Page 194 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

176 SLENDER STRUCTURES AND AXIAL FLOW

simply reflects that, in the absence of any support, a lateral displacement of the pipe is
not opposed by any restraint.
Now, if the analysis is conducted with (T = 0 from the start, it is easy to show that in
this case the condition of neutral stability, A1,2 = 0, requires
u2(1 - p) - f + n < 2h. (3.144b)

Clearly, since /3 < 1, this result is nonconservative; in particular, criterion (3.144b) predicts
a system to be stable when, in fact, through (3.144a) it is unstable (Roth 1964)! This is
a good demonstration of Bolotin & Zhinzher's (1969) thesis (Section 3.5.5).
It is Stein & Tobriner (1970) who consider wave propagation per se. They use the same
equation as Roth, but with r = 0. They obtain a general solution to initial conditions
q(c, 0) = f(6) and c(c, 0) = g(6) by means of Laplace transforms in time (denoted by
an overbar) and Fourier transforms in space (denoted by an asterisk), of the form

The Laplace transform over t is applied first, and then the Fourier transform over c, on
the resultant equation. After inversion, the general solution is

+- e-ur/2[2B1/2~a cos(4 - p) - (T sin(@ - p) cos 81 sinh 62
24-
+ [2/3%a sin(@ - p) + CT cos(@ - p)] sin 81 cosh 621
1
+- e-ur/2[sin(q5 - p) cos O1 cosh O2 - cos(@ - p) sin 81 cosh 1921 da,
4-
(3.145)

where
r = {[a2(a2 - (1 - B)u2 + n) + k - $72]2 + [B1/2u~a]2}1/2,

/3'/2UCT
p = - tan (3.146)
+
2 a2[a2 - (1 - B)u2 + n] k - io2
-
01 = zfi cos p, 02 = r,h sin p, 4 = w(~'/~ur c).
Numerical results in the case of (T = 0 and q = exp[i(Kc - wt)] are then considered. The
characteristic equation in this case is

w2 - 2B1'2UKW - K4 f (u2 - n)K2 - k = 0. (3.147)
Thus, for each wavenumber (wavelength) there is an associated frequency and, in the
absence of end constraints, all wavenumbers are permissible. The dominant wavelengths
depend on the spatial distribution of the initial disturbance and the propagation charac-
teristics of each of its Fourier components. Equation (3.147) may be solved for the phase