
11.2. Fourth-Order One-Dimensional Nonstationary Equations 971
11.2.6 Equations Containing Second Derivative in x and Mixed
Derivatives
1.
∂
2
w
∂t
2
−
∂
2
w
∂x
2
−
∂
4
w
∂t
2
∂x
2
= 0.
One-dimensional wave equation with strong dispersion. This equation describes the dy-
namics of interior one-dimensional wave motions in an exponentially stratified fluid as
well as one-dimensional longitudinal vibrations of a rigid Rayleigh bar of constant cross-
section. This is a special case of equation 11.2.6.4 with a(x) = b(x) = c(x) = 1 and
Φ(x, t) = 0.
1
◦
. Particular solutions:
w =
C
1
exp(−λt) + C
2
exp(λt)
C
3
exp
−
λx
√
1 + λ
2
+ C
4
exp
λx
√
1 + λ
2
,
w =
C
1
cos(λx) + C
2
sin(λx)
C
3
cos
λt
√
1 +