Second Computational Aeroacoustics (CAA) Workshop on ...

and the scatteredfield satisfiesthe homogeneousform of Eq. (I). -::_ ,
The solution for the incident field is obtained here through the useof a Hankel transform, given by
oc
G(s)= -2 R2 jo(sR)g(R)dR (3) _'
T" 0
g(R)--/s2j°(sR)G(s)dSo (4) i
where jn(z) is the spherical Bessel function of the first kind and order n. The properties of spherical
Bessel functions are given by Abramowitz and Stegun [1]. i
Now, integration by parts and the use of general expressions for the derivatives of spherical Bessel | |
functions [1], gives i
_ n _3o(ns) N_ n _ dR = -s2a(s) (5) .
So, the Hankel transform of Eq. (2) leads to i
where,
p_n_(R) = - jo(sR)P_(s) z
0 (s 2 _ k2o) ds (6) _--
P_(s) = -rr2/R2jo(sR)p,(R)d R
0
Now, with R = Cr 2 + x_ - 2rx_ cos 0, Abramowitz and Stegun [1] give an addition theorem for
spherical Bessel functions,
jo(_n) - _n
sin(sR)
-- -- _(2n + 1) j,,(sr) jn(sx,) P,(cos0)
where P, (cos 0) is the Legendre polynomial of order n. Thus, from Eqs. (6) and (8),
where,
oo
n=0
c_
p_,_(R) = - y_' (2n + 1) Pn(cos 0) I_(r)
n=0
f s_j.(_) j.(_) P_(_)
(s 2 _ k2 ) es
The general solution for the scattered field may be written in separable form as
oo
p_(r,O) = _ A_h_)(kor) P,(eos 0)
n_O
16
(7)
(8)
(9)
(_o)
(11)
z