THESE de DOCTORAT - cerfacs

4.1 Fundamental validation cases 59
e 2
r i
x
dV i
r j
y i
y j
e 1
dV j
Figure 4.1: Monopoles placed at y and listener placed at x
where k = ω/ ¯c 2 and Ŝ(y, ω) is a forcing source term. In this equation y represents the distance
from the source to the reference point and x represents the distance from the observer to the
reference point as illustrated in Fig. 4.1. The solution of this equation can be found by applying
Eq. (2.74)
ˆp(x, ω) =
∫
V 0 (y)
Ŝ(y, ω)
¯c 2 Ĝ(x|y, ω)dy (4.2)
The 2D spectral Green’s function, given by Eq. (2.82), is inserted into Eq. (4.2) resulting in
ˆp(x, ω) =
i
4¯c 2
∫
V 0 (y)
Ŝ(y, ω)H 2 0(kr)dy (4.3)
where r = |x − y| and H0 2 is the Hankel function of the second kind and order 0.
4.1.1 A Monopole in free space
One fundamental validation for an acoustic code is to test its ability to compute the radiation of
sound due to a source. As a consequence, for the present case only one monopole is considered
and no boundary conditions are taken into account. Moreover, as mentioned before, the mean
field is considered homogeneous. These conditions are imposed so that a perfect isotropic
radiation pattern is obtained. The source is pointwise and is defined as