Diffraction phenomena studied in electromagnetism, acoustics, and elastodynamics are often modeled using integrals, such as the well-known Sommerfeld integral. The far field asymptotic evaluation of such integrals obtained using the method of steepest descent leads to the classical Geometrical Theory of Diffraction (GTD). It is well known that the method of steepest descent is inapplicable when the integrand's stationary phase point coalesces with its pole, explaining why GTD fails in zones where edge diffracted waves interfere with incident or reflected waves. To overcome this drawback, the Uniform geometrical Theory of Diffraction (UTD) has been developed previously in electromagnetism, based on a ray theory, which is particularly easy to implement. In this paper, UTD is developed for the canonical elastodynamic problem of the scattering of a plane wave by a half-plane. UTD is then compared to another uniform extension of GTD, the Uniform Asymptotic Theory (UAT) of diffraction, based on a more cumbersome ray theory. A good agreement between the two methods is obtained in the far field.

1.
P.
Ya Ufimtsev
,
Fundamentals of the Physical Theory of Diffraction
(
Wiley
,
NJ
,
2007
), Chap. 3, pp.
59
68
; Chap. 4, pp. 71–76.
2.
B.
,
M.
Darmon
,
L.
Fradkin
, and
C.
Potel
, “
Numerical comparison of acoustic wedge models, with application to ultrasonic telemetry
,”
Ultrasonics
, in press (
2015
).
3.
V.
Zernov
,
L.
Fradkin
, and
M.
Darmon
, “
A refinement of the Kirchhoff approximation to the scattered elastic fields
,”
Ultrasonics
52
,
830
835
(
2012
).
4.
J. B.
Keller
, “
Geometrical theory of diffraction
,”
J. Opt. Soc. Am.
52
,
116
130
(
1962
).
5.
J. D.
Achenbach
and
A. K.
Gautesen
, “
Geometrical theory of diffraction for three-D elastodynamics
,”
J. Acoust. Soc. Am.
61
,
413
421
(
1977
).
6.
J. D.
Achenbach
,
A. K.
Gautesen
, and
H.
McMaken
,
Rays Methods for Waves in Elastic Solids
(
Pitman
,
New York
,
1982
), Chap. 5, pp.
109
148
.
7.
R. M.
Lewis
and
J.
Boersma
, “
Uniform asymptotic theory of edge diffraction
,”
J. Math. Phys.
10
,
2291
2305
(
1969
).
8.
S. W.
Lee
and
G. A.
Deschamps
, “
A uniform asymptotic theory of electromagnetic diffraction by a curved wedge
,”
IEEE Trans. Antennas. Propag.
24
,
25
34
(
1976
).
9.
D. S.
Ahluwalia
, “
Uniform asymptotic theory of diffraction by the edge of a three dimensional body
,”
SIAM J. Appl. Math.
18
,
287
301
(
1970
).
10.
B. L.
Van Der Waerden
, “
On the method of saddle points
,”
Appl. Sci. Res., Sect. B
2
,
33
45
(
1951
).
11.
P. H.
Pathak
and
R. G.
Kouyoumjian
, “
The dyadic diffraction coefficient for a perfectly-conducting wedge
,”
DTIC Document, Tech. Rep.
(
1970
).
12.
R. G.
Kouyoumjian
and
P. H.
Pathak
, “
A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface
,”
Proc. IEEE
62
,
1448
1461
(
1974
).
13.
P. C.
Clemmow
, “
Some extension to the method of integration by steepest descent
,”
Q. J. Mech., Appl. Math.
III
,
241
256
(
1950
).
14.
V. A.
Borovikov
and
B. Y.
Kinber
,
Geometrical Theory of Diffraction
(
Institution of Electrical Engineers
,
London
1994
), Sec. 3.7.
15.
H.
McMaken
, “
A uniform theory of diffraction for elastic solids
,”
J. Acoust. Soc. Am.
75
,
1352
1359
(
1984
).
16.
N.
Tsingos
,
T.
Funkhouser
,
A.
Ngan
, and
I.
Carlbom
, “
Modeling acoustics in virtual environments using the uniform theory of diffraction
,”
Proc. ACM SIGGRAPH
(
2001
), pp.
545
552
.
17.
M. F.
Catedra
,
J.
Perez
,
F.
Saez de Adana
, and
O.
Gutierrez
, “
Efficient ray-tracing techniques for three-dimensional analyses of propagation in mobile communications: Application to picocell and microcell scenarios
,”
IEEE Antennas Propag. Mag.
40
,
15
28
(
1998
).
18.
D.
Bouche
and
F.
Molinet
,
Méthodes Asymptotiques en Électromagnétisme (Asymptotic Methods in Electromagnetics)
(
Springer-Verlag
,
Berlin Heidelberg
,
1994
), Chap. 5, pp.
192
198
.
19.
D.
Bouche
,
F.
Molinet
, and
R.
Mittra
,
Asymptotic Methods in Electromagnetics
(
Springer-Verlag
,
Berlin Heidelberg
,
1997
), Chap. 5.
20.
V. A.
Borovikov
,
Uniform Stationary Phase Method
(
The Institution of Electrical Engineers
,
London
,
1994
), pp.
159
161
.
21.
F.
Molinet
,
Acoustic High-Frequency Diffraction Theory
(
Momentum Press
,
New York
,
2011
), Chap. 3, pp.
244
259
.
22.
L.
Ju. Fradkin
and
R.
Stacey
, “
The high-frequency description of scatter of a plane compressional wave by an elliptical crack
,”
Ultrasonics
50
,
529
538
(
2010
).
23.
D.
Gridin
, “
High-frequency asymptotic description of head waves and boundary layers surrounding critical rays in an elastic half-space
,”
J. Acoust. Soc. Am.
104
,
1188
1197
(
1998
).
24.
J. D.
Achenbach
and
A. K.
Gautesen
, “
Edge diffraction in acoustics and elastodynamics
,” in
Low and High Frequency Asymptotics 2
(
Elsevier
,
New York
,
1986
), Chap. 4, pp.
335
401
.
25.
L. W.
Schmerr
and
S.-J.
Song
,
Ultrasonic Nondestructive Evaluation Systems: Models and Measurements
(
Springer
,
New York
,
2007
).
26.
M.
Darmon
and
S.
Chatillon
, “
Main features of a complete ultrasonic measurement model - Formal aspects of modeling of both transducers radiation and ultrasonic flaws responses
,”
Open J. Acoust.
3A
,
43
53
(
2013
).
27.
M.
Darmon
,
V.
Dorval
,
A.
Kamta Djakou
,
L.
Fradkin
, and
S.
Chatillon
, “
A system model for ultrasonic NDT based on the physical theory of diffraction (PTD)
,”
Ultrasonics
64
,
115
127
(
2016
).
28.
G.
Toullelan
,
R.
Raillon
,
S.
Chatillon
,
V.
Dorval
,
M.
Darmon
, and
S.
Lonné
, “
Results of the 2015 UT modeling benchmark obtained with models implemented in CIVA
,”
AIP Conf. Proc.
in press (2016).
29.
A.
Kamta Djakou
,
M.
Darmon
, and
C.
Potel
, “
Elastodynamic models for extending GTD to penumbra and finite size scatterers
,”
Phys. Proc.
70
,
545
549
(
2015
).
30.
M.
Darmon
,
N.
Leymarie
,
S.
Chatillon
, and
S.
Mahaut
, “
Modelling of scattering of ultrasounds by flaws for NDT
,” in
Ultrasonic Wave Propagation in Non Homogeneous Media
(
Springer
,
Berlin
,
2009
), Vol.
128
, pp.
61
71
.
You do not currently have access to this content.