Introduction to Acoustics

[3.108], and Carslaw [3.109]. In the modern era with
digital computers, where the numerical calculation of
Fourier transforms is virtually instantaneous and routine,
an explicit analytical solution for the impulse response
is often a good starting point for determining diffraction
of waves of constant frequency.)
3.17.6 Constant-Frequency Diffraction
The transient solution (3.664) above yields the solution
for when the source is of constant frequency. One makes
the substitution
S(t) → ˆSe −iωt , (3.674)
and obtains
ˆpdiffr =− ˆS
β
�∞
L
e ikξ Kν(ξ) dξ. (3.675)
The integral that appears here can be reduced to standard
functions in various limits. A convenient first step is to
change the integration variable to the parameter s, so
that
ξ 2 = L 2 + (L 2 − Q 2 )sinh 2 (s/2) , (3.676)
dξ
[ξ2 − Q2 ] 1/2 [ξ2 − L2 ds
= ,
] 1/2 2ξ
(3.677)
and so that the range of integration on s is from 0 to ∞.
One notes that the integrand is even in s, sothatitis
convenient to extend the integration from −∞ to ∞,
and then divide by two. Also, one can use trigonometric
identities to combine the x1 term with the x2 term and
to combine the x3 term with the x4 term. All this yields
the result
where
ˆpdiffr = ˆS sin νπ
2β
�
�∞
+,−
−∞
e ikξ
ξ Fν(s,φ± φS)ds ,
(3.678)
Uν(φ) − J(νs)cosνφ
Fν(s,φ) =
J2 (νs) + 2J(νs)V 2 ν (φ) +U2 ,
ν (φ)
(3.679)
with the abbreviations
J(νs) = cosh νs − 1 , (3.680)
Uν(φ) = cos νπ − cos νφ , (3.681)
Vν(φ) = (1 − cos νφ cos νπ) 1/2 . (3.682)
Basic Linear Acoustics 3.17 Diffraction 103
The requisite integrals exist, except when either Uν(φ +
φS) orUν(φ − φS) should be zero. The angles at which
one or the other occurs correspond to boundaries between
different regions of the diffracted field.
3.17.7 Uniform Asymptotic Solution
Although the integration in (3.678) can readily be
completed numerically, considerable insight and useful
formulas arise when one considers the limit when both
the source and the listener are many wavelengths from
the edge. One argues that the dominant contribution to
the integration comes from values of s that are close to
0 (where the phase of the exponential is stationary), so
one approximates
ξ → L + (L2 − Q2 )
s
8L
2 = L + π
2k Γ 2 s 2 ,
(3.683)
e ikξ
ξ
→ eikL
L ei(π/2)Γ 2 s 2
, (3.684)
J(νs) = cosh νs − 1 → ν2
2 s2 , (3.685)
Uν(φ)
Fν(s,φ) →
U2 ν (φ) + ν2V 2 ,
ν (φ)s2
�
1 1
=
2νVν(φ) Mν(φ) + is +
�
1
.
Mν(φ) − is
(3.686)
The expression (3.686) uses the abbreviation
Mν(φ) = Uν(φ)
νVν(φ) =
while (3.684) uses the abbreviation
Γ 2 = k (L2 − Q 2 )
4πL
cos νπ − cos νφ
,
1/2 ν (1 − cos νφ cos νπ)
(3.687)
kwwS
= , (3.688)
πL
where the latter version follows from the definitions
(3.649)and(3.650).
One also notes that the symmetry in the exponential
factor allows one to identify
�∞
−∞
e i(π/2)Γ 2 s 2
Mν + is
ds =
�∞
−∞
e i(π/2)Γ 2 s 2
Mν − is
ds , (3.689)
so the number of required integral terms is halved.
A further step is to change the the integration variable to
u = (π/2) 1/2 Γ e −iπ/4 s . (3.690)
Part A 3.17