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Scattering of a Rayleigh wave by an elastic wedge whose angle is less than 180°
Wave Motion 36 (2002) 417–424
Scattering of a Rayleigh wave by an elastic wedge whose angle is less than 180◦夽 A.K. Gautesen∗ Department of Mathematics and Ames Laboratory, 136 Wilhelm Hall, Iowa State University, Ames, IA 50011, USA Received 8 September 2000; received in revised form 10 May 2001; accepted 10 May 2001
Abstract The steady-state problem of scattering of an incident Rayleigh wave by an elastic wedge whose angle is less than 180◦ is considered. The problem is reduced to the numerical solution of a pair of Fredholm integral equations of the second kind whose kernels are continuous functions. Numerical results are given for the amplitude and phase of the Rayleigh waves transmitted and reflected by the corner. Published by Elsevier Science B.V.
1. Introduction In [1], the author develops a new method to consider scattering of waves by a 90◦ elastic wedge. Here a slight modification of this method is used to study wedges whose wedge angle α is less than 180◦ . The incident wave is taken to be a Rayleigh surface wave. The method of solution also works for incidence of a plane longitudinal or transverse wave. Numerical results are given for the amplitude and phase of the Rayleigh waves transmitted and reflected by the corner for wedge angles between 63◦ and 180◦ and Poisson’s ratio ν = 1/4 and 1/3. Below this angle, the problem is complicated by multiple re-reflections of the incident wave. In principle this can be accounted for, but has not been done here. The results presented here are in general agreement, but more accurate, than the results [2] obtained by the author using a different method of solution. A review of early efforts to deal with this problem is given by Knopoff [3]. Li et al. [4] use a free-space Green’s function representation of the tractions to obtain boundary integral equations which they solve numerically. In an extension of his earlier work Fujii [5] gives results for a wide range of wedge angles. Budaev and Bogy [6,7] use the Sommerfeld–Malivzhinetz method to solve this problem. They give corrected numerical results in [8]. In [9], the author treats the case of wedge angles greater than 180◦ by a method similar to that used here. The method of solution is quite simple. First the problem is symmetrized to reduce the number of the unknown displacements on the traction-free surfaces from 4 to 2. Of course, the problem must then be solved twice— once for the symmetric case and once for the antisymmetric case. From a free-space Green’s function integral representation of the displacements the tractions, dilatation and rotation are computed on a line just below the x1 -axis 夽
It is with great pleasure that I dedicate this paper to my friend and colleague Dr. Jan Achenbach who has made so many contributions to the field of elastodynamics. ∗ Tel.: +1-515-294-9312; fax: +1-515-294-4491. E-mail address: [email protected] (A.K. Gautesen). 0165-2125/02/$ – see front matter. Published by Elsevier Science B.V. PII: S 0 1 6 5 - 2 1 2 5 ( 0 2 ) 0 0 0 3 3 - 1
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Fig. 1. Incident, transmitted and reflected surface waves.
Fig. 2. Branch cut and location of the Rayleigh poles of u¯ j (ξ ).
(x2 = 0)—see Fig. 1. Next the Fourier transform of these quantities is taken, yielding functional equations where the unknowns are the Fourier transform of the displacements u¯ j (ξ ) on the traction-free surface x2 = 0, x1 > 0. Then a suitable representation of u¯ j (ξ ) is taken. The branch cut and Rayleigh poles of u¯ j (ξ ) are shown in Fig. 2. A pair of numerically stable Fredholm integral equations of the second kind is achieved by computing the jump across the branch cut of u¯ j (ξ ). The kernels of these integral equations are continuous, and consist of elementary functions and the Wiener–Hopf product factorization of the Rayleigh function. In the next section, the governing Fredholm integral equations are derived. In the last section, the numerical results are presented and discussed.
2. Derivation of the integral equations The diffraction of steady-state waves by an elastic wedge whose wedge angle α is less than 180◦ is considered. The location of the wedge; the coordinate system; and the incident, transmitted and reflected surface waves are shown in Fig. 1. By using the free-space Green’s stress tensor, an integral representation of the total displacements uk (x1 , x2 ) is given by 2 ∞ uk (x1 , x2 )H (x) = uin τ G (x − y, x2 )ui (y, 0) k (x1 , x2 ) + i2;k 1 0 i=1 2 G + τij;k (x1 − t1 y, x2 − t2 y)nj ui (t1 y, t2 y) dy, k = 1, 2. (2.1) j =1
Here, H (x) = 1 if (x1 , x2 ) is a point in the elastic material and H (x) = 0 otherwise. The incident field is denoted in by uin k (x1 , x2 ) and since the incident field is a Rayleigh surface wave, uk (x1 , x2 ) = 0. The free-space Green’s stress
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G (x , x ). The normal n and tangent t to the upper traction-free surface are given by tensor is denoted by τij;k 1 2 j j
n1 = t2 = sin α,
n2 = −t1 = − cos α.
(2.2)
To reduce the number of unknowns the problem is divided into an antisymmetric ( = 1) and symmetric ( = 2) problem. For the antisymmetric (symmetric) problem the displacements are antisymmetric (symmetric) with respect to the line x2 = x1 tan 1/2α. This implies that u1 (y, 0) = (−1)
2
tj uj (t1 y, t2 y),
y > 0, = 1, 2,
(2.3)
nj uj (t1 y, t2 y),
y > 0, = 1, 2.
(2.4)
j =1
u2 (y, 0) = (−1)
2 j =1
Solving (2.3) and (2.4) for uj (t1 y, t2 y) and substituting the result into (2.1) reduces the number of unknowns by one half, but then the problem must be solved twice (once for = 1 and once for = 2). The importance of (2.1) is that it holds for all points (x1 , x2 ) and not only just for points lying within the elastic material. The normal and shear tractions vanish on the entire line x2 = 0, −∞ < x1 < ∞. These quantities are computed from (2.1) and then the Fourier transform of the resulting equations are taken to obtain the following equations: Uj (ξ ) R(ξ )u¯ i (ξ ) + Bij (ξ, γ1 , γ2 ) = 0, γi γi 2
i = 1, 2,
(2.5)
j =1
where u¯ i (ξ ) which denotes the Fourier transform of ui (x1 , 0) is defined by ∞ u¯ i (ξ ) = kL exp[ikL ξ x1 ]ui (x1 , 0) dx1 0
and kL denotes the wavenumber corresponding to longitudinal waves. The Rayleigh function R(ξ ) is defined by R(ξ ) = a 2 (ξ ) + 4ξ 2 γ1 γ2 ,
(2.6)
where γ1 (ξ ) = (κ + ξ )1/2 (κ − ξ )1/2 ,
κ=
cL , cT
(2.7)
γ2 (ξ ) = (1 + ξ )1/2 (1 − ξ )1/2 ,
(2.8)
a(ξ ) = κ 2 − 2ξ 2 .
(2.9)
It has the property that R(κR ) = 0, κR = cL /cR . The speed of longitudinal, transverse and Rayleigh waves have been denoted by cL , cT , and cR , respectively. In (2.5) and below, the summation convention is not used. The remaining quantities in (2.5) are defined by Bii (ξ, z1 , z2 ) = (−1) a(ξ ), +j
Bij (ξ, z1 , z2 ) = (−1) Ui (ξ ) =
2
2ξ zi ,
(2.10) i = j,
Qij (ξ, z1 , z2 )u¯ j (ξi ),
zi = γi (ξ ),
(2.11) (2.12)
j =1
ξi = ξ cos α + zi sin α,
(2.13)
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A.K. Gautesen / Wave Motion 36 (2002) 417–424
Qii (ξ, z1 , z2 ) = (−1)i a(ξi ), Qij (ξ, z1 , z2 ) = 2ξi (ξ sin α − zi cos α),
(2.14) i = j.
(2.15)
The rotation and dilatation vanish on the line x2 = −δ 2 , −∞ < x1 < ∞. Computing these quantities from (2.1), followed by taking the Fourier transform and the limit as δ → 0 leads to the following equation: 2
(−1)i+j Bij (ξ, γ1 , γ2 )u¯ j (ξ ) + Ui (ξ ) = 0,
i = 1, 2.
(2.16)
j =1
The branches and Rayleigh poles of u¯ i (ξ ) are shown in Fig. 2. The following representation of u¯ i (ξ ) is taken
ξ − y go y u¯ i (ξ ) = [Cui (ξ − y )−1 + ui (ξ ) + vi (ξ )]. (2.17) + K (ξ )(κR + ξ ) Here, K + (ξ ) is the Wiener–Hopf product factorization of K(ξ ) = R(ξ )/[2(κ 2 − 1)(κR2 − ξ 2 )] and is given by 2 2 1/2
1 κ + −1 4t γ1 (t)(t − 1) (t + ξ )−1 dt. tan (2.18) log K (ξ ) = − π 1 a 2 (t) This quantity is easily evaluated to 16 digit accuracy using a 32 point Gaussian quadrature. The pole at ξ = −κR yields the diffracted surface waves. The multiplicative factor in the right side of (2.17) approaches 1 as ξ approaches infinity. When = 1, α y y y = κ cos , (u1 , u2 ) = (− sin α, cos α) (2.19) 2 and when = 2, α , y = cos 2
y
y
(u1 , u2 ) = (κ 2 − 1 − cos α, − sin α).
(2.20)
These choices ensure that u¯ j (ξ ) satisfies (2.16) with i = . The constant C is determined below. go The term ui (ξ ) is defined by go
ui (ξ ) =
M m=0
[(ξ − pm )−1 + F + (ξ, pm )]um i +
N m=1
F + (ξ, qm )vim ,
where for rj = κexp[iπ(3 − 2j )/4], j = 1, 2 2 + + + + (r3−j − p)(γ2 (ξ ) − γ2 (rj )) 1 γ2 (ξ ) − γ2 (p) F + (ξ, p) = (−1)j + 2γ2 (p) ξ −p (r2 − r1 )(ξ − rj )
(2.21)
(2.22)
j =1
and γ2+ (ξ ) denotes the Wiener–Hopf sum splitting of γ2 (ξ ), γ2+ (ξ ) = −2iγ2 (ξ )
log[2−1/2 ((1 + ξ )1/2 + i(1 − ξ )1/2 )] . π
(2.23)
Here pm are the poles associated with the incident (m = 0), reflected (m = 1, 2) and re-reflected (m ≥ 3) waves. The wedge angle α is taken to be sufficiently large that re-re-reflected waves do not occur. Otherwise M (as well as N) increases rapidly. In particular M = 2, if α > 90◦ and M = 6 if α ≤ 90◦ , while N = 4, 100◦ > α > 90◦ and N = 0 otherwise. The second sum in (2.21) is included to remove the poles in the analytical continuation of the integral equation derived below which are near the contour of integration. When (2.21) is substituted for u¯ i (ξ )
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in (2.5), there are no poles at ξ = pm . For m = 0, (2κR γ1 (κR ), a(κR )) . (2.24) κ2 Since the definitions of the remaining constants in (2.21) are not needed in the present discussion, they are given at the end of this section. The last term in (2.17) is taken to be ∞ wj (z) 1 vj (ξ ) = − dz (2.25) 2πi 1 ξ + z and gives the remaining contribution to u¯ i (ξ ) from the branch cut. Note that ∞ wj (z) 1 2vj (−x ± i0) = ±wj (x) − dz, x > 1, (2.26) πi 1 z − x where the Cauchy principal-value of the integral is taken. Next the Fredholm integral of the second kind whose solution yields wi (x) is derived. Substitution of (2.17) into (2.5) yields p0 = κR ,
(u01 , u02 ) =
y
y
(κR − ξ )[vi (ξ ) + ui (ξ ) + Cui (ξ − y )−1 ] + Vi (ξ, γ1 , γ2 ) + CVi (ξ, γ1 , γ2 ) 2 1 ∞ − Vij (ξ, γ1 , γ2 , z)wj (z) dz = 0, i = 1, 2, 2πi 1 go
go
(2.27)
j =1
where Vij (ξ, z1 , z2 , z) =
2
Bik (ξ, z1 , z2 )Qkj (ξ, z1 , z2 )G(ξ )
k=1 go Vi (ξ, z1 , z2 )
y
=
Vi (ξ, z1 , z2 ) =
2 2 j =1 k=1
2 j =1
H (ξk ) , z + ξk
(2.28)
go
Bik (ξ, z1 , z2 )Qkj (ξ, z1 , z2 )G(ξ )H (ξk )uj (ξk ),
(2.29)
y
Vij (ξ, z1 , z2 , −y )uj ,
(2.30)
G(ξ ) = [2(κ 2 − 1)(ξ − y )K + (−ξ )]−1 , H (ξ ) =
(2.31)
(ξ − y ) R + ξ)
(2.32)
K + (ξ )(κ
and ξi is given by (2.13). Evaluation of (2.27) of ξ = κR yields y
go
Vi (κR , γR1 , γR2 )C = u0i − Vi (κR , γR1 , γR2 ) 2 1 ∞ + Vij (κR , γR1 , γR2 , z)wj (z) dz, 2πi 1
i=1
or 2,
(2.33)
j =1
where γRj = γj (κR ). The value C given by (2.33) is, of course, the same for i = 1 or 2, independently of the values of wj (z). The difference of (2.27) evaluated at ξ = −x + i0 and ξ = −x − i0, x > 1 yields the following integral equation: 2 1 ∞ go go y (κR + x)(wi (x) + Ui (x)) − Wij (x, z)wj (z) dz + Wi (x) + CWi (x) = 0, i = 1, 2, x > 1, 2πi 1 j =1
(2.34)
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A.K. Gautesen / Wave Motion 36 (2002) 417–424
where with zj = γj (x), s = sgn (x − κ) and Wij (x, z) = Vij (−x, z1 , z2 , z) − Vij (−x, −sz1 , −z2 , z),
(2.35)
Wiα (x) = Viα (−x, z1 , z2 ) − Viα (−x, −sz1 , −z2 ),
(2.36)
go
Ui (x) = −
M m=0
um i F (x, pm ) −
N n=1
α = go, y
vin F (x, qn ),
(2.37)
2 (−1)j (r3−j − p) γ2 (x) 1 . + F (x, p) = γ2 (p) x + p (r2 − r1 )(x + rj )
(2.38)
j =1
After substitution for C from (2.33), (2.34) becomes a coupled pair of Fredholm integral equations of the second kind with continuous kernels. It is numerically stable. The amplitude of diffracted surface wave is given by the ratio of the residues of u¯ i (ξ ) at ξ = −κR and ξ = κR , Rc + (−1) Tc =
(−1)j u+ j u0j
,
j =1
or
2,
(2.39)
where Rc and Tc denote the reflection and transmission coefficients for Rayleigh waves, u0j is defined by (2.24) and u± ¯ j (−x ± i0)], j = lim x→κR [(κR − x)u
j = 1, 2.
(2.40)
Unfortunately by (2.26) vj (−κR + i0) contains a Cauchy principal-value integral. The sum and difference of (2.16) evaluated at ξ = −x + i0 and ξ = −x − i0, followed by multiplication by (κR − x) yields 2κR γ1 (κR ) 3−2j + + − uj + uj = − (u3−j − u− j = 1, 2. (2.41) 3−j ), a(κR ) When (2.41) is used in (2.39) with j being either 1 or 2 to eliminate the Cauchy principal-value integral one finds κ 2 K + (κR )(y + κR ) 2κR γ1 (κR ) go go Rc + (−1) Tc = (w2 (κR ) + U2 (κR )) − (w1 (κR ) + U1 (κR )) . 2γ1 (κR )K + (−κR )(y − κR ) a(κR ) (2.42) Finally the remaining constants in (2.21) are defined. For k = 0, 1, ... , M/2 − 1 and j = 1, 2 pj +2k = pm cos α − γj (pm ) sin α, j +2k
ui
(2.43)
= −Dijm (pj +2k , z1 , z2 ),
(2.44)
where zn = γn (p2j +k ) and m = k[(j − 1)(5 − 3k) + 2]/2. For k = 0, . . . , N/2 − 1 and j = 1, 2 qj +2k = pm cos α − γj (pm ) sin α, j +2k
vi
(2.45)
= Dijm (qj +2k , sz1 , −z2 ),
(2.46)
where zn = γn (q2j +k ), m = j − 2kj + 3k and s = −(−1)k(j −1) . Here, Dijm (p, z1 , z2 ) = (κR − p)−1 zj (zj cos α − p sin α)−1 Bij (p, z1 , z2 )G(p)H (ξj )
2 n=1
Qjn (p, z1 , z2 )um n, (2.47)
where ξj = p cos α + zj sin α.
A.K. Gautesen / Wave Motion 36 (2002) 417–424
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3. Results and discussion An incident Rayleigh surface wave can be considered to be comprised of a plane longitudinal wave and a plane transverse wave with complex angles of incidence. Depending on the wedge angle, these waves can experience multiple re-reflections. Each reflection corresponds to a pole at ξ = pj in the Fourier transform of the geometrical optics field defined by (2.21). The number of such poles increases rapidly as the wedge angle decreases, e.g., there are over 4000 poles to be considered at a wedge angle of 16◦ . Therefore, to keep the number of poles down, the wedge angle has been taken to be greater than 63◦ . We remark that as the wedge angle decreases through 90◦ , the value of M (the number of poles) in (2.21) jumps from 2 to 6, while the value of N (extraneous poles) goes from 0 to 4. When the geometrical optics field is taken to have this form, the integral equation (2.34) for wj (x) remains the same whether the wedge angle approaches 90◦ from above or below. Once (2.34) has been solved for the antisymmetric ( = 1) and symmetric ( = 2) problems, the reflection and transmission coefficients are computed from (2.42). In Fig. 3 are plotted the amplitude and phase (◦ ) of the reflection and transmission coefficients versus wedge angle (◦ ) for Poisson’s ratio ν = 1/4 and 1/3. For a 90◦ wedge, the results are in very good agreement with those of Li et al. [4]. Very good agreement is also found with the results of Fujii [5] except for the phase of the reflection coefficient for wedge angles between 165◦ and 180◦ . Over this range of wedge angles, the amplitude of the reflection coefficient is less that 0.015. When these results are computed for a Poisson’s ratio of 0.294, very good agreement is found with the results given by Budaev and Bogy [8] except for the phase of the reflection coefficient. Note that the phase of the reflection and transmission coefficients defined in Fujii [5] and Budaev and Bogy [8] is the negative of that used here. We remark that the complex singularities of the displacements at the corner found by Karp and Karal [10] do not play an important part in the numerics. These singularities, which appear at infinity in the Fourier transform domain, are in agreement with the asymptotic solution to (2.34). Also, should a pole be omitted in the geometric optics field (2.21), the results do not conserve energy.
Fig. 3. Amplitude and phase (◦ ) of the reflection and transmission coefficients versus wedge angle (◦ ) for Poisson’s ratio ν = 1/4 and 1/3.
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A.K. Gautesen / Wave Motion 36 (2002) 417–424
Acknowledgements Ames Laboratory is operated for the US Department of Energy by Iowa State University under contract number W-7405-ENG-82. This research was supported in part by the Office of Basic Energy Science, Applied Mathematical Sciences Division. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
A.K. Gautesen, Scattering of a Rayleigh wave by an elastic quarter space: revisited, Wave Motion 35 (2002) 99–106. A.K. Gautesen, Scattering of a Rayleigh wave by an elastic wedge, Wave Motion 9 (1987) 51–59. L. Knopoff, Elastic wave propagation in a wedge, in: J. Miklowitz (Ed.), Wave Propagation in Solids, AMSE, New York, 1969, pp. 3–42. Z.L. Li, J.D. Achenbach, I. Komsky, Y.C. Lee, Reflection and transmission of obliquely incident surface waves by an edge of a quarter space: theory and experiment, ASME J. Appl. Mech. 59 (1992) 349–355. K. Fujii, Rayleigh-wave scattering at various corners: investigation in the wider range of wedge angles, Bull. Seism. Soc. Am. 84 (1994) 1916–1924. B.V. Budaev, D.B. Bogy, Rayleigh wave scattering by a wedge, Wave Motion 22 (1995) 239–257. B.V. Budaev, D.B. Bogy, Rayleigh wave scattering by a wedge II, Wave Motion 24 (1996) 307–314. B.V. Budaev, D.B. Bogy, Scattering of Rayleigh and Stonely waves by two adhering elastic wedges, Wave Motion 33 (2001) 321–337. A.K. Gautesen, Scattering of a Rayleigh wave by an elastic wedge whose angle is greater that 180◦ , ASME J. Appl. Mech. 68 (2001). S.N. Karp, F.C. Karal, The elastic-field behavior in the neighborhood of a crack of arbitrary angle, Commun. Pure Appl. Math. 15 (1962) 413–421.
Scattering of a Rayleigh wave by an elastic wedge whose angle is less than 180◦夽 A.K. Gautesen∗ Department of Mathematics and Ames Laboratory, 136 Wilhelm Hall, Iowa State University, Ames, IA 50011, USA Received 8 September 2000; received in revised form 10 May 2001; accepted 10 May 2001
Abstract The steady-state problem of scattering of an incident Rayleigh wave by an elastic wedge whose angle is less than 180◦ is considered. The problem is reduced to the numerical solution of a pair of Fredholm integral equations of the second kind whose kernels are continuous functions. Numerical results are given for the amplitude and phase of the Rayleigh waves transmitted and reflected by the corner. Published by Elsevier Science B.V.
1. Introduction In [1], the author develops a new method to consider scattering of waves by a 90◦ elastic wedge. Here a slight modification of this method is used to study wedges whose wedge angle α is less than 180◦ . The incident wave is taken to be a Rayleigh surface wave. The method of solution also works for incidence of a plane longitudinal or transverse wave. Numerical results are given for the amplitude and phase of the Rayleigh waves transmitted and reflected by the corner for wedge angles between 63◦ and 180◦ and Poisson’s ratio ν = 1/4 and 1/3. Below this angle, the problem is complicated by multiple re-reflections of the incident wave. In principle this can be accounted for, but has not been done here. The results presented here are in general agreement, but more accurate, than the results [2] obtained by the author using a different method of solution. A review of early efforts to deal with this problem is given by Knopoff [3]. Li et al. [4] use a free-space Green’s function representation of the tractions to obtain boundary integral equations which they solve numerically. In an extension of his earlier work Fujii [5] gives results for a wide range of wedge angles. Budaev and Bogy [6,7] use the Sommerfeld–Malivzhinetz method to solve this problem. They give corrected numerical results in [8]. In [9], the author treats the case of wedge angles greater than 180◦ by a method similar to that used here. The method of solution is quite simple. First the problem is symmetrized to reduce the number of the unknown displacements on the traction-free surfaces from 4 to 2. Of course, the problem must then be solved twice— once for the symmetric case and once for the antisymmetric case. From a free-space Green’s function integral representation of the displacements the tractions, dilatation and rotation are computed on a line just below the x1 -axis 夽
It is with great pleasure that I dedicate this paper to my friend and colleague Dr. Jan Achenbach who has made so many contributions to the field of elastodynamics. ∗ Tel.: +1-515-294-9312; fax: +1-515-294-4491. E-mail address: [email protected] (A.K. Gautesen). 0165-2125/02/$ – see front matter. Published by Elsevier Science B.V. PII: S 0 1 6 5 - 2 1 2 5 ( 0 2 ) 0 0 0 3 3 - 1
418
A.K. Gautesen / Wave Motion 36 (2002) 417–424
Fig. 1. Incident, transmitted and reflected surface waves.
Fig. 2. Branch cut and location of the Rayleigh poles of u¯ j (ξ ).
(x2 = 0)—see Fig. 1. Next the Fourier transform of these quantities is taken, yielding functional equations where the unknowns are the Fourier transform of the displacements u¯ j (ξ ) on the traction-free surface x2 = 0, x1 > 0. Then a suitable representation of u¯ j (ξ ) is taken. The branch cut and Rayleigh poles of u¯ j (ξ ) are shown in Fig. 2. A pair of numerically stable Fredholm integral equations of the second kind is achieved by computing the jump across the branch cut of u¯ j (ξ ). The kernels of these integral equations are continuous, and consist of elementary functions and the Wiener–Hopf product factorization of the Rayleigh function. In the next section, the governing Fredholm integral equations are derived. In the last section, the numerical results are presented and discussed.
2. Derivation of the integral equations The diffraction of steady-state waves by an elastic wedge whose wedge angle α is less than 180◦ is considered. The location of the wedge; the coordinate system; and the incident, transmitted and reflected surface waves are shown in Fig. 1. By using the free-space Green’s stress tensor, an integral representation of the total displacements uk (x1 , x2 ) is given by 2 ∞ uk (x1 , x2 )H (x) = uin τ G (x − y, x2 )ui (y, 0) k (x1 , x2 ) + i2;k 1 0 i=1 2 G + τij;k (x1 − t1 y, x2 − t2 y)nj ui (t1 y, t2 y) dy, k = 1, 2. (2.1) j =1
Here, H (x) = 1 if (x1 , x2 ) is a point in the elastic material and H (x) = 0 otherwise. The incident field is denoted in by uin k (x1 , x2 ) and since the incident field is a Rayleigh surface wave, uk (x1 , x2 ) = 0. The free-space Green’s stress
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G (x , x ). The normal n and tangent t to the upper traction-free surface are given by tensor is denoted by τij;k 1 2 j j
n1 = t2 = sin α,
n2 = −t1 = − cos α.
(2.2)
To reduce the number of unknowns the problem is divided into an antisymmetric ( = 1) and symmetric ( = 2) problem. For the antisymmetric (symmetric) problem the displacements are antisymmetric (symmetric) with respect to the line x2 = x1 tan 1/2α. This implies that u1 (y, 0) = (−1)
2
tj uj (t1 y, t2 y),
y > 0, = 1, 2,
(2.3)
nj uj (t1 y, t2 y),
y > 0, = 1, 2.
(2.4)
j =1
u2 (y, 0) = (−1)
2 j =1
Solving (2.3) and (2.4) for uj (t1 y, t2 y) and substituting the result into (2.1) reduces the number of unknowns by one half, but then the problem must be solved twice (once for = 1 and once for = 2). The importance of (2.1) is that it holds for all points (x1 , x2 ) and not only just for points lying within the elastic material. The normal and shear tractions vanish on the entire line x2 = 0, −∞ < x1 < ∞. These quantities are computed from (2.1) and then the Fourier transform of the resulting equations are taken to obtain the following equations: Uj (ξ ) R(ξ )u¯ i (ξ ) + Bij (ξ, γ1 , γ2 ) = 0, γi γi 2
i = 1, 2,
(2.5)
j =1
where u¯ i (ξ ) which denotes the Fourier transform of ui (x1 , 0) is defined by ∞ u¯ i (ξ ) = kL exp[ikL ξ x1 ]ui (x1 , 0) dx1 0
and kL denotes the wavenumber corresponding to longitudinal waves. The Rayleigh function R(ξ ) is defined by R(ξ ) = a 2 (ξ ) + 4ξ 2 γ1 γ2 ,
(2.6)
where γ1 (ξ ) = (κ + ξ )1/2 (κ − ξ )1/2 ,
κ=
cL , cT
(2.7)
γ2 (ξ ) = (1 + ξ )1/2 (1 − ξ )1/2 ,
(2.8)
a(ξ ) = κ 2 − 2ξ 2 .
(2.9)
It has the property that R(κR ) = 0, κR = cL /cR . The speed of longitudinal, transverse and Rayleigh waves have been denoted by cL , cT , and cR , respectively. In (2.5) and below, the summation convention is not used. The remaining quantities in (2.5) are defined by Bii (ξ, z1 , z2 ) = (−1) a(ξ ), +j
Bij (ξ, z1 , z2 ) = (−1) Ui (ξ ) =
2
2ξ zi ,
(2.10) i = j,
Qij (ξ, z1 , z2 )u¯ j (ξi ),
zi = γi (ξ ),
(2.11) (2.12)
j =1
ξi = ξ cos α + zi sin α,
(2.13)
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Qii (ξ, z1 , z2 ) = (−1)i a(ξi ), Qij (ξ, z1 , z2 ) = 2ξi (ξ sin α − zi cos α),
(2.14) i = j.
(2.15)
The rotation and dilatation vanish on the line x2 = −δ 2 , −∞ < x1 < ∞. Computing these quantities from (2.1), followed by taking the Fourier transform and the limit as δ → 0 leads to the following equation: 2
(−1)i+j Bij (ξ, γ1 , γ2 )u¯ j (ξ ) + Ui (ξ ) = 0,
i = 1, 2.
(2.16)
j =1
The branches and Rayleigh poles of u¯ i (ξ ) are shown in Fig. 2. The following representation of u¯ i (ξ ) is taken
ξ − y go y u¯ i (ξ ) = [Cui (ξ − y )−1 + ui (ξ ) + vi (ξ )]. (2.17) + K (ξ )(κR + ξ ) Here, K + (ξ ) is the Wiener–Hopf product factorization of K(ξ ) = R(ξ )/[2(κ 2 − 1)(κR2 − ξ 2 )] and is given by 2 2 1/2
1 κ + −1 4t γ1 (t)(t − 1) (t + ξ )−1 dt. tan (2.18) log K (ξ ) = − π 1 a 2 (t) This quantity is easily evaluated to 16 digit accuracy using a 32 point Gaussian quadrature. The pole at ξ = −κR yields the diffracted surface waves. The multiplicative factor in the right side of (2.17) approaches 1 as ξ approaches infinity. When = 1, α y y y = κ cos , (u1 , u2 ) = (− sin α, cos α) (2.19) 2 and when = 2, α , y = cos 2
y
y
(u1 , u2 ) = (κ 2 − 1 − cos α, − sin α).
(2.20)
These choices ensure that u¯ j (ξ ) satisfies (2.16) with i = . The constant C is determined below. go The term ui (ξ ) is defined by go
ui (ξ ) =
M m=0
[(ξ − pm )−1 + F + (ξ, pm )]um i +
N m=1
F + (ξ, qm )vim ,
where for rj = κexp[iπ(3 − 2j )/4], j = 1, 2 2 + + + + (r3−j − p)(γ2 (ξ ) − γ2 (rj )) 1 γ2 (ξ ) − γ2 (p) F + (ξ, p) = (−1)j + 2γ2 (p) ξ −p (r2 − r1 )(ξ − rj )
(2.21)
(2.22)
j =1
and γ2+ (ξ ) denotes the Wiener–Hopf sum splitting of γ2 (ξ ), γ2+ (ξ ) = −2iγ2 (ξ )
log[2−1/2 ((1 + ξ )1/2 + i(1 − ξ )1/2 )] . π
(2.23)
Here pm are the poles associated with the incident (m = 0), reflected (m = 1, 2) and re-reflected (m ≥ 3) waves. The wedge angle α is taken to be sufficiently large that re-re-reflected waves do not occur. Otherwise M (as well as N) increases rapidly. In particular M = 2, if α > 90◦ and M = 6 if α ≤ 90◦ , while N = 4, 100◦ > α > 90◦ and N = 0 otherwise. The second sum in (2.21) is included to remove the poles in the analytical continuation of the integral equation derived below which are near the contour of integration. When (2.21) is substituted for u¯ i (ξ )
A.K. Gautesen / Wave Motion 36 (2002) 417–424
421
in (2.5), there are no poles at ξ = pm . For m = 0, (2κR γ1 (κR ), a(κR )) . (2.24) κ2 Since the definitions of the remaining constants in (2.21) are not needed in the present discussion, they are given at the end of this section. The last term in (2.17) is taken to be ∞ wj (z) 1 vj (ξ ) = − dz (2.25) 2πi 1 ξ + z and gives the remaining contribution to u¯ i (ξ ) from the branch cut. Note that ∞ wj (z) 1 2vj (−x ± i0) = ±wj (x) − dz, x > 1, (2.26) πi 1 z − x where the Cauchy principal-value of the integral is taken. Next the Fredholm integral of the second kind whose solution yields wi (x) is derived. Substitution of (2.17) into (2.5) yields p0 = κR ,
(u01 , u02 ) =
y
y
(κR − ξ )[vi (ξ ) + ui (ξ ) + Cui (ξ − y )−1 ] + Vi (ξ, γ1 , γ2 ) + CVi (ξ, γ1 , γ2 ) 2 1 ∞ − Vij (ξ, γ1 , γ2 , z)wj (z) dz = 0, i = 1, 2, 2πi 1 go
go
(2.27)
j =1
where Vij (ξ, z1 , z2 , z) =
2
Bik (ξ, z1 , z2 )Qkj (ξ, z1 , z2 )G(ξ )
k=1 go Vi (ξ, z1 , z2 )
y
=
Vi (ξ, z1 , z2 ) =
2 2 j =1 k=1
2 j =1
H (ξk ) , z + ξk
(2.28)
go
Bik (ξ, z1 , z2 )Qkj (ξ, z1 , z2 )G(ξ )H (ξk )uj (ξk ),
(2.29)
y
Vij (ξ, z1 , z2 , −y )uj ,
(2.30)
G(ξ ) = [2(κ 2 − 1)(ξ − y )K + (−ξ )]−1 , H (ξ ) =
(2.31)
(ξ − y ) R + ξ)
(2.32)
K + (ξ )(κ
and ξi is given by (2.13). Evaluation of (2.27) of ξ = κR yields y
go
Vi (κR , γR1 , γR2 )C = u0i − Vi (κR , γR1 , γR2 ) 2 1 ∞ + Vij (κR , γR1 , γR2 , z)wj (z) dz, 2πi 1
i=1
or 2,
(2.33)
j =1
where γRj = γj (κR ). The value C given by (2.33) is, of course, the same for i = 1 or 2, independently of the values of wj (z). The difference of (2.27) evaluated at ξ = −x + i0 and ξ = −x − i0, x > 1 yields the following integral equation: 2 1 ∞ go go y (κR + x)(wi (x) + Ui (x)) − Wij (x, z)wj (z) dz + Wi (x) + CWi (x) = 0, i = 1, 2, x > 1, 2πi 1 j =1
(2.34)
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where with zj = γj (x), s = sgn (x − κ) and Wij (x, z) = Vij (−x, z1 , z2 , z) − Vij (−x, −sz1 , −z2 , z),
(2.35)
Wiα (x) = Viα (−x, z1 , z2 ) − Viα (−x, −sz1 , −z2 ),
(2.36)
go
Ui (x) = −
M m=0
um i F (x, pm ) −
N n=1
α = go, y
vin F (x, qn ),
(2.37)
2 (−1)j (r3−j − p) γ2 (x) 1 . + F (x, p) = γ2 (p) x + p (r2 − r1 )(x + rj )
(2.38)
j =1
After substitution for C from (2.33), (2.34) becomes a coupled pair of Fredholm integral equations of the second kind with continuous kernels. It is numerically stable. The amplitude of diffracted surface wave is given by the ratio of the residues of u¯ i (ξ ) at ξ = −κR and ξ = κR , Rc + (−1) Tc =
(−1)j u+ j u0j
,
j =1
or
2,
(2.39)
where Rc and Tc denote the reflection and transmission coefficients for Rayleigh waves, u0j is defined by (2.24) and u± ¯ j (−x ± i0)], j = lim x→κR [(κR − x)u
j = 1, 2.
(2.40)
Unfortunately by (2.26) vj (−κR + i0) contains a Cauchy principal-value integral. The sum and difference of (2.16) evaluated at ξ = −x + i0 and ξ = −x − i0, followed by multiplication by (κR − x) yields 2κR γ1 (κR ) 3−2j + + − uj + uj = − (u3−j − u− j = 1, 2. (2.41) 3−j ), a(κR ) When (2.41) is used in (2.39) with j being either 1 or 2 to eliminate the Cauchy principal-value integral one finds κ 2 K + (κR )(y + κR ) 2κR γ1 (κR ) go go Rc + (−1) Tc = (w2 (κR ) + U2 (κR )) − (w1 (κR ) + U1 (κR )) . 2γ1 (κR )K + (−κR )(y − κR ) a(κR ) (2.42) Finally the remaining constants in (2.21) are defined. For k = 0, 1, ... , M/2 − 1 and j = 1, 2 pj +2k = pm cos α − γj (pm ) sin α, j +2k
ui
(2.43)
= −Dijm (pj +2k , z1 , z2 ),
(2.44)
where zn = γn (p2j +k ) and m = k[(j − 1)(5 − 3k) + 2]/2. For k = 0, . . . , N/2 − 1 and j = 1, 2 qj +2k = pm cos α − γj (pm ) sin α, j +2k
vi
(2.45)
= Dijm (qj +2k , sz1 , −z2 ),
(2.46)
where zn = γn (q2j +k ), m = j − 2kj + 3k and s = −(−1)k(j −1) . Here, Dijm (p, z1 , z2 ) = (κR − p)−1 zj (zj cos α − p sin α)−1 Bij (p, z1 , z2 )G(p)H (ξj )
2 n=1
Qjn (p, z1 , z2 )um n, (2.47)
where ξj = p cos α + zj sin α.
A.K. Gautesen / Wave Motion 36 (2002) 417–424
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3. Results and discussion An incident Rayleigh surface wave can be considered to be comprised of a plane longitudinal wave and a plane transverse wave with complex angles of incidence. Depending on the wedge angle, these waves can experience multiple re-reflections. Each reflection corresponds to a pole at ξ = pj in the Fourier transform of the geometrical optics field defined by (2.21). The number of such poles increases rapidly as the wedge angle decreases, e.g., there are over 4000 poles to be considered at a wedge angle of 16◦ . Therefore, to keep the number of poles down, the wedge angle has been taken to be greater than 63◦ . We remark that as the wedge angle decreases through 90◦ , the value of M (the number of poles) in (2.21) jumps from 2 to 6, while the value of N (extraneous poles) goes from 0 to 4. When the geometrical optics field is taken to have this form, the integral equation (2.34) for wj (x) remains the same whether the wedge angle approaches 90◦ from above or below. Once (2.34) has been solved for the antisymmetric ( = 1) and symmetric ( = 2) problems, the reflection and transmission coefficients are computed from (2.42). In Fig. 3 are plotted the amplitude and phase (◦ ) of the reflection and transmission coefficients versus wedge angle (◦ ) for Poisson’s ratio ν = 1/4 and 1/3. For a 90◦ wedge, the results are in very good agreement with those of Li et al. [4]. Very good agreement is also found with the results of Fujii [5] except for the phase of the reflection coefficient for wedge angles between 165◦ and 180◦ . Over this range of wedge angles, the amplitude of the reflection coefficient is less that 0.015. When these results are computed for a Poisson’s ratio of 0.294, very good agreement is found with the results given by Budaev and Bogy [8] except for the phase of the reflection coefficient. Note that the phase of the reflection and transmission coefficients defined in Fujii [5] and Budaev and Bogy [8] is the negative of that used here. We remark that the complex singularities of the displacements at the corner found by Karp and Karal [10] do not play an important part in the numerics. These singularities, which appear at infinity in the Fourier transform domain, are in agreement with the asymptotic solution to (2.34). Also, should a pole be omitted in the geometric optics field (2.21), the results do not conserve energy.
Fig. 3. Amplitude and phase (◦ ) of the reflection and transmission coefficients versus wedge angle (◦ ) for Poisson’s ratio ν = 1/4 and 1/3.
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A.K. Gautesen / Wave Motion 36 (2002) 417–424
Acknowledgements Ames Laboratory is operated for the US Department of Energy by Iowa State University under contract number W-7405-ENG-82. This research was supported in part by the Office of Basic Energy Science, Applied Mathematical Sciences Division. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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