
11.2. Fourth-Order One-Dimensional Nonstationary Equations 981
2.
∂
2
w
∂t
2
− a
∂
2
w
∂x
2
− b
∂
4
w
∂
2
t∂x
2
+ c
∂
4
w
∂x
4
= Φ(x, t).
Rayleigh–Bishop equation with a > 0, b > 0, and c > 0. This equation describes one-
dimensional longitudinal vibrations of a thick short circular bar of constant cross-section.
This is a special case of equation 11.2.7.3.
1
◦
. Particular solutions for the homogeneous equation with Φ(x, t) = 0:
w = [A exp (−βx) + B exp(βx)][C cos(λt) + sin(λt)], λ = β
s
a − cβ
2
bβ
2
− 1
,
w = [A exp (−βx) + B exp(βx)][C exp(−λt) + D exp(λt)], λ = β
s
a − cβ
2
1 − bβ
2
,
w = [A cos(βx) + B sin(βx)][C cos(λt) + sin(λt)], λ = β
s
a + cβ
2
1 + bβ
2
,
where A, B, C, D, and β are