
1074 SYSTEMS OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS
◮ Using the Stokes–Helmholtz representation of the displacement vector.
Every solution of Eqs. (1) w ith f = 0 can be represented by the formulas
u = ∇ϕ + curl Ψ, (8)
where Ψ = (Ψ
1
, Ψ
2
, Ψ
3
) is a solution of the vector wave equation
Ψ
tt
− c
2
2
∆Ψ = 0 (9)
and the function ϕ is a solution of the scalar wave equation (6).
◮ Cauchy–Kovalevskaya solution.
Any solution of Eqs. (1) with f = 0 can be represented in the form
u =
1
[w] + (c
2
1
−c
2
2
)∇div w, (10)
where the vector function w satisfies the equation
2
1
[w] = 0. (11)
Here and in the following, the d’Alembert operators
1
and
2
are given by
1
≡ ∂
2
t
−c
2
1
∆,