Introduction to Acoustics

106 Part A Propagation of Sound
Part A 3.17
which is the incident wave only. Then, in the shadow
region, where φA >φ>0, one has
ˆp GA = 0 , (3.718)
and there is neither an incident wave nor a reflected
wave.
The diffracted wave, in this plane wave limit, becomes
ˆpdiffr = ˆpinc e ikz cos γ ikw sin γ eiπ/4
e √
2
× � sin νπ
Vν(φ ± φS) AD [Γ Mν(φ ± φS)] ,
+,−
(3.719)
where Γ is now understood to be given by (3.715). This
result is, as before, for the case when the listener is many
wavelengths from the edge, so that the parameter Γ is
large compared with unity.
3.17.10 Small-Angle Diffraction
Simple approximate formulas emerge from the above
general results when one limits one’s attention to the
diffracted field near the edge of the shadow zone boundary.
The incident wave is taken as having its direction
lying in the (x, y) plane and one introduces rotated coordinates
(x ′ , y ′ ), so that the y ′ -axis is in the direction
of the incident sound, with the coordinate origin remaining
at the edge of the wedge. One regards y ′ as
Incident
plane wave
Wedge
φs
– π
Shadow
x'
φ s – π – φ
Fig. 3.59 Geometry and parameters used in discussion of
small-angle diffraction of a plane wave by a rigid wedge
y'
being large compared to a wavelength. The magnitude
|x ′ | is regarded as substantially smaller than y ′ , but not
necessarily small compared to a wavelength.
In the plane wave diffraction expression (3.719) the
angle γ is π/2, and the only term of possible significance
is that corresponding to the minus sign, so one has
ikw eiπ/4 sin νπ
ˆpdiffr = ˆpinc e √
2 Vν(φ − φS)
× AD [Γ Mν(φ − φS)] . (3.720)
Also, because |x ′ | is small compared with y ′ , one can
assume that |φS − π − φ| is small compared with unity,
so that
cos ν(φ − φS) ≈ cos νπ + (ν sin νπ) (φ − φS + π) ,
(3.721)
Vν(φ − φS) ≈ sin νπ , (3.722)
Mν(φ − φS) ≈ φS − π − φ ≈ x′
. (3.723)
y ′
Further approximations that are consistent with this
small-angle diffraction model are to set w → y ′ in the
expression for Γ ,buttoset
w → y ′ + 1 (x
2
′ ) 2
y ′
(3.724)
in the exponent. The second term is needed if one needs
to take into account any phase shift relative to that of the
incident wave.
The approximations just described lead to the expression
iky′ 1 + i
ˆpdiffr = ˆpinc e AD(X) , (3.725)
2 e(π/2)X2
with
�
k
X =
π y ′
�
x ′ . (3.726)
A remarkable feature of this result is that it is independent
of the wedge angle β. It applies in the same
approximate sense equally for diffraction by a thin
screen and by a right-angled corner.
The total acoustic field in this region just above and
just where the shadow zone begins can be found by
adding the incident wave for x ′ < 0. In the shadow zone
there is no incident wave, and one accounts for this by
using a step function H(−X). Thus the total field is
approximated by
ˆp GA + ˆpdiffr → pinc e iky′
×
�
H(−X) +
(1 + i)
e
2
i(π/2)X2
AD(X)
�
. (3.727)