Wave Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 98

where S at stands for the spherical surface of a sphere of center r and radius at .
By performing the integration over Ω in Eq. (2.82) first over S r

|

and then over

r |

r −

r | | r − , we obtain

⎡
⎣

⎤
⎦ δ r

+ ∞

r − at d r r

r ) 1 ψ (

u ( r , t )= d S − −

| −

r |

4 π a r

0

S r
| r | r −

r )
at

1 ψ (

= d S .

4 π a

S at

Finally

1

M )

u ( M , t )= W ψ ( M , t )= ψ ( d S , (2.83)

a 2 t

4 π

S at

where M ∈ S at . The integration in Eq. (2.83) is a surface integration of the first kind

over the spherical surface S at .
Remark 1. Consider the u in wave equations as representing the string displace-
ment. The unit in Eq. (2.83) reads

]= a − 2 t − 1 [ ψ ][

T 2
L 2

1
T ·

L
T ·

L 2 [ u d S ]= · = L ,

which is correct.

Remark 2. By the solution structure theorem, the solution of
u tt =

a 2

Δ u , Ω × ( 0 , + ∞ ) ,

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

is

⎡
⎣ ∂
∂

⎛
⎝

⎞
⎠ +

⎤
⎦ ,

1 1

t

1
t

M ) M ) u ( M , t )= ψ ( d S ψ ( d S (2.84)

a 2 4 π t

S at S at

which is called the Poisson formula of three-dimensional wave equations .

Remark 3. The solution of
u tt

a 2 = Δ u + f ( M , t ) , Ω × ( 0 , + ∞ ) ,

( , )= ( , )= ,

u M 0 u t M 0 0