25 Elastic Displacements and Waves
273
With (344) Eqs.(342b) for the isotropic case can be greatly simplified. Insertion
gives
ρ
∂
2 s i
∂t 2 = [μ(δ ir δ ks + δ kr δ is ) + λ δ ik δ rs ] s r , sk + f i .
If we calculate the sums we receive
ρ
∂
2 s i
∂t 2 = (μ + λ) s r , ri +μ s i , rr + f i .
(347)
Notice that
s i , kk =
3
k=1
∂
2
∂x
2
k
= =s i ,
(348)
where represents the Laplacian. Equations (347) are now in fact equivalent to two
d’Alembertian wave equations that we get by appropriate differentiations from (347).
If we differentiate (347) with respect to x i and sum up over the index i, then
taking s i , i = div s into consideration, a wave equation for div s follows, namely
ρ
∂
2
∂t 2 div s = (2μ + λ) div s + f i , i
or
(div s) −
1
c
2
l
∂
2
∂t 2 (div s) = − f i , i .
(349)
According to (349) the elastic dilatation ε = div s propagates with the longitudinal
sound velocity c l ,
c l =
2μ + λ
ρ
.
(350)
In order to arrive at the second wave equation we need to do the following. We write
down Eq. (347) for an index j,
ρ
∂
2 s i
∂t 2 = (μ + λ) s r , r j +μ s j , rr + f j .
(347
)
We differentiate equation (347) with respect to x j , Eq. (347)
with respect to x i and
subtract both results from each other. Due to the fact that the second derivatives are
interchangeable, the first terms on the right-hand side drop out and the following
equation remains, once again using the Laplacian ,
ρ
∂
2
∂t 2 (s i , j −s j , i ) = μ (s i , j −s j , i ) + f i , j − f j , i .
(351)