
1104 SYSTEMS OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS
12.15.2 Decompositions of the Nondissipative Thermoelasticity
Equations with f = 0
◮ Decomposition based on two stream functions.
Each solution of the nondissipative thermoelasticity equations (2)–(3) with f = 0 can be
represented in the form
u = ∇ϕ + v, T =
ρ
α
c
2
1
∆ϕ −ϕ
tt
,
v = (v
1
, v
2
, v
3
), v
1
= ψ
(1)
y
, v
2
= −ψ
(1)
x
+ ψ
(2)
z
, v
3
= −ψ
(2)
y
,
(4)
where ψ
(1)
= ψ
(1)
(x, t) and ψ
(2)
= ψ
(2)
(x, t) are some solutions of the wave equation
ψ
tt
− c
2
2
∆ψ = 0, (5)
and the function ϕ = ϕ(x, t) is a solution of the fourth-order equation
ϕ
tttt
− [a + c
2
1
+ (αβ)/ρ]∆ϕ
tt
+ ac
2
1
∆∆ϕ = 0. (6)
Equation (6) can be represented in the form
(∂