Wave Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 41

Chapter 2
Wave Equations

We first develop the solution structure theorem for mixed problems of wave equa-
tions followed by methods of solving one-, two- and three-dimensional mixed prob-
lems. The solution structure theorem expresses contributions (to the solution of
wave equations) of the initial distribution and the source term by using that from
the initial rate of change of the solution, and hence considerably simplifies the de-
velopment of solutions. For the two- or three-dimensional cases, some knowledge
of special functions is required which can be found in Appendix A. The solution
structure theorem is also valid for Cauchy problems. Finally, we discuss methods
of solving one-, two- and three-dimensional Cauchy problems. The required know-
ledge of integral transformation is available in Appendix B.

2.1 The Solution Structure Theorem for Mixed Problems
and its Application

Cons id er mi xe d problems of three-dimensional wave equations in a closed re-
gion is the boundary surface. Three kinds of lin-

ear homogeneous boundary conditions can, therefore, be written in a unified form

Ω .Let Ω = Ω ∪ ∂ Ω ,where ∂ Ω

L u

∂ Ω =

, ∂

u

0. For example, for a one-dimensional bounded region 0 ≤ x ≤ l ,

∂ n

the boundary conditions at two ends are denoted by subscript 1 and 2 respectively
and become L 1 (

0.
The solution structure theorem is regarding the relation among the solutions of
the following four PDS
⎨

u , u x ) | x = 0 = 0and L 2 ( u , u x ) | x = l =

a 2

u tt = Δ u , Ω × ( 0 , + ∞ ) ,

L u

∂ Ω =

, ∂

u

0 ,

(2.1)

⎩ ∂ n

u ( M , 0 )= ϕ ( M ) , u t ( M , 0 )= 0 ,