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Pulse diffraction by a curved half plane
Wave Motion 17 (1993) 173-184 Elsevier
173
Pulse diffraction by a curved half plane Q. Zhang and E.V. Jull, G.R. Mellema and M.J. Yedlin Departments of Electrical Engineering and Geophysics and Astronomy, Canada V6T I24
University of British Cohanbia,
Vancouver, B.C.
Received 13 April 1992, Revised 23 July 1992
Diffraction of a pulsed point source near a hard half-plane cylindrically curved near its edge is analyzed by the geometrical theory of diffraction. Source and receiver are both on the convex side of the curved surface. The solution includes first and second order edge diffracted fields: those of the edge and creeping wave and those of the discontinuity in curvature at the junction between the cylindrical segment and the plane surface. The latter are particularly strong near the reflection boundary, as shown in numerical results for zero offset between a source receiver pair. Creeping waves are calculated across their transition boundary using Fock functions and into the shadow region where they are strong enough to be observed experimentally.
1. Introduction Pulse diffraction by edges, curves and discontinuities is of interest in remote sensing applications such as radar, acoustics and seismology. The characteristic signatures and relative amplitudes of the various diffracted fields are keys to interpretation of the target. Numerical models of canonical structures involving edges and plane surfaces, such as half planes, 90” steps and inclined steps, first constructed using Kirchhoff diffraction theory [I], have since been improved [2] with the application of the uniform geometrical theory of diffraction [3]. Structures involving both edges and curved surfaces present interesting new problems. Only recently have all the tools become available to provide a uniform solution for the situation examined here : high frequency pulse diffraction by a hard half plane curved cylindrically near its edge. We determine its pulse scattering signature and the significance and detectability of creeping waves on curved surfaces and of diffraction by discontinuities in curvature. The frequency domain solution is constructed for an isotropic point source above a rigid half plane tangent to and terminated by a cylindrical arc which curves down from both source and receiver. Reflected and edge diffracted fields follow from well known formulas [3,4]. Uniform diffraction by the discontinuity in curvature is also available [5]. Creeping wave launching and attenuation coefficients have long been available [6] but results for creeping wave diffraction by an edge [7] and the resulting space wave fields [8] outside the transition region are more recent. Newer still are uniform expressions for the creeping wave fields in the shadow boundary transition region [9]. All these results are needed to complete the requirements for the solution to the problem considered here. The frequency domain solution is converted to the time domain for a pulse with suppressed low frequency components and numerical results in which the various components of the diffracted field can be identified are shown. 0165-2125/93/$06.00 0 1993 - Elsevier Science. Publishers B.V. All rights reserved
Q. Zhang et al. / Diffraction by curved plane
174
2. Diffraction by a curved half plane 2.1. Incident and rejected fields Figure la shows the arrangement considered. A rigid half plane in y = 0 x> 0 is terminated with a cylindrically curved arc of angle a0 and radius of curvature a. The structure is uniform in the z direction with source and receiver in the z =0 plane. A point source at p’, C$’ above the half plane produces an incident field -jkr ui,!?-,
(1)
r
* X
a.
b. Fig. 1. Coordinates and paths of rays reflected from (a) the plane surface and (b) the cylindrical portion of a curved half plane.
Fig. 2. Coordinates and paths of rays diffracted from a curved half plane: (a) edge diffraction; (b, c) creeping wave diffraction.
Q. Zhang et al. / D@iaction by curoedplane
175
which together with reflected and diffracted fields is observed at p, C$also above the curved half plane and at a distance r from the source. The reflected field from the plane portion of the curved half plane is
in which r, is the path length between source and receiver by reflection. If reflection is from the cylindrical portion of the half plane, as in Fig. lb, the reflected field is ([4], eq. 2.139) (3) The parameters f, g, vl, are indicated in Fig. lb and /?=sin-’
gsin my, . (
1
2.2. Edge diffac ted jields Figure 2a shows a ray path for diffraction by the edge. This field is given by Ud =
exP(-.h) ri
J
Ti
D(v, v’) exp(-jkrd)y
where the uniform diffraction function of Kouyoumjian
WY’, WI=
(5)
rd(ri+rd)
1
-ew(-j(M) %jG
cos[(yl-
and Pathak [3] is + F’[kL’a( y + If)]
v')Pl
(6)
In (6) a(yffy’)=2cos
2
lyfyl’
i
-
2
)’
and for spherical wave incidence on a straight edge the distance parameters are Li_
rird (8)
ri+rd’
L’=
(ri +
a>rd
(9)
a(l+rd)+ri(l+2rdcos6i)’ The complex Fresnel integral co F’(x) = 2j JJ; exp(jx) s J;
exp(-jr2)
dr
(10)
provides continuity of the total field across shadow and retlection boundaries. The tangent to the half plane at the edge in Fig. 2a is the transition boundary for creeping wave diffracted fields. For near grazing incidence (v/’ + 0) or observation (ry + 0), (6) alone fails to provide a continuous
176
Q. Zhang et al. / Dzfraction by curved plane
transition to the creeping wave region. A solution due to Michaeli [9] is to multiply (6) by a transition coefficient c( 0’) if w’ + 0 or c( 6) if v -PO or both if both w’ and I,Vare small. Here c( 6) = l/2 e+“‘g(
o),
(11)
where o = ms/a and m = (ka/2)“3. The angular width of the transition region is m-’ radians. Here s is the distance along the arc from the edge to the point at which the ray path to the observation point is tangent, as shown in Fig. 2b. Similarly s’ is the distance along the curved surface from the tangent point for the incident ray to the edge and CT’= ms’/a. g( cr) is the Fock function [lo] :
s a,
e-j”’
&
(12)
_co Bi’(t) -j Ai’
and Ai’
and Bi’(t) are the derivatives, with respect to the argument, of the Airy function Ai Bi(t)=e
-jx/6
Ai(t
e-j2r/3)
+ @t/6
&ct
$=/‘>,
and (13)
Tabulated values and a plot of the Fock function g(o) are available [ 111. In the computations which follow a least squares polynomial approximation to the plotted curves of g(o) in [ 1l] was used, but more efficient procedures are available (e.g. [12], p. 467). 2.3. Creeping wave d@raction If either source or receiver are in the shadow of the edge diffracted field ( y’ or I < 0 in Fig. 2a) there is instead of a directly diffracted field a creeping wave diffraction component to the total field. This contribution depends on the relative location of source and receiver. Figure 2c illustrates a situation where both source and receiver are in the shadow of the edges. Then the incident ray at t’ from the source is tangent to the surface at P’ and excites a creeping wave. This travels around the curved surface a distance s’ shedding space rays tangentially. Diffraction at the edge results in a reflected creeping wave which at s from the edge sheds a space wave tangentially at Q’ to the observation point at a distance t. For source and receiver in the shadow of the edge but outside the transition region the diffracted field is
uidc= 02
R, exp[-%(s
inwhichn=l,2,....
+ s’)]
exp[-jk(s’ + t’ + s + t)] ~J(s’+t’+s+t)
’
(14)
The creeping wave diffraction and attenuation coefficients are from Levy and Keller
t61 (15)
where a; satisfies A’i(ab) =0 and A?(x) is the derivative of the Airy function. The creeping wave edge reflection coefficient is by Idemen and Erdogan [7] R,=
jka 4v~(v~- ka) ’
(17)
Q. Zhang et al. / D@raction
by curved plane
177
with
v;=ka-aA(F)i/l exp(-j 5).
(18)
If only the source is in the shadow of the edge, the creeping wave is launched with a diffraction coefficient Dn, and diffracted at the edge with a diffraction coefficient O:, yielding for the diffracted field exp[-jk(s’+
&=DDDD,exp(-A’)
t’+ rd)]
24
’
t’rd(tf+s’+rd)
(19)
where rd is the distance from the edge to the observation point and [S] kaJ_
D, = _ expkiW8) m (SK)“”
(20)
(vh + ka cos yl)dG’
Similarly if only the observation point is in the shadow of the edge (Fig. 2b) the diffracted field is, by reciprocity from ( 19),
u%=DAD,
exp( -As)
exp[ -jk(s + t + ri)] Jtri(t+S+ri)
’
(21)
The above expressions with m =n = 1 are valid for source and receiver in the shadow of the edge but outside the creeping wave transition region. For source or receiver positions within the transition region they are the leading terms in a series expression for the creeping wave diffracted fields. The full series is obtained by replacing the subscripts with m, n = 1, 2, . . . , in (14)-(21) and summing the contributions of these creeping wave modes. This series converges to the correct solution on the shadow side of the transition boundary. Michaelis’ solution [9] using Fock function transition coefficients may also be used on the shadow side of the transition boundary. Then u
cdc=
_
e-jn’4 g(o’)g(o) 4JzG7
e-j/c(s’ + t’ + s + t)
Jtt’(s’ + t’ + s + t)
(22)
would be used in place of (14) when source and receiver are in the transition region. Figure 3 shows some numerical examples of edge diffracted transition region fields of a curved half plane for zero offset between source and receiver at a height y = 4000 m and o. = 45” in Fig. 1. These are calculated for surfaces with radii a/A = 2, 20 and 100 and the corresponding angular widths of the transition regions are approximately 31”, 14”, and 8”. The geometrical theory of diffraction values are represented by the solid curves. On the shadow side of the transition boundary at x=4000 m, 15 terms were required in the series expression to provide agreement with Michaelis’ Fock function solutions, represented by the dotted curve. Values based on the Fock function corrections are continuous across the shadow boundary of the edge diffracted fields. 2.4. Curvature discontinuity
d@action
Diffraction also occurs at a discontinuity in curvature such as x = y = 0 in Fig. la. This situation has been examined more generally by Weston [ 131 and Senior [ 141. Here the diffracted field of this second order
178
Q. Zhang et al. / Dl@raction by curved plane
2
I ud1.10-s (a)
I
l\
-... ... --...__ -.-.__
0 -4.103
0
I Ud1*lo-6
cb)
t
Fig. 3. Diffracted field amplitudes in the creeping wave transition regions of the curved half plane with a,=45” of Fig. 1 for zero offset between source and receiver at a height y=4000 m. The solid curves represent geometrical theory of diffraction solutions and the dotted curves the Fock function solutions. Radii of curvature (a) a = 2;1; (b) a = 201; (c) a = 1001.
179
Q. Zhang et al. / Diffraction by curved plane
edge is
P’
uds= &), J
(P+P’)P
(23)
w(--jkP),
where ui is the incident field and D = -4( I+ cos q5cos f$‘) exp( -jn/4) S jku&E(cos &+cos c$‘)~ ’
(24)
is the diffraction coefficient used well away from the reflection boundary (u,,, u, defmed by (30), (31) >3). In the vicinity of this boundary a useful kffraction coefficient is ([5], p. 188)
Ds= (Dh- 0,) D
h
c
=
F(%) sin 4
-2 ew(--jd4)
x
1 +cos(~--‘)
-ew(-h/4) &it
=
+ZE--
D
2(l+cos~cos4’),
i
@z
exp(jvf)
v
v
<3
h,c
(25)
,
.
i
J Sgn(COS4 ’ + COS#)[2 sin f$+ 3 cot I$(cos 4 ‘+ cos $)] exp(j&)
(26)
3cot lp )
j sgn(cos $I’ + cos #)JZG
sin 4 +
cos C$(1 + (3a/2L) sin C$)(cos 4 ’ + cos Cp) sin#‘+sinq%(l+(a/L)sin#)
cos 4 (1 + (3a/2L) sin 4) F( 0,) Jsin#‘+sind(l+(a/L)sin#)-sin$‘+sin#(l+(a/L)sin#)
in which the Fresnel integral is defined by 03 F(x) = exp( -jf2) dt 1X
1 (27)
(28)
and the distance parameters are given by
(2%
L=-@-,
(30) ka
v,= J
1
lcos C#J ’ + cos (I31 &in qS’+sin 4(1 +(a/L)
sin 4)’
(31)
2.5. The totalJieId The total field above the half plane in the region illuminated by the edge diffracted field is U=Ui+Ur+Ud+UdS.
(32)
If either source or receiver or both are in the shadow of the edge the creeping wave diffracted field udc or ucd or u*‘, respectively, replaces the edge diffracted field ud, in (32).
Q. Zhang et al. / DifSraction by curved plane
180
3. Tie
domain solution
Pulse diffraction by the curved half plane can easily be constructed by Fourier transforming the frequency domain solution (32). The time behavior of the pulse used is
sintwdf- to)1e-acr-ro32 f(4
J_\
=
(33)
’
which has suppressed low frequency components typical of seismic pulses. The parameters used were wI = 20& rad/s, 01= 2.2 x lo3 and r = to is the time of the pulse at the source. The frequency spectrum of the pulse; i.e. the Fourier transform F(w) of (33), then has its maximum at about 35 Hz. The time domain solution is u3 u(w)F(w) 2”” dw, g(t) = (34) s -Cc where u(w) is the total field given by (32) and w is the angular frequency.
4. Numerical results Figures 4a and 4b show computed results for both source and receiver together at a height of y = 4000 m above the plane surface of the half plane with a cylindrical arc of angle a0 = 45” and radius of curvature
a.
. r-
b.
+-
;
t
IIlllIII 10 38 36 34 32
Fig. 4. Calculated pulses reflected and diffracted by the curved half plane of Fig. 1 with a = 800 m, a0 = 45”, p = p’, $= 4 ’ and source receiver heights y = 4000 m. The wave velocity is 4000 m/s. (a) Reflected and diffracted fields;(b) Diffracted fields only.
Q. Zhang et al. / Dtflraction by curved plane
181
800 m. The horizontal scale is distance and separation betwen the traces is 346 m. Traces 1 and 41 correspond to x = *6928 m or 4 = 4 ’ = 30” and 150” respectively. The vertical scale is relative time for a pulse velocity of 4000 m/s, so the large amplitude horizontal event for x > 0, which represents reflection from the plane portion of the curved half plane, occurs two seconds after the pulse leaves the source. For XC 0 (traces 22-41) edge diffraction and curved surface reflection are merged in the large amplitude hyperbolic event (traces 22-34) but events on traces 35-41 are due to diffraction at the edge of the curved half plane only. Events on the other (X > 0) side of this hyperbola are weaker because they represent only back scattered diffraction, which is less in this direction, and because source and receiver are further from the diffracting edge. For traces l-10 source and receiver are in the shadow of the diffracting edge and hyperbolic events on these traces, shown in expanded form in Fig. 5, are due to creeping wave diffraction. An earlier hyperbolic event centered at x=0 is due to diffraction at the discontinuity in curvature at x=y= 0 in Figs. 1 and 2. This is clearly seen in Fig. 4b where the reflected fields have been removed. The amplitude of curvature discontinuity diffraction is surprisingly large at its center on trace 2 1. This situation seems to be analogous to that on a half plane shadow boundary where diffracted fields and incident fields are of comparable magnitude. It is evident in Fig. 4b that reception of this diffraction decreases very rapidly as source and receiver move away from directly over the curvature discontinuity. Figure 6 shows frequency domain plots of the diffracted fields only of half planes with curve edges of radii a=4;1 and 4Od for the arrangement of Fig. 4. The transition region fields were not calculated here and so at the transition boundary near C$= 4 ’ = 45” there is a pattern discontinuity. With source and receiver a=
‘Used
1 1
0.75bJ%Il 0.5.
0.25-
0.25
7
i
Fig. 5. Edge diffracted and creeping wave pulses in the creeping wave transition region of Fig. 4.
Fig. 6. The diffracted field of the curved half plane of Fig. 1 with (a) a =4A; (b) a=401, (x0=45”, and zero offset between source and receiver.
Q. Zhang et al. / Diffraction by curved plane
182
directly over the curvature discontinuity (4 = 4 ’ = 90”) the diffracted field is a maximum and half the total field amplitude. Around this peak interference between edge and discontinuity diffraction produces an oscillating pattern for a = 41 with edge diffraction being the dominant effect away from the peak. It increases to a maximum as the source and receiver pass over the edge near C#J = 4 ’ = 135”. For the curved half plane with radius of curvature a = 4il the curvature discontinuity diffraction seems to be much more localized.
5. Experimental results The plots in Fig. 7 are from acoustic measurements, details of which are given in a thesis [ 141. For each trace microphone and loudspeaker are together 57 cm above an aluminum half plane which is flat for
2 -
8 -
0
50
150
Ice
200
2M
x (a)
Fig. 7. Mapping of acoustic pulse diffraction by a cylindrically curved half plane. The horizontal position of the source and receiver together at y = 57 cm above the flat portion of the half plane is indicated by the x scale. The vertical scale is the time of the received pulse. (b) is a median filtered version of (a).
183
Q. Zhang et al. / Diffraction by curved plane
80
120
100
140
160
x (cm)
Fig. 8. Enlargement of Fig. 7b for 70 c x < 160 cm, 4 < r < 9 s. The diagonal event increasing with x represents creeping wave diffraction for x4 117 cm and edge diffraction for x> 117 cm.
x = 147 cm and cylindrically curved over an arc of 45” for 147 < X-C190 cm, with a radius of curvature of about 50 cm. The horizontal events in Fig. 7a are the initial pulse, reflection from the flat portion of the half plane, echoes of this reflection and reflection from the floor. Figure 7b is a median filtered image of the same measurements and emphasizes the non horizontal events, the strongest of which are reflection from the curved surface and its echo. Also evident as a hyperbolic event is diffraction by the edge of the curved half plane. The shadow boundary of the edge at x = 117 cm is indicated by the vertical arrows. This hyperbolic event continues into x < 117 cm, as is clearly evident in Fig. 8, an expanded portion of Fig. 7b. This provides experimental evidence of creeping wave diffraction well into the shadow region.
6. Concluding remarks An interesting feature of these results are the creeping wave diffracted fields. By using their series expansion they can be accurately calculated from the shadow boundary onwards. The edge diffracted field expressions fail near the shadow boundary however and must be replced in the transition region with the Fock function solution. This kind of situation arises in a variety of applications, for example in calculating the rear patterns of paraboloidal reflector, antennas, and efficient methods of calculating these transition region fields are important. Another notable feature of this investigation is the remarkably strong diffraction from the junction of the plane and curved surfaces of the half plane. This effect is local for zero offset between source and receiver as illustrated in Fig. 4b, but it was believed strong enough to observe. In the measurements made with audio frequency acoustic pulses of Fig. 7a it has not been possible by signal processing to separate
Q. Zhang et al. / Dtflraction by curved plane
184
reflection from the plane surface from diffraction by the discontinuity as in Fig. 4b. However it is possible to observe the creeping wave diffraction pulses well into the shadow region with the experimental arrangement, confirming the predictions of Figs. 4 and 5.
References [ 1] F.J. Hilterman, “Three dimensional seismic modelling”, Geophysics 35, 1020-1037 (1970). [2] Q. Zheng, E.V. Jull and M.J. Yedlin, “Acoustic pulse diffraction by step discontinuities on a plane”, Geophysics 55, 749-756 (1990). [3] 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). [4] J.J. Bowman, T.B.A. Senior and P.L.E. Uslenghi (Eds.), Electromagnetic and Acoustic Scattering by Simple Shapes, NorthHolland, Amsterdam (1969). [5] G.L. James, Geometrical Theory of Drrraction for Electromagnetic Waves, Peter Peregrinus, London (1976). [6] B.R. Levy and J.B. Keller, “Diffraction by a smooth object”, Commun. Pure AppLMath. 12, 159-209 (1959). [7] M. Idemen and E. Erdogan, “Diffraction of the creeping waves generated on a perfectly conducting spherical scatterer by a ring source”, IEEE Trans. on Antennas and Propagat. AP-31, 776-784 (1983). [8] A.H. Serbest, A. Buyukaksoy and G. Uzgoren, “Diffraction of high frequency electromagnetic waves by curved strips”, IEEE Trans. Antennas and Propagat. AP-37, [9] A. Michaeli, “Transition functions for Trans. Antennas and Propagat. AP-37, [IO] V.A. Fock, Electromagnetic Dtjraction
592-600 (1989).
high-frequency diffraction by a curved perfectly conducting wedge, Parts I-III”, IEEE 1073-1092 (1989). and Propagation Problems, Pergamon, New York (1965).
[11] N.A. Logan, General research in diffraction theory, Vol. 1, 2, LMSD-288087, 288088, Missiles and Space Div., Lockheed Aircraft Corp., 1959. [12] D.A. McNamara, C.W.I. Pistorius and J.A.G. Malherbe., Introduction to the Uniform Geometrical Theory of Diffraction, Artech, Boston (1990). [ 131 V.H. Weston, “The effect of a discontinuity in curvature in high-frequency scattering”, IRE Trans. on Antennas and Propagat. Ap-IO, 775-780 (1962). 1141 T.B.A. Senior, “The diffraction matrix for a discontinuity in curvature”, IEEE Trans. on Antennas and Propagat. AP-20, 326333 (1972). [ 151 G.R. Mellema, An acoustic scatter-mapping imaging system, M.A. SC. thesis, Dept. of Electrical Engineering, University of
British Columbia, February 1990.
173
Pulse diffraction by a curved half plane Q. Zhang and E.V. Jull, G.R. Mellema and M.J. Yedlin Departments of Electrical Engineering and Geophysics and Astronomy, Canada V6T I24
University of British Cohanbia,
Vancouver, B.C.
Received 13 April 1992, Revised 23 July 1992
Diffraction of a pulsed point source near a hard half-plane cylindrically curved near its edge is analyzed by the geometrical theory of diffraction. Source and receiver are both on the convex side of the curved surface. The solution includes first and second order edge diffracted fields: those of the edge and creeping wave and those of the discontinuity in curvature at the junction between the cylindrical segment and the plane surface. The latter are particularly strong near the reflection boundary, as shown in numerical results for zero offset between a source receiver pair. Creeping waves are calculated across their transition boundary using Fock functions and into the shadow region where they are strong enough to be observed experimentally.
1. Introduction Pulse diffraction by edges, curves and discontinuities is of interest in remote sensing applications such as radar, acoustics and seismology. The characteristic signatures and relative amplitudes of the various diffracted fields are keys to interpretation of the target. Numerical models of canonical structures involving edges and plane surfaces, such as half planes, 90” steps and inclined steps, first constructed using Kirchhoff diffraction theory [I], have since been improved [2] with the application of the uniform geometrical theory of diffraction [3]. Structures involving both edges and curved surfaces present interesting new problems. Only recently have all the tools become available to provide a uniform solution for the situation examined here : high frequency pulse diffraction by a hard half plane curved cylindrically near its edge. We determine its pulse scattering signature and the significance and detectability of creeping waves on curved surfaces and of diffraction by discontinuities in curvature. The frequency domain solution is constructed for an isotropic point source above a rigid half plane tangent to and terminated by a cylindrical arc which curves down from both source and receiver. Reflected and edge diffracted fields follow from well known formulas [3,4]. Uniform diffraction by the discontinuity in curvature is also available [5]. Creeping wave launching and attenuation coefficients have long been available [6] but results for creeping wave diffraction by an edge [7] and the resulting space wave fields [8] outside the transition region are more recent. Newer still are uniform expressions for the creeping wave fields in the shadow boundary transition region [9]. All these results are needed to complete the requirements for the solution to the problem considered here. The frequency domain solution is converted to the time domain for a pulse with suppressed low frequency components and numerical results in which the various components of the diffracted field can be identified are shown. 0165-2125/93/$06.00 0 1993 - Elsevier Science. Publishers B.V. All rights reserved
Q. Zhang et al. / Diffraction by curved plane
174
2. Diffraction by a curved half plane 2.1. Incident and rejected fields Figure la shows the arrangement considered. A rigid half plane in y = 0 x> 0 is terminated with a cylindrically curved arc of angle a0 and radius of curvature a. The structure is uniform in the z direction with source and receiver in the z =0 plane. A point source at p’, C$’ above the half plane produces an incident field -jkr ui,!?-,
(1)
r
* X
a.
b. Fig. 1. Coordinates and paths of rays reflected from (a) the plane surface and (b) the cylindrical portion of a curved half plane.
Fig. 2. Coordinates and paths of rays diffracted from a curved half plane: (a) edge diffraction; (b, c) creeping wave diffraction.
Q. Zhang et al. / D@iaction by curoedplane
175
which together with reflected and diffracted fields is observed at p, C$also above the curved half plane and at a distance r from the source. The reflected field from the plane portion of the curved half plane is
in which r, is the path length between source and receiver by reflection. If reflection is from the cylindrical portion of the half plane, as in Fig. lb, the reflected field is ([4], eq. 2.139) (3) The parameters f, g, vl, are indicated in Fig. lb and /?=sin-’
gsin my, . (
1
2.2. Edge diffac ted jields Figure 2a shows a ray path for diffraction by the edge. This field is given by Ud =
exP(-.h) ri
J
Ti
D(v, v’) exp(-jkrd)y
where the uniform diffraction function of Kouyoumjian
WY’, WI=
(5)
rd(ri+rd)
1
-ew(-j(M) %jG
cos[(yl-
and Pathak [3] is + F’[kL’a( y + If)]
v')Pl
(6)
In (6) a(yffy’)=2cos
2
lyfyl’
i
-
2
)’
and for spherical wave incidence on a straight edge the distance parameters are Li_
rird (8)
ri+rd’
L’=
(ri +
a>rd
(9)
a(l+rd)+ri(l+2rdcos6i)’ The complex Fresnel integral co F’(x) = 2j JJ; exp(jx) s J;
exp(-jr2)
dr
(10)
provides continuity of the total field across shadow and retlection boundaries. The tangent to the half plane at the edge in Fig. 2a is the transition boundary for creeping wave diffracted fields. For near grazing incidence (v/’ + 0) or observation (ry + 0), (6) alone fails to provide a continuous
176
Q. Zhang et al. / Dzfraction by curved plane
transition to the creeping wave region. A solution due to Michaeli [9] is to multiply (6) by a transition coefficient c( 0’) if w’ + 0 or c( 6) if v -PO or both if both w’ and I,Vare small. Here c( 6) = l/2 e+“‘g(
o),
(11)
where o = ms/a and m = (ka/2)“3. The angular width of the transition region is m-’ radians. Here s is the distance along the arc from the edge to the point at which the ray path to the observation point is tangent, as shown in Fig. 2b. Similarly s’ is the distance along the curved surface from the tangent point for the incident ray to the edge and CT’= ms’/a. g( cr) is the Fock function [lo] :
s a,
e-j”’
&
(12)
_co Bi’(t) -j Ai’
and Ai’
and Bi’(t) are the derivatives, with respect to the argument, of the Airy function Ai Bi(t)=e
-jx/6
Ai(t
e-j2r/3)
+ @t/6
&ct
$=/‘>,
and (13)
Tabulated values and a plot of the Fock function g(o) are available [ 111. In the computations which follow a least squares polynomial approximation to the plotted curves of g(o) in [ 1l] was used, but more efficient procedures are available (e.g. [12], p. 467). 2.3. Creeping wave d@raction If either source or receiver are in the shadow of the edge diffracted field ( y’ or I < 0 in Fig. 2a) there is instead of a directly diffracted field a creeping wave diffraction component to the total field. This contribution depends on the relative location of source and receiver. Figure 2c illustrates a situation where both source and receiver are in the shadow of the edges. Then the incident ray at t’ from the source is tangent to the surface at P’ and excites a creeping wave. This travels around the curved surface a distance s’ shedding space rays tangentially. Diffraction at the edge results in a reflected creeping wave which at s from the edge sheds a space wave tangentially at Q’ to the observation point at a distance t. For source and receiver in the shadow of the edge but outside the transition region the diffracted field is
uidc= 02
R, exp[-%(s
inwhichn=l,2,....
+ s’)]
exp[-jk(s’ + t’ + s + t)] ~J(s’+t’+s+t)
’
(14)
The creeping wave diffraction and attenuation coefficients are from Levy and Keller
t61 (15)
where a; satisfies A’i(ab) =0 and A?(x) is the derivative of the Airy function. The creeping wave edge reflection coefficient is by Idemen and Erdogan [7] R,=
jka 4v~(v~- ka) ’
(17)
Q. Zhang et al. / D@raction
by curved plane
177
with
v;=ka-aA(F)i/l exp(-j 5).
(18)
If only the source is in the shadow of the edge, the creeping wave is launched with a diffraction coefficient Dn, and diffracted at the edge with a diffraction coefficient O:, yielding for the diffracted field exp[-jk(s’+
&=DDDD,exp(-A’)
t’+ rd)]
24
’
t’rd(tf+s’+rd)
(19)
where rd is the distance from the edge to the observation point and [S] kaJ_
D, = _ expkiW8) m (SK)“”
(20)
(vh + ka cos yl)dG’
Similarly if only the observation point is in the shadow of the edge (Fig. 2b) the diffracted field is, by reciprocity from ( 19),
u%=DAD,
exp( -As)
exp[ -jk(s + t + ri)] Jtri(t+S+ri)
’
(21)
The above expressions with m =n = 1 are valid for source and receiver in the shadow of the edge but outside the creeping wave transition region. For source or receiver positions within the transition region they are the leading terms in a series expression for the creeping wave diffracted fields. The full series is obtained by replacing the subscripts with m, n = 1, 2, . . . , in (14)-(21) and summing the contributions of these creeping wave modes. This series converges to the correct solution on the shadow side of the transition boundary. Michaelis’ solution [9] using Fock function transition coefficients may also be used on the shadow side of the transition boundary. Then u
cdc=
_
e-jn’4 g(o’)g(o) 4JzG7
e-j/c(s’ + t’ + s + t)
Jtt’(s’ + t’ + s + t)
(22)
would be used in place of (14) when source and receiver are in the transition region. Figure 3 shows some numerical examples of edge diffracted transition region fields of a curved half plane for zero offset between source and receiver at a height y = 4000 m and o. = 45” in Fig. 1. These are calculated for surfaces with radii a/A = 2, 20 and 100 and the corresponding angular widths of the transition regions are approximately 31”, 14”, and 8”. The geometrical theory of diffraction values are represented by the solid curves. On the shadow side of the transition boundary at x=4000 m, 15 terms were required in the series expression to provide agreement with Michaelis’ Fock function solutions, represented by the dotted curve. Values based on the Fock function corrections are continuous across the shadow boundary of the edge diffracted fields. 2.4. Curvature discontinuity
d@action
Diffraction also occurs at a discontinuity in curvature such as x = y = 0 in Fig. la. This situation has been examined more generally by Weston [ 131 and Senior [ 141. Here the diffracted field of this second order
178
Q. Zhang et al. / Dl@raction by curved plane
2
I ud1.10-s (a)
I
l\
-... ... --...__ -.-.__
0 -4.103
0
I Ud1*lo-6
cb)
t
Fig. 3. Diffracted field amplitudes in the creeping wave transition regions of the curved half plane with a,=45” of Fig. 1 for zero offset between source and receiver at a height y=4000 m. The solid curves represent geometrical theory of diffraction solutions and the dotted curves the Fock function solutions. Radii of curvature (a) a = 2;1; (b) a = 201; (c) a = 1001.
179
Q. Zhang et al. / Diffraction by curved plane
edge is
P’
uds= &), J
(P+P’)P
(23)
w(--jkP),
where ui is the incident field and D = -4( I+ cos q5cos f$‘) exp( -jn/4) S jku&E(cos &+cos c$‘)~ ’
(24)
is the diffraction coefficient used well away from the reflection boundary (u,,, u, defmed by (30), (31) >3). In the vicinity of this boundary a useful kffraction coefficient is ([5], p. 188)
Ds= (Dh- 0,) D
h
c
=
F(%) sin 4
-2 ew(--jd4)
x
1 +cos(~--‘)
-ew(-h/4) &it
=
+ZE--
D
2(l+cos~cos4’),
i
@z
exp(jvf)
v
v
<3
h,c
(25)
,
.
i
J Sgn(COS4 ’ + COS#)[2 sin f$+ 3 cot I$(cos 4 ‘+ cos $)] exp(j&)
(26)
3cot lp )
j sgn(cos $I’ + cos #)JZG
sin 4 +
cos C$(1 + (3a/2L) sin C$)(cos 4 ’ + cos Cp) sin#‘+sinq%(l+(a/L)sin#)
cos 4 (1 + (3a/2L) sin 4) F( 0,) Jsin#‘+sind(l+(a/L)sin#)-sin$‘+sin#(l+(a/L)sin#)
in which the Fresnel integral is defined by 03 F(x) = exp( -jf2) dt 1X
1 (27)
(28)
and the distance parameters are given by
(2%
L=-@-,
(30) ka
v,= J
1
lcos C#J ’ + cos (I31 &in qS’+sin 4(1 +(a/L)
sin 4)’
(31)
2.5. The totalJieId The total field above the half plane in the region illuminated by the edge diffracted field is U=Ui+Ur+Ud+UdS.
(32)
If either source or receiver or both are in the shadow of the edge the creeping wave diffracted field udc or ucd or u*‘, respectively, replaces the edge diffracted field ud, in (32).
Q. Zhang et al. / DifSraction by curved plane
180
3. Tie
domain solution
Pulse diffraction by the curved half plane can easily be constructed by Fourier transforming the frequency domain solution (32). The time behavior of the pulse used is
sintwdf- to)1e-acr-ro32 f(4
J_\
=
(33)
’
which has suppressed low frequency components typical of seismic pulses. The parameters used were wI = 20& rad/s, 01= 2.2 x lo3 and r = to is the time of the pulse at the source. The frequency spectrum of the pulse; i.e. the Fourier transform F(w) of (33), then has its maximum at about 35 Hz. The time domain solution is u3 u(w)F(w) 2”” dw, g(t) = (34) s -Cc where u(w) is the total field given by (32) and w is the angular frequency.
4. Numerical results Figures 4a and 4b show computed results for both source and receiver together at a height of y = 4000 m above the plane surface of the half plane with a cylindrical arc of angle a0 = 45” and radius of curvature
a.
. r-
b.
+-
;
t
IIlllIII 10 38 36 34 32
Fig. 4. Calculated pulses reflected and diffracted by the curved half plane of Fig. 1 with a = 800 m, a0 = 45”, p = p’, $= 4 ’ and source receiver heights y = 4000 m. The wave velocity is 4000 m/s. (a) Reflected and diffracted fields;(b) Diffracted fields only.
Q. Zhang et al. / Dtflraction by curved plane
181
800 m. The horizontal scale is distance and separation betwen the traces is 346 m. Traces 1 and 41 correspond to x = *6928 m or 4 = 4 ’ = 30” and 150” respectively. The vertical scale is relative time for a pulse velocity of 4000 m/s, so the large amplitude horizontal event for x > 0, which represents reflection from the plane portion of the curved half plane, occurs two seconds after the pulse leaves the source. For XC 0 (traces 22-41) edge diffraction and curved surface reflection are merged in the large amplitude hyperbolic event (traces 22-34) but events on traces 35-41 are due to diffraction at the edge of the curved half plane only. Events on the other (X > 0) side of this hyperbola are weaker because they represent only back scattered diffraction, which is less in this direction, and because source and receiver are further from the diffracting edge. For traces l-10 source and receiver are in the shadow of the diffracting edge and hyperbolic events on these traces, shown in expanded form in Fig. 5, are due to creeping wave diffraction. An earlier hyperbolic event centered at x=0 is due to diffraction at the discontinuity in curvature at x=y= 0 in Figs. 1 and 2. This is clearly seen in Fig. 4b where the reflected fields have been removed. The amplitude of curvature discontinuity diffraction is surprisingly large at its center on trace 2 1. This situation seems to be analogous to that on a half plane shadow boundary where diffracted fields and incident fields are of comparable magnitude. It is evident in Fig. 4b that reception of this diffraction decreases very rapidly as source and receiver move away from directly over the curvature discontinuity. Figure 6 shows frequency domain plots of the diffracted fields only of half planes with curve edges of radii a=4;1 and 4Od for the arrangement of Fig. 4. The transition region fields were not calculated here and so at the transition boundary near C$= 4 ’ = 45” there is a pattern discontinuity. With source and receiver a=
‘Used
1 1
0.75bJ%Il 0.5.
0.25-
0.25
7
i
Fig. 5. Edge diffracted and creeping wave pulses in the creeping wave transition region of Fig. 4.
Fig. 6. The diffracted field of the curved half plane of Fig. 1 with (a) a =4A; (b) a=401, (x0=45”, and zero offset between source and receiver.
Q. Zhang et al. / Diffraction by curved plane
182
directly over the curvature discontinuity (4 = 4 ’ = 90”) the diffracted field is a maximum and half the total field amplitude. Around this peak interference between edge and discontinuity diffraction produces an oscillating pattern for a = 41 with edge diffraction being the dominant effect away from the peak. It increases to a maximum as the source and receiver pass over the edge near C#J = 4 ’ = 135”. For the curved half plane with radius of curvature a = 4il the curvature discontinuity diffraction seems to be much more localized.
5. Experimental results The plots in Fig. 7 are from acoustic measurements, details of which are given in a thesis [ 141. For each trace microphone and loudspeaker are together 57 cm above an aluminum half plane which is flat for
2 -
8 -
0
50
150
Ice
200
2M
x (a)
Fig. 7. Mapping of acoustic pulse diffraction by a cylindrically curved half plane. The horizontal position of the source and receiver together at y = 57 cm above the flat portion of the half plane is indicated by the x scale. The vertical scale is the time of the received pulse. (b) is a median filtered version of (a).
183
Q. Zhang et al. / Diffraction by curved plane
80
120
100
140
160
x (cm)
Fig. 8. Enlargement of Fig. 7b for 70 c x < 160 cm, 4 < r < 9 s. The diagonal event increasing with x represents creeping wave diffraction for x4 117 cm and edge diffraction for x> 117 cm.
x = 147 cm and cylindrically curved over an arc of 45” for 147 < X-C190 cm, with a radius of curvature of about 50 cm. The horizontal events in Fig. 7a are the initial pulse, reflection from the flat portion of the half plane, echoes of this reflection and reflection from the floor. Figure 7b is a median filtered image of the same measurements and emphasizes the non horizontal events, the strongest of which are reflection from the curved surface and its echo. Also evident as a hyperbolic event is diffraction by the edge of the curved half plane. The shadow boundary of the edge at x = 117 cm is indicated by the vertical arrows. This hyperbolic event continues into x < 117 cm, as is clearly evident in Fig. 8, an expanded portion of Fig. 7b. This provides experimental evidence of creeping wave diffraction well into the shadow region.
6. Concluding remarks An interesting feature of these results are the creeping wave diffracted fields. By using their series expansion they can be accurately calculated from the shadow boundary onwards. The edge diffracted field expressions fail near the shadow boundary however and must be replced in the transition region with the Fock function solution. This kind of situation arises in a variety of applications, for example in calculating the rear patterns of paraboloidal reflector, antennas, and efficient methods of calculating these transition region fields are important. Another notable feature of this investigation is the remarkably strong diffraction from the junction of the plane and curved surfaces of the half plane. This effect is local for zero offset between source and receiver as illustrated in Fig. 4b, but it was believed strong enough to observe. In the measurements made with audio frequency acoustic pulses of Fig. 7a it has not been possible by signal processing to separate
Q. Zhang et al. / Dtflraction by curved plane
184
reflection from the plane surface from diffraction by the discontinuity as in Fig. 4b. However it is possible to observe the creeping wave diffraction pulses well into the shadow region with the experimental arrangement, confirming the predictions of Figs. 4 and 5.
References [ 1] F.J. Hilterman, “Three dimensional seismic modelling”, Geophysics 35, 1020-1037 (1970). [2] Q. Zheng, E.V. Jull and M.J. Yedlin, “Acoustic pulse diffraction by step discontinuities on a plane”, Geophysics 55, 749-756 (1990). [3] 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). [4] J.J. Bowman, T.B.A. Senior and P.L.E. Uslenghi (Eds.), Electromagnetic and Acoustic Scattering by Simple Shapes, NorthHolland, Amsterdam (1969). [5] G.L. James, Geometrical Theory of Drrraction for Electromagnetic Waves, Peter Peregrinus, London (1976). [6] B.R. Levy and J.B. Keller, “Diffraction by a smooth object”, Commun. Pure AppLMath. 12, 159-209 (1959). [7] M. Idemen and E. Erdogan, “Diffraction of the creeping waves generated on a perfectly conducting spherical scatterer by a ring source”, IEEE Trans. on Antennas and Propagat. AP-31, 776-784 (1983). [8] A.H. Serbest, A. Buyukaksoy and G. Uzgoren, “Diffraction of high frequency electromagnetic waves by curved strips”, IEEE Trans. Antennas and Propagat. AP-37, [9] A. Michaeli, “Transition functions for Trans. Antennas and Propagat. AP-37, [IO] V.A. Fock, Electromagnetic Dtjraction
592-600 (1989).
high-frequency diffraction by a curved perfectly conducting wedge, Parts I-III”, IEEE 1073-1092 (1989). and Propagation Problems, Pergamon, New York (1965).
[11] N.A. Logan, General research in diffraction theory, Vol. 1, 2, LMSD-288087, 288088, Missiles and Space Div., Lockheed Aircraft Corp., 1959. [12] D.A. McNamara, C.W.I. Pistorius and J.A.G. Malherbe., Introduction to the Uniform Geometrical Theory of Diffraction, Artech, Boston (1990). [ 131 V.H. Weston, “The effect of a discontinuity in curvature in high-frequency scattering”, IRE Trans. on Antennas and Propagat. Ap-IO, 775-780 (1962). 1141 T.B.A. Senior, “The diffraction matrix for a discontinuity in curvature”, IEEE Trans. on Antennas and Propagat. AP-20, 326333 (1972). [ 151 G.R. Mellema, An acoustic scatter-mapping imaging system, M.A. SC. thesis, Dept. of Electrical Engineering, University of
British Columbia, February 1990.