The Uniform geometrical Theory of Diffraction for elastodynamics: Plane wave scattering from a half-plane

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1 The Uniform geometrical Theory of Diffraction for elastodynamics: Plane wave scattering from a half-plane Audrey Kamta Djakou a) and Michel Darmon Department of Imaging and Simulation for Nondestructive Testing, CEA, LIST, Gif-sur-Yvette, France Larissa Fradkin Sound Mathematics Ltd., Camridge CB4 AS, United Kingdom Catherine Potel ) Laoratoire d Acoustique de l Universite du Maine (LAUM), UMR CNRS 6613, 7085 Le Mans cedex 9, France (Received 1 April 015; revised 5 Septemer 015; accepted 0 Octoer 015; pulished online 4 Novemer 015) 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 otained 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 inapplicale 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 drawack, the Uniform geometrical Theory of Diffraction (UTD) has een developed previously in electromagnetism, ased on a ray theory, which is particularly easy to implement. In this paper, UTD is developed for the canonical elastodynamic prolem of the scattering of a plane wave y a half-plane. UTD is then compared to another uniform extension of GTD, the Uniform Asymptotic Theory (UAT) of diffraction, ased on a more cumersome ray theory. A good agreement etween the two methods is otained in the far field. VC 015 Acoustical Society of America. [ [GH] Pages: I. INTRODUCTION The scattering of elastic waves from an ostacle is of great interest in ultrasonic non-destructive evaluation (NDE), the main scattering phenomena eing specular reflection and diffraction. When oth the wavefront of an incident wave and the oundary of the scattering oject can e modeled as locally plane, the scattered field is usually simulated using the Kirchhoff Approximation (KA). This represents the field as an integral over the scattering surface. 1 KA gives a reliale and continuous description of specular reflections and fictitious fields compensating the incident field in the ostacle shadow, which together form the socalled Geometrico-Elastodynamic (GE) field. However, the diffracted fields are not always well descried.,3 In the asence of interaction with other waves, the est description of diffracted fields is otained via the Geometrical Theory of Diffraction (GTD). 4 This postulates existence of rays diffracted from the structure irregularities such as edge or tip, additional to the incident and reflected rays and also gives a recipe for calculating the amplitudes carried y these rays. In elastodynamics the underlying canonical prolem is the scattering of a plane wave y a stress-free half-plane. Its a) Also at: Laoratoire d Acoustique de l Universite du Maine (LAUM), UMR CNRS 6613, 7085 Le Mans cedex 9, France. Electronic mail: audrey.kamta-djakou@cea.fr ) Also at: Federation Acoustique du Nord Ouest (FANO), FR CNRS 3110, France. solution is represented in the form of the Sommerfeld integral 5,6 and the classical GTD solution is otained using the steepest descent method. The resulting contriution of the stationary phase points represents the diffracted waves in the far field using the so-called diffraction coefficients, otherwise known as the directivity patterns. It is well known that the method of steepest descent is inapplicale when the stationary phase point of the integrand coalesces with its pole, explaining why the classical GTD fails in the zones where edge diffracted waves interfere with incident or reflected waves. For this reason the classical GTD solution is said to e non-uniform. Several uniform extensions of GTD have een developed in electromagnetism, such as the Uniform Asymptotic Theory (UAT) 7 9 ased on the Van Der Waerden method, 10 the Uniform Theory of Diffraction (UTD) 11,1 ased on the Pauli Clemmow method 13 and the Physical Theory of Diffraction (PTD), 1,,14 which comines GTD and Kirchhoff Approximation (KA) and applies to aritrary large locally plane scatterers. The elastodynamic versions of UAT (Ref. 6) and PTD (Ref. 3) have een reported efore; in the leading order when the scatterer is a half-plane PTD is identical to UAT. 3 Due to the integral nature of the method, PTD appears to e computationally expensive, especially when the scatterers are large. A procedure similar to PTD has een confusingly called y its author the Uniform Theory of Diffraction. 15 It relies on another integral method, which is difficult to implement. Whenever applicale, ray methods are often preferred 37 J. Acoust. Soc. Am. 138 (5), Novemer /015/138(5)/37/10/$30.00 VC 015 Acoustical Society of America

2 FIG. 1. A plane wave of propagation vector k a incident on a semi-infinite stress free crack. Thick lack arrow, direction of the incident wave; thick gray arrow, direction of the diffracted wave (k ). to integral methods since highly optimized and therefore extremely fast ray tracing algorithms have ecome availale. Both UAT and UTD are ray methods, producing continuous approximations to total fields. However, UAT involves artificial extension of the scattering surface and fictitious reflected rays (see Fig. 3 and the end of Sec. III), while UTD does not. For this reason, UTD is more used than UAT in acoustics 16 and electromagnetism. 17 The paper is structured as follows: in Sec. II, the exact solution of the canonical prolem of a plane wave scattering from a stress-free half-plane is presented in the form of the Sommerfeld integral and its non-uniform asymptotics are given in the form convenient for use in the UTD recipe. In Sec. III, uniform asymptotics of the integral are derived using the Pauli Clemmow procedure. As a result, the elastodynamic UTD is otained, with the total field involving no fictitious rays and with diffraction coefficients, which are a simple modification of the GTD diffraction coefficients. Comparison of UTD and UAT is carried out in Sec. IV. II. SCATTERING OF A PLANE ELASTIC WAVE BY A HALF-PLANE CRACK: NON-UNIFORM ASYMPTOTICS In the following, the symols a and are used to denote the wave type, i.e., a; ¼ L, TV, or TH (longitudinal, transverse vertical, or transverse horizontal, respectively). In general, a is used for the incident wave and for reflected and diffracted waves. Scalar quantities are generally laelled y taking a and as suscripts, while a is used as superscripts for vectors related to the incident wave and as suscripts for vectors related to scattered waves. The geometry of the prolem is presented in Fig. 1, using the Cartesian system ased on an orthonormal asis fe 1 ; e ; e 3 g and the origin O at the crack edge. The crack is emedded in an elastic homogeneous and isotropic space and its face lies in the half-plane fx ¼ 0; x 1 0g. The edge coincides with the x 3 axis and is irradiated y a plane wave u a ðxþ ¼A d a e ið xtþka xþ ; (1) where A is the wave displacement amplitude; d a is its polarization; k a is its wave vector defined thanks to the angles X a and h a (descried in Fig. 1), whose magnitude k a ¼ x=c a, with x, the circular frequency and c a, the speed of the corresponding mode; t is time; and x is the position vector, which is expressed in the Cartesian coordinates as ðx 1 ; x ; x 3 Þ and in the cylindrical coordinates as ðr; h; x 3 Þ. The exponential factor expð ixtþ is implied ut omitted everywhere. The exact scattered field generated when a plane elastic wave irradiates a stress-free half-plane crack is expressed y Achenach 6 as an angular spectral decomposition. The scattered field at the oservation point u a ðxþ can e rewritten as the Sommerfeld integral ð u a ðxþ ¼eik a cos X a x 3 f ½ q cos k; sgnðsin hþš e if cos ðk hþ C d ½ q cos k; sgnðsin hþš dk; () where f ½ q cos k; sgnðsin hþš is f q cos k; sgnðsin hþ ¼ i q j p sin k g q cos k; sgnðsin hþ ; (3) q a cos h a q cos k with the suscript denoting the scattered wave mode, q ¼ k sin X ; j ¼ c L =c the dimensionless slowness of the scattered wave, and g ½ q cos k; sgnðsin hþš, the numerator of the scattered wave potential eing an analytical function defined in Ref. 6 (see sections 5.1 to 5.4 for definition of all the involved terms). The integrand f ½ q cos k; sgnðsin hþš has then two poles k ¼ h and k ¼ p h where h ½0; pš is defined as q cos h ¼ q a cos h a : (4) In Eq. (), f is the far-field parameter f ¼ k L ; (5) where L, the distance parameter, can e written as L ¼ r sin X ; (6) with X eing the angle of the diffraction cone (see Fig. 1) linked to the incidence angle X a y the Snell s law k cos X ¼ k a cos X a ; h is related to the position vector x (see Fig. 1) y ( h ¼ h if 0 h p h ¼ p h if p h p ; (7) d is the polarization vector of the scattered wave and C is the integration contour in the complex k-plane depicted in Fig.. The resulting total field is u tot ðxþ ¼u a ðxþþ X u a ðxþ: (8) J. Acoust. Soc. Am. 138 (5), Novemer 015 Kamta Djakou et al. 373

3 In the high frequency approximation, f 1, the integral Eq. () can e evaluated asymptotically. The main contriutions are due to the integral critical points such as singularities of the amplitude (poles or ranch points) and the stationary points of the phase function. The contriutions are easy to calculate in the so-called geometrical regions where the critical points are isolated (far apart), so that the resulting asymptotics are non-uniform. The only integrand singularities considered elow are the poles. A. Contriutions of isolated poles: GE To evaluate the integral Eq. () asymptotically, the integration contour ðcþ is deformed to the steepest descent contour ðcþ which passes through the phase stationary point h ½0; pš. Among the two poles k ¼ h and k ¼ p h, only the isolated pole k ¼ h (with 0 h p) can lead to a non-zero contriution to the integral () since it might e crossed when deforming the contour. It is only crossed when 0 h h ðsee Fig: Þ: (9) Therefore, its contriution must e taken into account for the asymptotic evaluation of the integral Eq. (). The GE field is the sum of this pole contriution and of the incident field. Comining Eqs. (7) and (9), this pole k ¼ h gives rise to two different physical contriutions depending on the oservation angle h. When h ¼ p h [i.e., when p h p according to (7)], the pole k ¼ h is only crossed when p h h p and its contriution then leads to the reflected field. For h ¼ h, the pole h is crossed when 0 h h. It gives rise to the so-called compensating field. This cancels the incident field when ¼ a and is zero otherwise. Equation (9) ensuring non-zero pole contriution is checked when the oservation point lies in irradiated areas of the GE field: i.e., in the insonified zone of the reflected field (p h h p and h ¼ p h) and in that of the compensating field (0 h h and h ¼ h). Then, applying the residue theorem to integral Eq. () and using the clockwise contour around the pole (see Fig. ), give the amplitudes of the reflected and compensating fields, lim k! h pif ½ q cos k; sgnðsin hþšðk h Þ¼j g ð q cos h ; 1Þ ¼R a ; (10) h ½p h ; pš lim pif ½ q cos k; sgnðsin hþšðk h Þ¼j g ð q cos h ; 1Þ ¼ 1; a ¼ ; (11) k! h h ½0; h Š lim pif ½ q cos k; sgnðsin hþš ¼ j g ð q cos h ; 1Þ ¼0; a 6¼ ; k! h h ½0; h Š (1) where R a is the reflection coefficient in terms of displacement. Therefore, adding the incident field to the poles contriution results in the Geometrico-Elastodynamic field u ðgeþ ðxþ ¼Hðg a Þu a ðxþþ X Hðg Þu aðrefþ ðxþ; (13) u aðgtdþ ðþ¼ x u a x a D aðgtdþ ðx a ; h a ; hþ e d q cos h; sgnðsin hþ ik S p ffiffiffiffiffiffiffiffiffi ; k L (15) where Hð:Þ is the Heaviside function, and where u aðrefþ ðxþ ¼AR a d ð q cos h ; 1Þ e ik p x (14) is the reflected field; the compensating field has een comined with the incident field to produce the first term of Ref. 13. InEq.(14), p ¼ðsin X cos h ; sin X sin h ; cos X Þ is the unit reflected wave vector and d ð q cos h ; 1Þ its polarisation vector. The arguments g a ¼ sgnðh h a Þ and g ¼ sgnðh p þ h Þ of the respective Heaviside functions determine whether the oservation point is in the illuminated region or shadow of incident and reflected waves (see Fig. 3). B. Contriutions of isolated stationary points: GTD The contriution of each isolated stationary point k s ¼ h can e found y applying the steepest descent method to Eq. (), producing the classical GTD recipe where u a ðx a Þ¼ua ðx a Þda ; x a ¼ð0; 0; x 3 S cos X Þ is the diffraction point on the scattering edge (see Fig. 1); S is the distance etween the diffraction point x a and the oservation point x. Also, D aðgtdþ ðx a ; h a ; hþ ¼ q p j ffiffiffiffiffi g q cos h; sgnðsin hþ j sin hje ip=4 (16) p q a cos h a q cos h are GTD diffraction coefficients. They contain poles h ¼ h and h ¼ p h, which descrie the shadow oundaries (see Fig. 3). Note that at the incident and reflected shadow oundaries the total GE field Eq. (13) is finite while the GTD diffracted field is infinite. Therefore, the approximate GTDased total field 374 J. Acoust. Soc. Am. 138 (5), Novemer 015 Kamta Djakou et al.

4 FIG. 3. Interaction of a plane wave with a semi-infinite crack emedded in an elastic homogeneous solid. Thick arrows, incident waves; dash arrows, reflected waves. where f 1ands is a complex-valued parameter, hðk; sþ and qðkþ are k holomorphic functions, and c is the steepest descent path. Note too that h cannot coalesce with the simple pole k ¼ p h, ecause h is always smaller than p, whilep h is always larger than p. Therefore the Pauli Clemmow approximation to the scattered field is u aðutdþ ¼ Fðk L aþu aðgtdþ ; (19) where a is the phase difference etween the rays propagating in direction h and h a ¼ sin h h ; (0) and F is a transition function pffiffiffipffiffiffiffi pffiffiffi FX ð Þ ¼ i X ip e ix F X ; p < arg X < 3p ; with F eing the Fresnel function, F ðxþ ¼ p 1 ffiffiffiffi ip ð þ1 X (1) e it dt: () FIG.. C and the steepest descent path c in the complex plane k ¼ r þ s. h is the real pole of the integral [Eq. ()] and h is its phase stationary point. u totðgtdþ ðxþ ¼u ðgeþ ðxþþ X is not spatially uniform. u aðgtdþ ðxþ; (17) III. SCATTERING OF A PLANE ELASTIC WAVE BY A HALF-PLANE CRACK: UTD To overcome the non-uniformity of the approximate total field Eq. (17), UTD is derived in this section in elastodynamics. When the phase stationary point k s ¼ h in Eq. () coalesces with the simple pole k ¼ k 0 ðsþ ¼h (h p), Eq. () can e approximated using the Pauli Clemmow procedure. 18,19 Note that the integral Eq. () has the form ð Iðf; sþ ¼ hðk; sþ e f qðkþ dk; (18) c The transition function is the complex conjugate p ffiffiffi of the Kouyoumjian function. 1 Due to the presence of X, F(X) is multivalued and can e rendered single-valued in a standard manner, using, e.g., the ranch cut along the negative imaginary axis fim X < 0; Re X ¼ 0g. Using Eq. (19), a UTD diffraction coefficient can e defined as D aðutdþ ¼ Fðk L aþ D aðgtdþ : (3) Sustituting Eqs. (15) and (3) in Eq. (19), the UTD diffracted field can e expressed as u aðutdþ ¼ u a x a D aðutdþ expðik S Þ pffiffiffiffiffiffiffiffiffi k L d q cos h; sgnðsin hþ : (4) It has the same form as the GTD diffracted field and the UTD diffraction coefficient differs from the GTD one y the transition function. When the oservation point is far from the shadow oundaries the approximation Eq. (4) is J. Acoust. Soc. Am. 138 (5), Novemer 015 Kamta Djakou et al. 375

5 equivalent to GTD. Indeed, the asymptotic value of the Fresnel function in the illuminated region given y Borovikov 0 permits to find Fðk L aþ! 1: (5) j h h j!þ1 Since Fð0Þ ¼1=, near the shadow oundaries, rffiffiffi i p=4 Fk ð L aþ e ð Þ pp ffiffiffiffiffiffiffiffiffi ðsgnþ k L ; (6) where ¼ h h is a small numer (jj 1). One can use Eqs. (6), (10) (1) in the asymptotics of Eq. (3) to otain D aðutdþ 8 0 ifh p and a 6¼ >< sgn p ffiffiffiffiffiffiffiffiffi k L ; if h p and a ¼ R a ð sgn p ffiffiffiffiffiffiffiffiffi >: Þ k L ; if h > p: (7) The UTD diffraction coefficients do not diverge at shadow oundaries as the GTD coefficients do, however the sign function makes them discontinuous. It follows that near the shadow oundaries, the leading-order asymptotics of the diffracted field Eq. (4) are u aðutdþ 8 0; if h p and a 6¼ >< A sgn e ik a cos X a ðx 3 S a cos X a Þ e ik as a d a ; if h p and a ¼ ðþ x A Ra >: ð sgn Þe ik a cos X a ðx 3 S cos X Þ e ik S d ð q cos h ; 1Þ; if h > p; (8) where the exponential term exp ½ik a cos X a ðx 3 S a; cos X a; ÞŠ is the phase of the incident wave at the diffraction point x a. Near the incident shadow oundary, when a 6¼ the UTD diffracted field does not exist and due to the presence of the sign function when a ¼, it is discontinuous. The UTD diffracted field is discontinuous at the reflected shadow oundary too. However, since the approximation UTD Eq. (4) works in the vicinity of shadow oundaries and descries diffracted fields outside them, adding to it the GE field produces the UTD approximation to the total field, u totðutdþ ðxþ ¼u aðgeþ ðxþþ X u aðutdþ ðxþ: (9) It is easy to show that this approximate total field is continuous. Indeed, since r ¼ S a; sin X a;,eq.(13) can e rewritten as u GE ðxþ ¼AðHðg a Þe ik a½s a þcos X a ðx 3 S a cos X a ÞŠ cosðh h a Þ e ik a cos X a x 3 ð1 cosðh h a ÞÞ d a þ Hðg ÞR a e ik ½S þcos X ðx 3 S cos X ÞŠ cosðhþh Þ e ik a cos X x 3 ½1 cosðhþh ÞŠ d ð q cos h ; 1ÞÞ; (30) where the arguments of the Heaviside functions are, respectively, g a ¼ sgnðh h a Þ (31) g ¼ sgnðh p þ h Þ: (3) Therefore near the shadow oundaries, the GE filed can e approximated using ( u GE ðxþ Aeik a½s a þcos X a ðx 3 S a cos X a ÞŠ HðÞd a if h p AR a eik ½S þcos X ðx 3 S cos X ÞŠ Hð Þ d ð q cos h ; 1Þþu a ð h a Þ if h > p: (33) Finally, expressing the Heaviside function as HðÞ¼ 1 ð 1 þ sgn Þ; (34) it can e seen that near oth the incident and reflected shadow oundaries the GE field (13) is discontinuous and its discontinuities cancel those of the UTD diffracted field. Thus, the UTD total field Eq. (9) is continuous. The contriution of coalescing stationary phase point and pole can also e calculated y UAT (Ref. 6) (seeappendix). It is well understood in electromagnetism 1 that unlike UAT, the UTD asymptotics do not include all terms of the order ðk L Þ ð1=þ. IV. COMPARISON OF UTD AND UAT AND DISCUSSION Let us first discuss the difference in analytical properties of various UTD and UAT fields. When comparing the UTD 376 J. Acoust. Soc. Am. 138 (5), Novemer 015 Kamta Djakou et al.

6 FIG. 4. Extension of the reflected field to its shadow zone using fictitious rays (a) half-plane (thick line) illuminated y a plane wave and () curved face (thick line) illuminated y a spherical wave. total field Eq. (9) to the GTD-ased non-uniform approximation Eq. (17), the GE field is unchanged while the diffraction coefficient is multiplied y the transition function Eq. (1). The argument k L a of the transition function descries the proximity of the oservation point to a shadow oundary. When the oservation point moves away from such oundary k L a increases, the transition function approaches 1 [see Eq. (5)] and UTD diffraction coefficient approaches the GTD diffraction coefficient. When the oservation point moves towards such oundary, k L a and the transition function oth vanish ut as shown in Sec. III the UTD coefficient is discontinuous and the UTD total field continuous. By contrast, when UAT is presented in the form Eq. (A1), it is the diffracted field that remains the same as in the GTD-ased non-uniform approximation Eq. (17), while the GE field is modified. In UAT oth the modified GE fields and GTD fields diverge at shadow oundaries, ut the corresponding singularities cancel each other, and the total UAT field is again continuous. Since in Eq. (A1) the incident and reflected fields are defined on the whole space, oth incident and reflected fields have to e extended to their corresponding shadow region. This can e achieved y tracing the incident rays through the ostacle as if it was asent. Moreover, the reflected field has to e extended to the shadow region of the reflected field too. This can e achieved y extending the illuminated face eyond the crack edge and tracing the resulting fictitious reflected rays (see Fig. 4). While reliance on the fictitious rays is a definite disadvantage, it has een mentioned aove that, unlike UTD, UAT is more consistent in that it contains all terms of order ðk L Þ ð1=þ. In electromagnetism, the terms missing in UTD are small. Therefore we move on to a numerical comparison of UTD and UAT to estalish whether the same is true in elastodynamics. Note that the elastodynamic UAT has een tested efore, using the finite difference numerical method (see section.3.1 in Ref. ). FIG. 5. Directivity pattern of the total field predicted y different models (GTD, UAT, and UTD) at r ¼ k L ; X L ¼ 90 ; h L ¼ 30. (a) UTD model is the one found using Pauli Clemmow method. () Modified UTD model. Each circle represents amplitude of the total field normalized y the incident amplitude. To compare UAT and UTD, oth two-dimensional (D) and three-dimensional (3D) configurations are considered elow, with X a ¼ 90 and X a 6¼ 90, respectively. In a D configuration, the incident wave is normal to the edge crack and thus the diffracted waves have cylindrical fronts, while in a 3D configuration the incidence is olique and the diffracted waves have conical fronts. The numerical results are presented in the ðe 1 ; e Þ plane, which is perpendicular to the edge crack, since the prolem is invariant in the x 3 direction (see Fig. 1). The oservation is specified using the cylindrical coordinates ðr; hþ associated with the ðe 1 ; e Þ plane. In all simulations, the solid material is ferritic steel with Poisson s ratio ¼ 0:9, longitudinal speed c L ¼ 5900 m s 1 and transversal speed c T ¼ 330 ms 1. In far field from the flaw, the scattering patterns exhiit a lot of oscillation loes. We present first results for r ¼ k a and 3k a and then at r ¼ 8k a (k a eing the wavelength of the J. Acoust. Soc. Am. 138 (5), Novemer 015 Kamta Djakou et al. 377

7 FIG. 8. Reflection of a transversal wave y a half-plane crack. An incident transversal wave gives rise to reflected transversal and longitudinal waves, which respect the Snell s law ðh L < h TV Þ. For a transversal incidence elow the longitudinal critical angle h c (h TV < h c ), the longitudinal waves are no more reflected ut laid on the crack surface. FIG. 6. Directivity pattern of the total field predicted y different models (GTD, UAT, and UTD) at r ¼ 3k TV for h TV ¼ 30. (a) X TV ¼ 90, () X TV ¼ 60. Each circle represents amplitude of the total field normalized y the incident amplitude. incident wave) so that the patterns have less loes and lend themselves to a clear interpretation (see Figs. 5, 6, and 9). In Figs. 5 and 6 all approximate total fields are calculated using all scattered modes, with the GTD-ased non-uniform asymptotics given y Eq. (17), UAT given y Eq. (A1), and UTD given y (9). In Figs. 5 and 6 the (a) configurations FIG. 7. Waves diffracted from a crack edge: L stands for longitudinal wave, TV for transversal, and H for head wave; h c is the longitudinal critical angle. are D and in Fig. 6, the () configuration is 3D. As expected, oth UTD and UAT are continuous at shadow oundaries and practically coincide far away from the shadow oundaries. Near the shadow oundaries they differ ut not y much. There are three shadow oundaries in Fig. 5, the incident L shadow oundary h ¼ 30, reflected TV shadow oundary h 300 and reflected L shadow oundary h ¼ 330. The peaks near critical angles h c 56:8 and p h c 303; are due to the interference of the diffracted T wave with the corresponding head waves generated y diffraction at the edge (see Fig. 7). The head waves are contriutions to the exact solution Eq. () of the integrand s ranch points. 3 The coalescence etween the stationary phase points and ranch points lies outside the scope of this paper. However, it is worth noting that while GTD and its uniform corrections are not valid near the critical angle, since the head wave attenuates with the distance, in far-field the spikes oserved in the diffraction coefficient D a TV at the critical angles are physical in nature. In Fig. 5(a), UAT reproduces the critical GTD peak exactly, while UTD has a smaller amplitude. The discrepancy can e explained y noting that each UTD diffraction coefficient has four poles, one at the incident angle h a, one at the reflection angle p h a, one at the reflection angle p h, and one at the reflection angle h. When 6¼ a; p h is the propagation angle of a mode-converted reflected wave and h is an angle associated with a non-physical mode-converted transmitted wave. It is non-physical, ecause the residue at this pole is zero and the diffraction coefficient D a ; 6¼ a does not diverge (see Figs. 4.5 and 4.6 in Ref. 4). In the vicinity of this pole, the transition function tends to 0 [see Eq. (6)] and the UTD diffraction coefficient differs from the corresponding GTD diffraction coefficient. In Fig. 5(a), such pole h TV 61:70 is close to the critical angle h c 57 and therefore near this critical angle the peak amplitude is reduced. The performance of UTD can e improved y taking into account that since the residue of the pole h ; 6¼ a is zero and therefore there is no need to consider its coalescence with the stationary points, it is possile to introduce the following modified UTD (MUTD): 378 J. Acoust. Soc. Am. 138 (5), Novemer 015 Kamta Djakou et al.

8 for 6¼ a, if h a p; 8 >< >: D aðutdþ D aðutdþ ð Þ ¼ D a GTD ¼ F k L sin h þ h D aðgtdþ if 0 h p if p < h p (35) and if h a > p; 8 >< >: ¼ F k L sin h h D aðgtdþ if 0 h p ð Þ ¼ D aðgtdþ if p < h p: D aðutdþ D a UTD (36) The UTD curve in Fig. 5() confirms that the Modified UTD reproduces the GTD critical peaks In Figs. 6(a) and 6(), there are only two shadow oundaries, the incident TV shadow oundary h ¼ 30 and reflected TV shadow oundary h ¼ 330. Since the incidence angle h TV is sucritical there are no reflected longitudinal waves. In the configuration used in Fig. 8 the incident angle h TV is supercritical. The simulations reported in Fig. 9 have een performed using parameters similar to Fig. 5() ut for a larger far-field parameter f ¼ 16p sin X (so that r ¼ 8k L ). The resulting UAT and UTD fields are much closer than in Fig. 6(). As the oservation point moves away from the crack, near the shadow oundaries the error etween the approximations decreases [see Figs. 10(a), 10(), and 10(c)]. The modified UTD produces a much smaller error near the critical angle h c 57 than UTD. The aove results are envisaged as particularly useful in NDT of cracks in materials. Indeed, UTD has already proved an efficient method for simulating the scattering of an elastic plane wave y a half-plane. In NDT, the flaws are usually FIG. 9. Directivity pattern of the total field predicted y different models (GTD, UAT, and UTD) at r ¼ 8k L ; X L ¼ 90 ; h L ¼ 30. Each circle represents an amplitude of the total field normalized y the incident amplitude. inspected in the far field of the used proes or else in their focal areas if they are focused. Consequently, often, at each point of the meshed flaw the incident wave-fields can e approximated y a wave that is locally plane 5,6 and GE ray methods and GTD-ased models can e easily applied in the proes far field. 7 Since the proposed UTD model relies on the same high-frequency assumptions as GTD, it should prove to e efficient in the far field of the flaw. If inspection is carried out in the near or intermediate field, the approximation of the locally plane wave for the incident field can e withdrawn, as has een shown in the recent report on the new promising GTD-ased NDT system model which uses each incident ray as an input of the flaw scattering model. 8 The proposed UTD model can e used to model scattered echoes from any locally plane flaw, since according to the GTD locality principle, a curved edge of the flaw contour can e approximated y the edge of the half-plane, which is tangent to it. In modelling specular reflections, the finite size of the flaw is routinely accounted for in the GE ray method: only the incident rays impacting on the finite flaw surface are reflected. In modelling edge diffraction, the finite extent of the edge is accounted for y using incremental methods. 9 The limits of validity of GTD-ased NDT system models are descried, e.g., in Ref. 7 for a rectangular flaw. It has een shown that GTD predictions are valid for flaw heights of more than one wavelength for longitudinal waves and more than two wavelengths for shear waves (whether incident and diffracted). The NDT models ased on UTD can e extended to employ solutions of canonical prolems of wedge diffraction to model 3D multi-facetted or ranched flaws (which comprise planar facets). V. CONCLUSIONS The elastodynamic UTD has een derived to descrie the scattering of a plane wave from a stress-free half-plane. Unlike GTD, at the shadow oundaries, each UTD diffracted field contains no singularities, only discontinuities, and each such discontinuity cancels the discontinuity in the corresponding GE field. Just like the total UAT field, the total UTD field is continuous. In the far field UTD practically coincides with UAT and the modified version of UTD, J. Acoust. Soc. Am. 138 (5), Novemer 015 Kamta Djakou et al. 379

9 comparisons carried out in this paper confirm that near the shadow oundaries, where the missing terms play a part, the resulting differences do not appear to e significant for practical applications (see, for example, Ref. 30). APPENDIX: THE ELASTODYNAMIC UAT The total field in elastodynamic UAT can e presented in different ways. Here we present it in the same form as in Ref. 6, h tot UAT u ð Þ ðxþ ¼ A F ðn a Þ ^F i ðn a Þ e ikax d a þ X h AR a i F n ^F n e ik p x d q cos h ; sgnðsin hþ where þ X u a x a D aðgtdþ e ik S pffiffiffiffiffiffiffiffiffi k L d q cos h; sgnðsin hþ ; (A1) ^F ðxþ ¼ e i p=4 ð Þ e ix p X ffiffiffi ; p and the parameters sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n a ¼ sgnðh h a Þ k a L a sin h a h sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n ¼ sgnðh þ h pþ k L sin h þ h and (A) (A3) (A4) are the detour parameters. 8 FIG. 10. Asolute error etween UTD and UAT total fields in percentage of the incident displacement amplitude [noted A in Eq. (1)] versus the oservation angle at X L ¼ 90 ; h L ¼ 30 in percent of the incident amplitude (a) r ¼ k L, () r ¼ 8k L, (c) r ¼ 500k L. Solid line represents the asolute error etween initial UTD and UAT total fields and dashed line represents asolute error etween modified UTD and UAT. (a), (), and (c) use different scales along the vertical axis. MUTD, proposed in this paper, reproduces the critical peaks in UAT too. In ray tracing implementations UTD is more convenient than UAT, ecause unlike UAT, it does not rely on fictitious rays. It is well understood that, unlike UAT, UTD misses terms of the same order it includes, however the numerical 1 P. Ya Ufimtsev, Fundamentals of the Physical Theory of Diffraction (Wiley, NJ, 007), Chap. 3, pp ; Chap. 4, pp B. L u, M. Darmon, L. Fradkin, and C. Potel, Numerical comparison of acoustic wedge models, with application to ultrasonic telemetry, Ultrasonics, in press (015). 3 V. Zernov, L. Fradkin, and M. Darmon, A refinement of the Kirchhoff approximation to the scattered elastic fields, Ultrasonics 5, (01). 4 J. B. Keller, Geometrical theory of diffraction, J. Opt. Soc. Am. 5, (196). 5 J. D. Achenach and A. K. Gautesen, Geometrical theory of diffraction for three-d elastodynamics, J. Acoust. Soc. Am. 61, (1977). 6 J. D. Achenach, A. K. Gautesen, and H. McMaken, Rays Methods for Waves in Elastic Solids (Pitman, New York, 198), Chap. 5, pp R. M. Lewis and J. Boersma, Uniform asymptotic theory of edge diffraction, J. Math. Phys. 10, (1969). 8 S. W. Lee and G. A. Deschamps, A uniform asymptotic theory of electromagnetic diffraction y a curved wedge, IEEE Trans. Antennas. Propag. 4, 5 34 (1976). 9 D. S. Ahluwalia, Uniform asymptotic theory of diffraction y the edge of a three dimensional ody, SIAM J. Appl. Math. 18, (1970). 10 B. L. Van Der Waerden, On the method of saddle points, Appl. Sci. Res., Sect. B, (1951). 380 J. Acoust. Soc. Am. 138 (5), Novemer 015 Kamta Djakou et al.

10 11 P. H. Pathak and R. G. Kouyoumjian, The dyadic diffraction coefficient for a perfectly-conducting wedge, DTIC Document, Tech. Rep. (1970). 1 R. G. Kouyoumjian and P. H. Pathak, A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface, Proc. IEEE 6, (1974). 13 P. C. Clemmow, Some extension to the method of integration y steepest descent, Q. J. Mech., Appl. Math. III, (1950). 14 V. A. Borovikov and B. Y. Kiner, Geometrical Theory of Diffraction (Institution of Electrical Engineers, London 1994), Sec H. McMaken, A uniform theory of diffraction for elastic solids, J. Acoust. Soc. Am. 75, (1984). 16 N. Tsingos, T. Funkhouser, A. Ngan, and I. Carlom, Modeling acoustics in virtual environments using the uniform theory of diffraction, Proc. ACM SIGGRAPH (001), pp M. F. Catedra, J. Perez, F. Saez de Adana, and O. Gutierrez, Efficient ray-tracing techniques for three-dimensional analyses of propagation in moile communications: Application to picocell and microcell scenarios, IEEE Antennas Propag. Mag. 40, 15 8 (1998). 18 D. Bouche and F. Molinet, Methodes Asymptotiques en Electromagnetisme (Asymptotic Methods in Electromagnetics) (Springer- Verlag, Berlin Heidelerg, 1994), Chap. 5, pp D. Bouche, F. Molinet, and R. Mittra, Asymptotic Methods in Electromagnetics (Springer-Verlag, Berlin Heidelerg, 1997), Chap V. A. Borovikov, Uniform Stationary Phase Method (The Institution of Electrical Engineers, London, 1994), pp F. Molinet, Acoustic High-Frequency Diffraction Theory (Momentum Press, New York, 011), Chap. 3, pp L. Ju. Fradkin and R. Stacey, The high-frequency description of scatter of a plane compressional wave y an elliptical crack, Ultrasonics 50, (010). 3 D. Gridin, High-frequency asymptotic description of head waves and oundary layers surrounding critical rays in an elastic half-space, J. Acoust. Soc. Am. 104, (1998). 4 J. D. Achenach and A. K. Gautesen, Edge diffraction in acoustics and elastodynamics, in Low and High Frequency Asymptotics (Elsevier, New York, 1986), Chap. 4, pp L. W. Schmerr and S.-J. Song, Ultrasonic Nondestructive Evaluation Systems: Models and Measurements (Springer, New York, 007). 6 M. Darmon and S. Chatillon, Main features of a complete ultrasonic measurement model - Formal aspects of modeling of oth transducers radiation and ultrasonic flaws responses, Open J. Acoust. 3A, (013). 7 M. Darmon, V. Dorval, A. Kamta Djakou, L. Fradkin, and S. Chatillon, A system model for ultrasonic NDT ased on the physical theory of diffraction (PTD), Ultrasonics 64, (016). 8 G. Toullelan, R. Raillon, S. Chatillon, V. Dorval, M. Darmon, and S. Lonne, Results of the 015 UT modeling enchmark otained with models implemented in CIVA, AIP Conf. Proc. in press (016). 9 A. Kamta Djakou, M. Darmon, and C. Potel, Elastodynamic models for extending GTD to penumra and finite size scatterers, Phys. Proc. 70, (015). 30 M. Darmon, N. Leymarie, S. Chatillon, and S. Mahaut, Modelling of scattering of ultrasounds y flaws for NDT, in Ultrasonic Wave Propagation in Non Homogeneous Media (Springer, Berlin, 009), Vol. 18, pp J. Acoust. Soc. Am. 138 (5), Novemer 015 Kamta Djakou et al. 381