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Comment on “Scattering of elastic waves by a movable rigid sphere embedded in an infinite elastic solid”
Journal o f Sound and Vibration (1974) 33(3), 372-375
COMMENT ON "SCATTERING OF ELASTIC WAVES BY A MOVABLE RIGID SPHERE EMBEDDED IN AN INFINITE ELASTIC SOLID" [1] The inverse problem of determining the strength and profile of an incident elastic wave in an isotropic medium, caused by the motion of a rigid spherical inclusion, is one of a class of analogous problems that has interested geophysicists and mechanical engineers. Mow [2] effectively worked out the problem for the case of an incident P-wave and the relationship between the Fourier transform of the motion of the sphere and the strength of the incident wave was found to be simple enough to invert easily in closed form. It would seem that Iwashimizu's very concise paper should include Mow's result as well as the analogous result for an incident S-wave. However, Mow's result is not derivable from the third of equations (28) in reference [1]. In fact, equations (28) and (29) do not seem to follow from equations (26) and (27). (There is a typographical error in equation (27).) Equally important, the expressions in equations (28) and (29) in this paper are too complicated to recognize, by inspection, the behaviour of the sphere. It might be useful, therefore, to derive the Fourier transforms of the motion functions in the S-wave case in a form simple enough to invert easily. The two types of incident waves striking a sphere of radius a to be considered are, respectively, a plane shear wave with front perpendicular to the z-axis and a spherical wave produced by a center of rotation lying on the z-axis at (0,0,-ro). In Cartesian co-ordinates, these incident disturbances are a t = ( O , f ( t - (z + a)/c2),O), where f ( t ) = 0 for t <~0, for the plane wave, and a l = (O,O[Oz,-O[Oy)(l[Rg(t + c - p[c2)), where c = (ro - a)[c2, R = (x 2 + y2 + (z + to)2) 1/2, and c2 is the shear wave velocity. If U . Uz are the displacement components of the sphere, 9 the rotation about the x-axis, fis the scattered field, and (*) indicates the Fourier transform of the starred functions, these problems are completely described by (V 2 + k22)fi~* = (1 - k 2 / k ~ ) V(V. fi~*),
outside r = a
(1)
9the continuity conditions at r = a, u* = U* cos 0 + U* sin q5sin O,
(2)
u* = - U * sinO+ U* sin~ s i n O - cb* a sin 0 sin ~ cos ~b,
(3)
u~ = U* cos ~b- ~b* a sin 0 cos 2 ~,
(4)
the dynamical conditions at r = a, -o9 2 M U * = I f (T*,cos 0 - z* sin O) a 2 sin 0 dO d~b,
(5)
r~a
- o 9 2 M U * = f f (z~sinOsin~ + ~**cos q5 + "r*ocosOsingp) a2sinOdOdc~,
(6)
r~a
- w z I ~ * = - f f (x*0sintk + z*ocos ~ cosO)a3sinOdOdc~, l)=tl
372
(7)
373
LETTERS T O T H E E D I T O R
the Sommerfeld radiation condition and ~. = ~ t , + ~s,.
(8)
M and I are the mass and moment of inertia of the sphere about the x-axis; 2,/t and p are, respectively, the Lam6 constants and density of the medium; the functions z are the components of the stress tensor in spherical co-ordinates, and kl and k2 are, respectively, the compressional and shear wave numbers. The method of solution is straightforward (see, in addition, the paper by Eringen [3]). The vector fi~* can be written in terms of three scalar functions, ~'1, lP2 and ~b3, as ~s, = V(ar
-
v • v • (a~2~)
-
v x (~3~),
where r is the radius vector. These functions, as solutions of reduced wave equations, can be expanded in spherical surface harmonics. When the resulting expansions are substituted into equation (8) and then into the continuity conditions (2)-(4) the coefficients of the series for fis* can be determined. These will contain the functions of motion U*, U*, and tb*.When, finally, the stress tensor, T*, is computed from ~*, these motion functions can be found from the dynamical conditions (5)-(7). If one normalizes "time" and "length" so that a = 1 and c, = 1 (where c, is the compressional wave velocity of the medium), and sets co = iv and K = P/Po, the ratio of the density of the medium to that of the sphere, the tedious calculations outlined in the previous paragraph will yield, for the plane S-wave problem, the Laplace transforms of the motion functions: L(Uy) = 3 L ( f ) c~{(v + I) + e-Vt*"(v2 + 2v + 2)(cosh(v/c2) + sinh(v[c2))} + + {v" + (2 + c2 + K(I + 2c2)) va + (2 + c 2 + K(I + 2c22+ 9c2)) v2 + + 9Kc2(1 + c2) v + 9Kc2~}, or, more compactly, L(U,) = L(f) [Px(v) + e -2"/~2P2(v)]/P3(v), for polynomials P~, Pz and P3, and 15 K 2c L(O) = ~ ~-~4 2 L(f) e-wc~ {(v3 - 2c~ v - 2c~) sinh (v/c2)+ (v3 + 2c2 v 2 + 2c~ v) cosh (v/c2)} or
LfD L(q)) = - - ~ ( P a ( v ) + e -2./c* Ps(v)). Also U, = 0. It is easy to show that the roots of the fourth degree polynomial,/'3, all have negative real parts (by applying the Routh-Hurwitz stability criterion, for example). Once these roots are determined for a particular medium, both Ur and q, can be found by an easy application of the Convolution Theorem (provided, of course, that f decays sufficiently rapidly So that the inverse transforms exist). Similar results are obtainable for the center of rotation problem with different polynomials in place of P x , / 2 , P , and Ps, and the function g in place o f f The rate of decay of Ur is governed in both cases by the roots of P3. If all the spherical Bessel and Hankel functions that occur in Iwashimizu's results are written out, the same denominator appears in his formulas (28).
Department of 3lathematics, University of Connecticut, Storrs, Connecticut 06268, U.S.A. (Received 22 October 1973)
R . D . SIDMAN
374
LEt lt:RS TO THE EDITOR REFERENCES
1. Y. IwAsm~zu 1972 Journal of Sound and Vibration 21, 463-469. Scattering of elastic waves by a movable rigid sphere embedded in an infinite elastic solid. 2. C. C. Mow 1966 Journal of Applied Mechanics 33, 807-813. On the transient motion of a rigid spherical inclusion in an elastic medium and its inverse problem. 3. A. C-'EMALERINGrN 1957 Quarterly Journal of Mechanics and Applied Mathematics, X, 257-270. Elasto-dynamic problem concerning the spherical cavity.
AUTHOR'S REPLY, AND ERRATA In the preceding letter [1] R. D. Sidman has commented on a paper of the author [2]. He states that in reference [2], equations (28) and (29) do not seem to follow from equations (26) and (27) and, moreover, equation (28)3 is not coincident with Mow's result [3]. (Although he refers to reference [3], equation (28)3 should be compared with equation (I 8) in the previous paper of C. C. Mow [4].) However, this is irrelevant. Since the derivation of equations (28) and (29) is not difficult, only the equivalence of equation (28)3 and Mow's result will be shown below. By equation (19)t and the relation jr(z) -- (sinz - zcosz)/z 2 equation (28)3 becomes
.j., (i)
[-6ih~(ka)/(Ka) 2 + At(a)G,{j~(Ka), ht(ka))ht(Ka)/Ka - 3A~(a)j~(Ka)/Ka]
From equations (17) and (19)2, one has t
Al(a) --- 2hl(ra) hx(ka) - a
dhl(Ka) d da. da {ahx(ka)} =
= kaht(Ka ) ha(ka) + 2Kah2(Ka) hl(ka) - (Ka) (ka) h2(Ka) h2(ka), a~(a)G,{j~(Ka),h,(ka)} = 3 [2j,(Ka)ht(ka)-
..... ]]= a djt(Ka) &d .an,tKaj
= 3[kajx(Ka) h,(ka) + 2Kaj2(Ka) hl(ka) - (Ka)(ka)j2(Ka) h2(ka)]. When these are substituted into the above equation (1), it can be written as
= (~, x[A) [-6ihl(ka)[(Ka)" + 6ht(ka){j2(Ka)hl(Ka) - J a (Ka)h2(Ka)} - 3kah,(ka) {j,(Ka) h l(Ka) - j~(Ka) ha(Ka))],
(2)
where
A = --2(1 -- r.)Kah2(Ka)ht(ka) - (l - g)kaht(Ka)h,(ka) + (Ka)(ka)h2(Ka)h2(ka).
(3)
The above equation (2) can be simplified to
r
- 3 i x [ k~
(4)
by u s e of the relation j,_t(z)h,(z)- h,_t(z)j,(z)=-i]z 2. Since ~:, K, k, and k / K are, respectively, denoted by ~, ~,/~, and k in reference [4], A newly defined by equation (3) is the s a m e as A 1 defined by equation (17) in reference [4]. Moreover, it is seen from the definitions that q'o and [/in reference [4] are A[iKand A~e-l~t, respectively. Thus the above equation (4), i.e., equation (28)3, is proved to be identical with Mow's result: i.e., equation (18) in reference
[4].
COMMENT ON "SCATTERING OF ELASTIC WAVES BY A MOVABLE RIGID SPHERE EMBEDDED IN AN INFINITE ELASTIC SOLID" [1] The inverse problem of determining the strength and profile of an incident elastic wave in an isotropic medium, caused by the motion of a rigid spherical inclusion, is one of a class of analogous problems that has interested geophysicists and mechanical engineers. Mow [2] effectively worked out the problem for the case of an incident P-wave and the relationship between the Fourier transform of the motion of the sphere and the strength of the incident wave was found to be simple enough to invert easily in closed form. It would seem that Iwashimizu's very concise paper should include Mow's result as well as the analogous result for an incident S-wave. However, Mow's result is not derivable from the third of equations (28) in reference [1]. In fact, equations (28) and (29) do not seem to follow from equations (26) and (27). (There is a typographical error in equation (27).) Equally important, the expressions in equations (28) and (29) in this paper are too complicated to recognize, by inspection, the behaviour of the sphere. It might be useful, therefore, to derive the Fourier transforms of the motion functions in the S-wave case in a form simple enough to invert easily. The two types of incident waves striking a sphere of radius a to be considered are, respectively, a plane shear wave with front perpendicular to the z-axis and a spherical wave produced by a center of rotation lying on the z-axis at (0,0,-ro). In Cartesian co-ordinates, these incident disturbances are a t = ( O , f ( t - (z + a)/c2),O), where f ( t ) = 0 for t <~0, for the plane wave, and a l = (O,O[Oz,-O[Oy)(l[Rg(t + c - p[c2)), where c = (ro - a)[c2, R = (x 2 + y2 + (z + to)2) 1/2, and c2 is the shear wave velocity. If U . Uz are the displacement components of the sphere, 9 the rotation about the x-axis, fis the scattered field, and (*) indicates the Fourier transform of the starred functions, these problems are completely described by (V 2 + k22)fi~* = (1 - k 2 / k ~ ) V(V. fi~*),
outside r = a
(1)
9the continuity conditions at r = a, u* = U* cos 0 + U* sin q5sin O,
(2)
u* = - U * sinO+ U* sin~ s i n O - cb* a sin 0 sin ~ cos ~b,
(3)
u~ = U* cos ~b- ~b* a sin 0 cos 2 ~,
(4)
the dynamical conditions at r = a, -o9 2 M U * = I f (T*,cos 0 - z* sin O) a 2 sin 0 dO d~b,
(5)
r~a
- o 9 2 M U * = f f (z~sinOsin~ + ~**cos q5 + "r*ocosOsingp) a2sinOdOdc~,
(6)
r~a
- w z I ~ * = - f f (x*0sintk + z*ocos ~ cosO)a3sinOdOdc~, l)=tl
372
(7)
373
LETTERS T O T H E E D I T O R
the Sommerfeld radiation condition and ~. = ~ t , + ~s,.
(8)
M and I are the mass and moment of inertia of the sphere about the x-axis; 2,/t and p are, respectively, the Lam6 constants and density of the medium; the functions z are the components of the stress tensor in spherical co-ordinates, and kl and k2 are, respectively, the compressional and shear wave numbers. The method of solution is straightforward (see, in addition, the paper by Eringen [3]). The vector fi~* can be written in terms of three scalar functions, ~'1, lP2 and ~b3, as ~s, = V(ar
-
v • v • (a~2~)
-
v x (~3~),
where r is the radius vector. These functions, as solutions of reduced wave equations, can be expanded in spherical surface harmonics. When the resulting expansions are substituted into equation (8) and then into the continuity conditions (2)-(4) the coefficients of the series for fis* can be determined. These will contain the functions of motion U*, U*, and tb*.When, finally, the stress tensor, T*, is computed from ~*, these motion functions can be found from the dynamical conditions (5)-(7). If one normalizes "time" and "length" so that a = 1 and c, = 1 (where c, is the compressional wave velocity of the medium), and sets co = iv and K = P/Po, the ratio of the density of the medium to that of the sphere, the tedious calculations outlined in the previous paragraph will yield, for the plane S-wave problem, the Laplace transforms of the motion functions: L(Uy) = 3 L ( f ) c~{(v + I) + e-Vt*"(v2 + 2v + 2)(cosh(v/c2) + sinh(v[c2))} + + {v" + (2 + c2 + K(I + 2c2)) va + (2 + c 2 + K(I + 2c22+ 9c2)) v2 + + 9Kc2(1 + c2) v + 9Kc2~}, or, more compactly, L(U,) = L(f) [Px(v) + e -2"/~2P2(v)]/P3(v), for polynomials P~, Pz and P3, and 15 K 2c L(O) = ~ ~-~4 2 L(f) e-wc~ {(v3 - 2c~ v - 2c~) sinh (v/c2)+ (v3 + 2c2 v 2 + 2c~ v) cosh (v/c2)} or
LfD L(q)) = - - ~ ( P a ( v ) + e -2./c* Ps(v)). Also U, = 0. It is easy to show that the roots of the fourth degree polynomial,/'3, all have negative real parts (by applying the Routh-Hurwitz stability criterion, for example). Once these roots are determined for a particular medium, both Ur and q, can be found by an easy application of the Convolution Theorem (provided, of course, that f decays sufficiently rapidly So that the inverse transforms exist). Similar results are obtainable for the center of rotation problem with different polynomials in place of P x , / 2 , P , and Ps, and the function g in place o f f The rate of decay of Ur is governed in both cases by the roots of P3. If all the spherical Bessel and Hankel functions that occur in Iwashimizu's results are written out, the same denominator appears in his formulas (28).
Department of 3lathematics, University of Connecticut, Storrs, Connecticut 06268, U.S.A. (Received 22 October 1973)
R . D . SIDMAN
374
LEt lt:RS TO THE EDITOR REFERENCES
1. Y. IwAsm~zu 1972 Journal of Sound and Vibration 21, 463-469. Scattering of elastic waves by a movable rigid sphere embedded in an infinite elastic solid. 2. C. C. Mow 1966 Journal of Applied Mechanics 33, 807-813. On the transient motion of a rigid spherical inclusion in an elastic medium and its inverse problem. 3. A. C-'EMALERINGrN 1957 Quarterly Journal of Mechanics and Applied Mathematics, X, 257-270. Elasto-dynamic problem concerning the spherical cavity.
AUTHOR'S REPLY, AND ERRATA In the preceding letter [1] R. D. Sidman has commented on a paper of the author [2]. He states that in reference [2], equations (28) and (29) do not seem to follow from equations (26) and (27) and, moreover, equation (28)3 is not coincident with Mow's result [3]. (Although he refers to reference [3], equation (28)3 should be compared with equation (I 8) in the previous paper of C. C. Mow [4].) However, this is irrelevant. Since the derivation of equations (28) and (29) is not difficult, only the equivalence of equation (28)3 and Mow's result will be shown below. By equation (19)t and the relation jr(z) -- (sinz - zcosz)/z 2 equation (28)3 becomes
.j., (i)
[-6ih~(ka)/(Ka) 2 + At(a)G,{j~(Ka), ht(ka))ht(Ka)/Ka - 3A~(a)j~(Ka)/Ka]
From equations (17) and (19)2, one has t
Al(a) --- 2hl(ra) hx(ka) - a
dhl(Ka) d da. da {ahx(ka)} =
= kaht(Ka ) ha(ka) + 2Kah2(Ka) hl(ka) - (Ka) (ka) h2(Ka) h2(ka), a~(a)G,{j~(Ka),h,(ka)} = 3 [2j,(Ka)ht(ka)-
..... ]]= a djt(Ka) &d .an,tKaj
= 3[kajx(Ka) h,(ka) + 2Kaj2(Ka) hl(ka) - (Ka)(ka)j2(Ka) h2(ka)]. When these are substituted into the above equation (1), it can be written as
= (~, x[A) [-6ihl(ka)[(Ka)" + 6ht(ka){j2(Ka)hl(Ka) - J a (Ka)h2(Ka)} - 3kah,(ka) {j,(Ka) h l(Ka) - j~(Ka) ha(Ka))],
(2)
where
A = --2(1 -- r.)Kah2(Ka)ht(ka) - (l - g)kaht(Ka)h,(ka) + (Ka)(ka)h2(Ka)h2(ka).
(3)
The above equation (2) can be simplified to
r
- 3 i x [ k~
(4)
by u s e of the relation j,_t(z)h,(z)- h,_t(z)j,(z)=-i]z 2. Since ~:, K, k, and k / K are, respectively, denoted by ~, ~,/~, and k in reference [4], A newly defined by equation (3) is the s a m e as A 1 defined by equation (17) in reference [4]. Moreover, it is seen from the definitions that q'o and [/in reference [4] are A[iKand A~e-l~t, respectively. Thus the above equation (4), i.e., equation (28)3, is proved to be identical with Mow's result: i.e., equation (18) in reference
[4].