Second Computational Aeroacoustics (CAA) Workshop on ...

where
d - A2 AI
A x 2
Error limits and the expansion coefficients are used to determine the integral limits Z±.
The integral near the branch points can be replaced by
where
B±=IL°g_Q_(a)daa-y -211 Log_(a-q_)da,a_y
Q±(a)=x(a)._.
Q4 (o0 are free of singularity and zeroes within the respective chosen integral limits.
The simple poles of K(a) are separated as follows:
where
s-+
t" Loge =I L_(a) P Loge (a - v_. )
a-y _ oe-y
L±(a)=K(a). (a- v_).
L± (a) are free of singularity and zeroes within the respective integral limits.
The zeroes of K(a) are similarly separated as
where
Z. = I L°geU(°_) d_ + I L°ge(a-Z") doe,
o_- y o_- y
U(a) is free of singularity and zeroes within the respective integral limit.
Now consider the integral
K(o0
U(a) - (43)
H(y) =If L°ge G(a) da,
o_- y
Here G(a) is free of singularity and zeroes within the integral limit, and thus represents Q_(a) in Eq. (38),
L+(a) in Eq. (40), U(o0 in Eq. (42), or K(o0 in Eq. (34) if K(a) is free of singularity in that region. This
integral can be written as
b da
H(y) =i_ Loge[G(oOIG(y)]ot_y da + LogeG(y) I£ ot-y
34
(37)
(38)
(39)
(40)
(41)
(42)
(44)
(45)
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