THESE de DOCTORAT - cerfacs

2.3 Hybrid computation of noise: Acoustic Analogies 37
2.3.4 Solving Lighthill’s analogy analytically
In Lighthill’s analogy the D’Alembertian is the operator which describes the acoustic radiation
due to all the different sources present (Eq. 2.41). This inhomogeneous wave equation can be
expressed as
1 ∂ 2 p
c ∞ ∂t 2 − ∂2 p
∂xi
2 = q (2.64)
where q is an arbitrary source and c ∞ is the propagation speed of the acoustic perturbations.
Explicit integral solutions for this hyperbolic equation are available by means of Green’s theorem.
The integral solutions obtained account for the effect of sources, boundary conditions and
initial conditions in a relative simple formula. A Green function is written as G(x, t|y, τ) and
should be read “as the measurement of G at the observation point x at time t due to a pulse δ
produced at position y at time τ". Mathematically it yields
1 ∂ 2 G
c ∞ ∂t 2 − ∂2 G
∂xi
2
= δ(x − y)δ(t − τ) (2.65)
The delta function δ(t) is not a common function with a pointwise meaning, but a generalized
function formally defined by its filter property:
∫ ∞
−∞
F(x)δ(x − x 0 )dx = F(x 0 ) (2.66)
for any well-behaving function F(x). Moreover, Green functions have two important properties
that must be taken into account for further developments. The first one is called ‘causality’
which states that at a time before the pulsation occurs (t < τ), G(x, t|y, τ) is equal to zero as
well as its temporal derivative. The second property is called ‘reciprocity’ and is defined as
G(x, t|y, τ) = G(y, −τ|x, −t) (2.67)
Using now the reciprocity relation and interchanging the notation x ↔ y and t ↔ τ it can be
proved that the Green’s function also satisfies the equation:
1 ∂ 2 G
c ∞ ∂τ 2 − ∂2 G
∂y 2 i
A formal solution of the inhomogeneous wave equation
= δ(x − y)δ(t − τ) (2.68)