulk modulus” B and the “fluid compressibility” are related by
B = 1/ (5.44)
As in the Lagrangian description, the speed of sound c satisfies c 2 = B / m .
Thus, Equation 5.43 takes the form 2 p'/ t 2 - c 2 2 p'/ x 2 = 0. We could have
eliminated pressure as the working variable and obtained the wave equation for
velocity

2 u' / t 2 - (B/ m ) 2 u' / x 2 = 0

(5.45)