Second Computational Aeroacoustics (CAA) Workshop on ...

The solution for pressure, p, can be written as
Substitution of (10) into (7), (9) gives the problem for _(x,y)
__OP=0 at z_+y2=0.25 (9) _;
p(x,y,t) = !m(_(x,g)e -i_') (10)
02D 02P -ia;e -b[(x-=')_+y_] (11)
Ox---- 7 + _ + _2i_ =
015 _ g2
On -0 at x _+ =0.25 (12)
radiation boundary conditions as x, y --4 oc (13)
(11) is a non-homogeneous I-Ielmholtz equation. The problem (11) - (13) can be solved by the
method of superposition. Let
_(_,y) = pi(x,_) + p_(x,y) (14)
where pi is the incident wave generated by the source and p_ is the wave reflected off the cylin-
der. pi satisfies
-4- w2pi --" -iwe -b[(_-_')_ +y_]
and the outgoing wave condition. When pi is found the problem for p_ becomes
02pr 02pr
Ox--- 7- + _ + W2p_ = 0 (16)
Op____Z__ = Opi
On On
(15)
at x 2 + g2 = 0.25 (17)
To solve for p,(z, y) we use polar coordinates (r=, 0_) with the origin at x = z=, y = 0; thus the
solution is independent of 0_. (16) reduces to the non-homogeneous Bessel equation
with boundary conditions
d2Pi d- 1 dpi -4-wZpi --iwe -br_ (18)
dr] r s dr=
pi(r=) is bounded at r= = 0 (19)
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