Wave Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 84

where F [ u tt ]= u tt ( ω , t ) has been used. The solution of the above equation is

u ( ω , t )= A ( ω ) cos ω at + B ( ω ) sin ω at ,

A ( ω ) and B ( ω ) can be determined by initial conditions u ( ω , 0 )= 0and u t ( ω , 0 )=

¯

ψ ( ω )
ω

ψ ( ω )

¯ ( ω )= ( ω )=

as A 0, B . Thus

a

¯

e − iω at

ψ ( ω )
ω

¯ 1

2 a

ψ ( ω )
i

ψ ( ω )
i

¯

e iω at u ( ω , t )= sin ω at = − .

a ω ω

Using the integral property and the shifting property of Fourier transformations and
taking an inverse transformation leads to

x + at

1
2 a

u ( x , t )= W ψ ( x , t )= ψ ( ξ ) d ξ .

x − at

The solution of PDS (2.60) thus follows from the solution structure theorem

t

= ∂
∂

u

t W ϕ +

W ψ ( x , t )+ W f τ ( x , t − τ ) d τ , (2.61)

0

where f τ = f ( x , τ ) .Since

x + at

∂ W ϕ

∂

2 a ∂

1

ξ = ϕ (

x + at )+ ϕ ( x − at )

= ϕ ( ξ ) d ,

t ∂ t 2

x − at

the solution of

u tt

a 2 u xx = , − ∞ < x < + ∞ , 0 < t ,

( , )= ϕ ( ) , ( , )= ψ ( )

u x 0 x u t x 0 x

reads

x + at

)= ϕ (

x + at )+ ϕ ( x − at ) 1

2 a

u ( x , t + ψ ( ξ ) d ξ . (2.62)

2

x − at

This is called the D'Alembert formula of one-dimensional wave equation .
The solution of
u tt =

a 2 u xx + f ( x , t ) , − ∞ < x < + ∞ , 0 < t ,

u ( x , 0 )= 0 , u t ( x , 0 )= 0

is

t x + a ( t − τ )

1
2 a

u ( x , t )= f ( ξ , τ ) d ξ d τ , (2.63)

0 x − a ( t − τ )

which is called the Kirchhoff formula of one-dimensional wave equation .