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The (dynamic) equilibrium is expressed by the following equation (equation of
motion):
2
⎛ ∂σ
⎞ ∂ u
− σA+ ⎜σ + dx⎟A=
ρAdx
2
⎝ ∂x
⎠ ∂t
from which we obtain:
2
∂σ
∂ u
= ρ
2
∂x
∂t
For an elastic material we have σ = Eε
, and since the longitudinal deformation is
ε =∂u/
∂ x, we may re-write the last equation as:
2 2
∂ u ∂ u
E = ρ x
2 t
2
∂ ∂
or, finally:
2 2
∂ u 2 ∂ u
= c
2 0
with c
2 0
= E/
ρ
∂t
∂x
This is known as the (1-D) wave equation and the constant c0
is the sound speed in the
elastic material.
The general solution (D’Alembert’s solution) to this equation reads:
uxt ( , ) = f( x− ct) + gx ( + ct)
0 0
2