P. A. Martin b) Department of Mathematics, University of Manchester, Manchester M13 9PL, United Kingdom

Size: px
Start display at page:

Download "P. A. Martin b) Department of Mathematics, University of Manchester, Manchester M13 9PL, United Kingdom"

Transcription

1 Time-harmonic torsional waves in a composite cyliner with an imperfect interface J. R. Berger a) Division of Engineering, Colorao School of Mines, Golen, Colorao P. A. Martin b) Department of Mathematics, University of Manchester, Manchester M13 9PL, Unite Kingom S. J. McCaffery c) Division of Engineering, Colorao School of Mines, Golen, Colorao Receive 4 June 1999; accepte for publication 1 November 1999 In this paper, the propagation of time-harmonic torsional waves in composite elastic cyliners is investigate. An imperfect interface is consiere where tractions are continuous across the interface an the isplacement jump is proportional to the stress acting on the interface. A frequency equation is erive for the ro an ispersion curves of normalize frequency as a function of normalize wave number for elastic bimaterials with varying values for the interface constant F are presente. The analysis is shown to recover the ispersion curves for a bimaterial ro with a perfect wele interface (F 0), an has the correct limiting behavior for large F. It is shown that the moes, at any given frequency, are orthogonal, an it is outline how the problem of reflection of a torsional moe by a planar efect such as a circumferential crack can be treate. 000 Acoustical Society of America. S PACS numbers: 43.0.Mv, 43.0.Gp, Cg, Dv ANN INTRODUCTION The motivation for this stuy comes from the application of electromagnetic acoustic transucers EMATs to the nonestructive testing of reinforce cables. We moel the cable as an infinitely long bimaterial cyliner, with a core of circular cross section surroune by a coaxial claing; the core an the claing are ifferent homogeneous isotropic elastic solis. Applications of EMATs are reviewe by Frost 1979 an by Hirao an Ogi We are intereste in the use of time-harmonic torsional waves in the composite cyliner. Johnson et al have use EMATs to stuy staning torsional moes in a single-material circular cyliner. This is a classical problem originally stuie by Pochhammer see, for example, Achenbach, 1973, sec. 6.10, or Miklowitz, 1978, sec Propagation of time-harmonic torsional waves in a ro compose of two or more elastic layers has also been stuie; see Thurston s paper 1978 for a comprehensive review. Perhaps the earliest work is by Armenàkas 1965, 1967, He stuie the ispersion of harmonic waves an establishe the isplacements an stresses at the interface of each layer analytically. A frequency equation was obtaine by enforcing continuity conitions at the interface an a stress-free bounary conition on the lateral surfaces of the cylinrical ro. Charalambopoulos et al have consiere the free a Electronic mail: jberger@mines.eu b Present aress: Department of Mathematical an Computer Sciences, Colorao School of Mines, Golen, CO 80401; electronic mail: pamartin@mines.eu c Present aress: CIRES, University of Colorao, Bouler, CO vibration of a bimaterial elastic ro of finite length. The problem was solve for time-harmonic waves using the Helmholtz ecomposition of the three-imensional elasticity equations. The interface between the layers was consiere as perfect, proviing continuity of isplacement an traction. The frequency equation for the full three-imensional ro was foun in terms of a 9 9 eterminantal equation whose roots yiel the ispersion relations for the ro. Rattanawangcharoen an Shah 199 have also consiere the layere cylinrical ro, but they stuie the problem from a more general perspective in that their formulation allowe many layers. A propagator matrix approach was use which relate the stresses an isplacements of one layer to the next. The propagator matrix was foun to implicitly generate the frequency equation for the ro. The main motivation for the paper was to arrive at an efficient computational scheme for the many-layer problem which i not rely on a homogenization metho such as integrating through the layers. In this paper, we consier the bimaterial elastic cyliner with an imperfect interface between the core an the claing. We o this because it is unrealistic to assume a perfectly bone wele interface for our intene application to reinforce cables. We moel the imperfect interface using a linear moification to the stanar perfect-interface conitions, allowing some slippage. The interface conitions involve a single imensionless parameter F. We stuy the effect of varying F on the ispersion relations. Note that the results for a perfectly bone interface can be recovere by setting F 0. EMATs can be use to excite propagating moes with a specifie axial wavelength, where is etermine by the physical spacing between the magnets of alternating polarity J. Acoust. Soc. Am. 107 (3), March /000/107(3)/1161/7/$ Acoustical Society of America 1161

2 One then ajusts the frequency until one of the propagating torsional moes is excite. When such a moe interacts with a efect in the composite cyliner, other allowable moes at the frequency, but with various wavelengths, will be stimulate; evanescent moes ecaying exponentially with istance from the efect will also be present, in general. We show that the torsional moes at a given frequency are orthogonal, extening a proof ue to Gregory We also iscuss the evanescent moes an their computation. Finally, we outline how our knowlege of the moal structure for the composite cyliner can be use to moel the problem of reflection of a torsional moe by a thin efect in a cross-sectional plane. The EMAT system can only receive waves with the same wavelength as the incient moe, so that some information at the excitation frequency is lost; but the experiment can be repeate at other moal frequencies. I. FORMULATION Let (r,,z) be cylinrical polar coorinates. We consier the infinite isotropic elastic bimaterial cyliner shown in Fig. 1. The cyliner consists of a soli core, r a, surroune by an annular claing, a r b; the core an claing are mae of materials 1 an, respectively. Material m has Lamé mouli m an m, m 1,. The analysis presente here generally follows Armenàkas In general, the isplacement fiel u (u,v,w) in each portion of the bimaterial can be written using the Lamé scalar potential an vector potential ( r,, z ); see, for example, Achenbach, 1973, sec..13. We are intereste in torsional waves, for which the only non zero isplacement component is the tangential isplacement v, an v itself is require to be inepenent of. Hence, the only potential neee is the z-component of the vector potential, z, say. In terms of, we have v r. The only nontrivial stress components are an r v r v r z v z. FIG. 1. Geometry of the bimaterial cyliner. The potential satisfies c t, where is the Laplace operator an c is the shear wave spee. For waves propagating in the positive z-irection, the appropriate solution of 4 can be written as r,z,t r e i kz t, where i is 1, k an are real, an solves 1 r r r /c k 0. This is Bessel s equation of orer zero. Its solutions epen on the sign of k c. Thus, efine Z n J n, W n Y n, an q /c k if k c, 7 an Z n 1 n I n, W n K n, an q k /c if k c, where J n an Y n are Bessel functions an I n an K n are moifie Bessel functions. The factor ( 1) n will allow a unifie treatment for all frequencies. The behavior of the solution as q 0 will be examine in some etail later; for now we assume that q 0 ( k c ). So, the appropriate solution of 6 is r q AZ 0 qr b BW 0 qr, where A an B are arbitrary constants, an the factors q an b have been introuce for later convenience, implying that A an B are imensionless; recall that b is the outer raius. The isplacement fiel obtaine by substituting 5 an 9 in 1 is v q 1 AZ 1 qr qb BW 1 qr e i kz t, as Z 0 (x) Z 1 (x) an W 0 (x) W 1 (x). Note that I 0 I 1. From, we obtain for the stress, r AZ qr qb BW qr e i kz t, 11 as Z 1 (x) x 1 Z 1 (x) Z (x) an W 1 (x) x 1 W 1 (x) W (x). Let us now use the expressions above, using subscripts 1 an to inicate quantities in the core an claing, respectively. Thus, from 10, the isplacement in the claing is v q 1 A Z 1 q r q b B W 1 q r e i kz t. 1 For the core, the solution for v 1 must be boune at the origin so we have v 1 q 1 1 A 1 Z 1 q 1 r e i kz t. In these expressions, q j is efine by q j k j k if k j k, k k j if k j k j 1,,, where k j /c j. Note that the wave number, k, is the same in the expressions for q 1 an q ; this observation gives a 116 J. Acoust. Soc. Am., Vol. 107, No. 3, March 000 Berger et al.: Torsional waves in composite cyliners 116

3 relation between q 1 an q, which we will iscuss later. With reference to Fig. 1, we now consier bounary an interface conitions on the isplacement fiel given by 1 an 13. At the outer surface, we have the traction-free bounary conition r 0 at r b. 15 We consier the interface conitions in the following section. A. Interface conitions A variety of conitions may be taken on the interface r a in orer to represent imperfect interface conitions. A review of interface conitions for elastic wave problems has been presente by Martin 199. For most moels, the isplacement u an traction t on one sie of the interface are assume to be linearly relate to the isplacement u an traction t on the other sie of the interface. For example, the moel of Rokhlin an Wang 1991, originally erive for plane interfaces, takes interface conitions of the form t Gu Bt, u Ft Au, where A, B, F, an G are 3 3 matrices, an the square brackets inicate a jump in the quantity across the interface; for example, if the interface is at r a, we have u u u u a, u a,, 16 suppressing the epenence on z an t. If the coupling term G can be neglecte, an furthermore if A an B are set equal to zero, we recover the moel of Jones an Whittier 1967 for a flexibly bone interface, t 0, u Ft, where F is a constant iagonal matrix. For simplicity, we will use the Jones Whittier moel for the analysis presente here. For thin, elastic interfacial layers, the elements of F have been relate to the thickness an elastic constants of the layer by, for example, Jones an Whittier 1967, Mal an Xu 1989, an Pilarski an Rose For torsional waves, u reuces to a scalar for the tangential isplacement v an t reuces to a scalar for the tangential shear stress r. The interface conitions are then an r a r a v a/ 1 F r a, 19 0 where v v (a ) v 1 (a ) an F is a imensionless scalar. Our goal is to investigate solutions which satisfy 15, 19, an 0 as the interface parameter F is varie. We note that if F 0, the perfect interface conitions of continuity of traction an isplacement are recovere. II. FREQUENCY EQUATION FOR THE ROD We now present the etails for the set of equations which will etermine the ispersion relations in the bimaterial ro. Substituting the isplacement fiel of 1 in the bounary conition, 15, yiels A Z q b q b B W q b 0. 1 Following the Jones-Whittier moel, we have for continuity of traction across the interface, from 1, 13, an 19, 1 / A 1 Z q 1 a A Z q a q b B W q a 0. Note that neither of these equations changes in the case of the perfectly bone interface. The isplacement jump across the interface is given by 0. We then have q 1 b 1 A 1 Z 1 q 1 a Fq 1 az q 1 a q b 1 A Z 1 q a q bb W 1 q a 0. 3 Equations 1 3 provie three equations in the three unknown constants A 1, A, an B. In matrix form, the system of equations is Db 0, 4 where the elements of the nonsymmetric matrix D are obtaine irectly from 1 3 an b (A 1,A,B ) T. For a nontrivial solution we then require et D 0. 5 This is the frequency equation for the ro. The quantity et D seems to epen on only five imensionless parameters, namely q 1 b, q b, a/b, 1 /, an F; 6 in particular, the ensity ratio or, equivalently, c 1 /c oes not appear explicitly. However, this is illusory: we have to know how to choose Z n (J n or ( 1) n I n?) an W n (Y n or K n? in each material, an these choices epen on the relative sizes of k, k 1, an k, information that we cannot extract from a knowlege of 6 alone. Thus we procee as follows. Assume that we are given values for a/b, 1 /, F, an c /c 1 k 1 /k, 7 say. Choose a value for the axial wave number kb. We now seek values of k b, say, so that 5 is satisfie. Note that k 1 b k b, an then q 1 b an q b are efine by 14, with the associate selections of Z n an W n ictate by 7 an 8. In fact, the relations between q 1, q, k 1, k, k, an are complicate, because they epen on the relative magnitues of k, k 1, an k ; there are four cases, as summarize in Table I. In this table, the secon column specifies the four cases in terms of the shear wave spees of the two materials these are material constants an the axial wave spee c a /k. A similar table was given by Kleczewski an Parnes 1987 in their stuy of torsional moes when the claing is unboune (b in our notation. In orer to compare with Armenàkas 1965 for F 0, we have etermine the ispersion curves of normalize frequency, 1163 J. Acoust. Soc. Am., Vol. 107, No. 3, March 000 Berger et al.: Torsional waves in composite cyliners 1163

4 TABLE I. Relations between q 1 an q, in which c /c 1 k 1 /k an c a /k. Wave numbers Wave spees q 1 q Relation between q 1 an q k k 1 k k k k 1 c 1 c c a k 1 k k k q 1 q k (1 ) c c 1 c a k k k 1 c 1 c a c k 1 k k k q 1 q k (1 ) k 1 k k c c a c 1 k k 1 k k q 1 q k (1 ) k 1 k k c a c 1 c k k 1 k k k k 1 k c a c c 1 q 1 q k (1 ) k b a / b a / c, as a function of normalize axial wave number, k b a /, for a given value of the interface parameter F. We note that the frequency equation etermine here cannot be written in terms of a single argument such as qa, which can be one in the case of a ro mae from a single material. As such, we stuy numerical solutions to 5 in the next section for values of a/b, F, an the elastic constants. III. DISPERSION CURVES WITH VARYING INTERFACE CONDITIONS To benchmark the analysis presente here, we first present results which can be irectly compare with Armenàkas 1965 in the case of a perfect interface, F 0. We take a/b 0.5, 1 / 10, an c 1 /c 1.83 so that the ensity ratio, 1 / 3. We show the ispersion curves of frequency,, versus wave number,, for F 0, F 1, F 10, an F 100 for the secon moe in Fig. an the thir FIG.. Dispersion curves for the secon moe in the bimaterial cyliner. FIG. 3. Dispersion curves for the thir moe in the bimaterial cyliner. moe in Fig. 3. The first moe will be analyze in a subsequent section. The ispersion curve for F 0 agrees exactly with the analysis of Armenàkas As the interface parameter is increase, note the ecrease in, especially at the smaller values of. At higher values of, the loss of perfect continuity at the interface has a reuce effect. One feature of note in Fig. 3 is the curve for F 100, which exhibits a corner at 0.8. This is not a numerical artifact: the figure was prouce using very small increments in. Similar behavior was foun for other large values of F. A secon way of visualizing the behavior of the ispersion curves as the interface parameter is varie is illustrate in Fig. 4. In the figure, we show results for the secon moe an plot frequency,, versus the interface parameter, F, as the wave number is varie. As expecte, we see a much greater effect on by F at the smaller wave numbers. This suggests a possible measurement approach for etermining F where is fixe, is measure, an then F is etermine from the figure. The curves shown in Fig. 4 appear to be approaching asymptotic values for large F. With reference to the interface conitions given by 0, we see that in the limit as F we recover the bounary conition r a 0. This is the appropriate bounary conition for the outer bounary of a soli ro of raius a an for the inner bounary conition for a hollow tube with inner raius a. Inthe case of F, the frequency equation given by 5 reflects this change in bounary conition, where Z q 1 a g q a,q b 0, J. Acoust. Soc. Am., Vol. 107, No. 3, March 000 Berger et al.: Torsional waves in composite cyliners 1164

5 frequency,, for the thir moe in the composite ro correspon to the secon moe frequencies in the hollow tube. Some iscussion on the interface parameter F is perhaps in orer. As mentione in the Introuction, the application motivating the analysis presente here is the nonestructive evaluation of reinforce cables. Typically these cables are fabricate with a steel core surroune by an aluminum claing. Because of the unerlying wire-rope structure of the core an the claing, the interface conitions are imperfect. Our approach here is to treat the interface parameter in a phenomenological manner to account for the imperfect interface. As such, we o not stipulate a strict physical interpretation to the numerical value of F, nor o we attempt to relate the value of F to elastic constants. IV. FIRST TORSIONAL MODE Armenàkas 1965 note that the first torsional moe is not properly escribe by the solution of the Bessel equation, unlike the higher torsional moes analyze above. Therefore, special consieration of the first torsional moe is necessary, an this is carrie out next. For example, when q 0, there can be a nonispersive moe propagating in the claing of the ro with ispersive moes in the core. Suppose that k c, so that q 0 an 6 reuces to FIG. 4. Normalize frequency vs interface parameter at fixe wave number in the bimaterial cyliner for the secon moe. g q a,q b Z q a W q b Z q b W q a. The frequency equation 8 has two sets of solutions, one given by Z (q 1 a) 0 an the other given by g(q a,q b) 0. The first of these is the frequency equation for a soli ro of raius a, whereas the secon is the frequency equation for a hollow tube of inner raius a an outer raius b. So, the imperfect interface formulation has the expecte behavior for large values of F. Numerically, one must be somewhat careful in hanling the limit as F an check if the relations between q 1 an q given in Table I still hol. We present some typical results in Table II where we have use F Note in the table that we report values of normalize frequency as we have throughout this paper, even for the soli ro moes in material 1 reporte in the table. From the table, we see that the asymptotic values of frequency,, for the secon moe in the composite ro, are the nonispersive first moes (q 1 a 0) in the soli ro. Also, the asymptotic values of TABLE II. Frequencies for the secon an thir moes in the composite ro with F , an corresponing frequencies for the first moe in a soli ro an the secon moe in a hollow tube. secon moe thir moe, hollow tube secon moe, soli ro first moe r r 0, with general solution r A B log r. This gives a solution for v proportional to B/r, which cannot be the general solution as it involves only one arbitrary constant, B. As we are intereste in rather than, we return to 6 ; ifferentiation with respect to r gives r 1 r r r /c k 0, a secon-orer orinary ifferential equation for. When k c, the general solution of this equation is r Ar B/r. Hence, in imensionless form, we have an v r,z,t Ar Bb /r e i kz t r r,z,t B b/r e i kz t Note that the expressions 9 an 30 can also be obtaine by taking the limit q 0 in 10 an 11 apart from some numerical factors which can be absorbe into A an B. This accounts for the various q-factors in 10 : they lea to meaningful boune expressions for small q. Let us assume that q 0. The outer bounary conition 15 implies that B 0, whence r 0 in the claing an v A re i kz t. 31 Within the core, v 1 is given by 13. For continuity of tractions across r a, 19 gives 1165 J. Acoust. Soc. Am., Vol. 107, No. 3, March 000 Berger et al.: Torsional waves in composite cyliners 1165

6 A 1 Z q 1 a 0, whereas the imperfect-interface conition 0 gives A 1 Z 1 q 1 a q 1 aa Now, for nontrivial solutions, we require k k 1, so that 3 gives q 1 a j,s, the s-th zero of J (x). Then A 1 is arbitrary with A given by 33 with Z 1 J 1. It is interesting to note that the interface parameter F oes not enter into any of the equations, so that the ispersion curves for the first torsional moe when q 0 are ientical, regarless of whether the interface is perfect or imperfect. This is consistent with the fact that since q 0, a nonispersive moe is propagating in the claing an 3 is simply the frequency equation for the ispersive moes in the core. Alternatively, let us assume that q 1 0; for boune isplacements in the core, we obtain v 1 r,z,t A 1 re i kz t. Within the claing, v is given by 1, so that the outer bounary conition gives 1. The interface conition 19 gives A Z q a q b B W q a 0; the frequency equation is then obtaine by combining this equation with 1 : it is the same equation as for a hollow cylinrical tube. The other interface conition, 0, then etermines A 1 as A 1 q a 1 A Z 1 q a q b /a B W 1 q a. Again, these equations o not involve F. V. EVANESCENT MODES So far, we have only consiere propagating torsional moes. However, cyliners can also support evanescent moes, which ecay exponentially with z. Such moes can be constructe by writing r,z,t r e kz i t, where solves 1 r whence r r /c k 0, r q AJ 0 qr b BY 0 qr with q ( /c) k. Then, proceeing exactly as before, we arrive at the frequency equation 5 in which Z n an W n are to be replace by J n an Y n, respectively. VI. DISCUSSION ON MODE ORTHOGONALITY We have constructe various torsional moes for the composite cyliner in the general form u r,,z,t Re U r, e i kz t. In our computations, we have fixe the axial wave number k an then calculate the frequencies of the allowable moes. This is convenient for comparisons with Armenàkas 1965 an it is appropriate for the application to EMATs; these can be use to excite propagating moes of a specifie axial wavelength. However, once such a moe has been excite, we are intereste in stuying its reflection by efects in the cyliner. This is most conveniently one by specifying the frequency an then etermining all the allowable moes at that frequency. With this in min, we write a typical moe as u n r,,z,t Re U n r, e i k n z t, where the wave number k (n) nee not be real. These moes are biorthogonal. To be more explicit, enote the stresses corresponing to u (n) by n r,,z,t Re S n r, e i k n z t. Then, if k (n) k (m), we have U m z S n zz S m rz U n r S m z U n rr 0, A 34 where A is the cross section of the composite cyliner. This relation can be prove by a simple extension of the proof given by Gregory One applies the elastic reciprocal theorem twice, once in the core an once in the claing, an then as the results; the interface conitions imply that the contributions from integrating over the two sies of the interface cancel. In fact, 34 hols for all moes in composite cyliners of any cross section, an with any number of imperfect cylinrical interfaces. For our problem, with torsional moes given by v n r,z,t Re V n r e i k n z t, Eq. 34 reuces to b V m r V n r r r 0, 0 m n, 35 so that torsional moes are actually orthogonal. This orthogonality relation is useful when the reflection of a torsional moe by certain efects is examine. For example, we may consier a bimaterial cyliner with a planar break crack perpenicular to the cyliner s axis, giving an iealize moel of a amage cable. Specifically, we partition the cross-section A into a broken part A b an an unbroken part A u, so that A A b A u. The bounaries of A b an A u are concentric circles; for example, we might take A u to be the circle 0 r c, with A b as the annulus c r b, so that the cable is circumferentially cracke. Then, if a torsional moe is incient on the efect, the reflecte an transmitte fiels can be written as moal sums. This is a stanar approach for planar obstacles in waveguies. In the context of torsional waves, it has been use recently by Engan 1998 to analyze the effect of a step-change in raius of homogeneous circular cyliners. For the present problem, application of the bounary conitions at the efect plane leas to a system of equations for the reflection an transmission coefficients; of particular interest are the reflecte an transmitte moes with the same wavelength as the incient moe, because these are the only moes that can be etecte by the EMAT. Again, in a stanar way, one can erive integral equations an/or variational expressions for the reflection an transmis J. Acoust. Soc. Am., Vol. 107, No. 3, March 000 Berger et al.: Torsional waves in composite cyliners 1166

7 sion coefficients; see, for example, Schwinger an Saxon 1968 for a etaile iscussion on relate scattering problems. VII. CONCLUSIONS We have presente an analysis for torsional waves propagating in a bimaterial ro with imperfect interface conitions. To the authors knowlege, the effect of imperfect interface conitions on ispersive wave motion has not been stuie for ros. We fin the expecte behavior for waves in the ro when we take the interface parameter F 0 an F. When F 0 we fin that the frequency ecreases with increasing F at a given wave number in the ispersion relations. This effect was shown to be more pronounce at small wave numbers. The propagation of nonispersive moes in the claing was also investigate, an the frequency equation for ispersive moes in the core was recovere. We also showe that, at any given frequency, the moes are orthogonal. This fact can be exploite in the solution of a scattering problem, where an incient torsional moe interacts with an annular efect in the bimaterial ro. The moel evelope here shoul be useful in analyzing nonestructive evaluation measurements in reinforce cables where perfect interface conitions may not exist. ACKNOWLEDGMENTS Two of us J.R.B. an S.J.M. gratefully acknowlege the support receive from the Center for Avance Control of Energy an Power Systems, a National Science Founation Inustry/University Cooperative Research Center, at the Colorao School of Mines. J.R.B. also acknowleges the aitional support provie by the Engineering an Physical Sciences Research Council as a visiting fellow in the Department of Mathematics at the University of Manchester. Achenbach, J. D Wave Propagation in Elastic Solis North- Hollan, New York. Armenàkas, A. E Torsional waves in composite ros, J. Acoust. Soc. Am. 38, Armenàkas, A. E Propagation of harmonic waves in composite circular cylinrical shells. I: Theoretical investigation, AIAA J. 5, Armenàkas, A. E Propagation of harmonic waves in composite circular cylinrical shells. Part II: Numerical analysis, AIAA J. 9, Charalambopoulos, A., Fotiais, D. I., an Massalas, C. V Free vibrations of a ouble layere elastic isotropic cylinrical ro, Int. J. Eng. Sci. 36, Engan, H. E Torsional wave scattering from a iameter step in a ro, J. Acoust. Soc. Am. 104, Frost, H. M Electromagnetic-ultrasoun transucers: Principles, practice, an applications, in Physical Acoustics, eite by W. P. Mason an R. N. Thurston Acaemic, New York, Vol. 14, pp Gregory, R. D A note on bi-orthogonality relations for elastic cyliners of general cross section, J. Elast. 13, Hirao, M., an Ogi, H Electromagnetic acoustic resonance an materials characterization, Ultrasonics 35, Johnson, W., Aul, B. A., an Alers, G. A Spectroscopy of resonant torsional moes in cylinrical ros using electromagnetic-acoustic transuction, J. Acoust. Soc. Am. 95, Jones, J. P., an Whittier, J. S Waves at a flexibly bone interface, J. Appl. Mech. 34, Kleczewski, D., an Parnes, R Torsional ispersion relations in a raially ual elastic meium, J. Acoust. Soc. Am. 81, Mal, A. K., an Xu, P. C Elastic waves in layere meia with interface features, in Elastic Wave Propagation, eite by M. F. McCarthy an M. A. Hayes North-Hollan, Amsteram, pp Martin, P. A Bounary integral equations for the scattering of elastic waves by elastic inclusions with thin interface layers, J. Nonestruct. Eval. 11, Miklowitz, J The Theory of Elastic Waves an Waveguies North- Hollan, Amsteram. Pilarski, A., an Rose, J. L A transverse-wave ultrasonic obliqueincience technique for interfacial weakness etection in ahesive bons, J. Appl. Phys. 63, Rattanawangcharoen, N., an Shah, A. H Guie waves in laminate isotropic circular cyliner, Comput. Mech. 10, Rokhlin, S. I., an Wang, Y. J Analysis of bounary conitions for elastic wave interaction with an interface between two solis, J. Acoust. Soc. Am. 89, Schwinger, J., an Saxon, D. S Discontinuities in Waveguies Goron & Breach, New York. Thurston, R. N Elastic waves in ros an cla ros, J. Acoust. Soc. Am. 64, J. Acoust. Soc. Am., Vol. 107, No. 3, March 000 Berger et al.: Torsional waves in composite cyliners 1167

Generalization of the persistent random walk to dimensions greater than 1

Generalization of the persistent random walk to dimensions greater than 1 PHYSICAL REVIEW E VOLUME 58, NUMBER 6 DECEMBER 1998 Generalization of the persistent ranom walk to imensions greater than 1 Marián Boguñá, Josep M. Porrà, an Jaume Masoliver Departament e Física Fonamental,

More information

Table of Common Derivatives By David Abraham

Table of Common Derivatives By David Abraham Prouct an Quotient Rules: Table of Common Derivatives By Davi Abraham [ f ( g( ] = [ f ( ] g( + f ( [ g( ] f ( = g( [ f ( ] g( g( f ( [ g( ] Trigonometric Functions: sin( = cos( cos( = sin( tan( = sec

More information

05 The Continuum Limit and the Wave Equation

05 The Continuum Limit and the Wave Equation Utah State University DigitalCommons@USU Founations of Wave Phenomena Physics, Department of 1-1-2004 05 The Continuum Limit an the Wave Equation Charles G. Torre Department of Physics, Utah State University,

More information

Simulation of Angle Beam Ultrasonic Testing with a Personal Computer

Simulation of Angle Beam Ultrasonic Testing with a Personal Computer Key Engineering Materials Online: 4-8-5 I: 66-9795, Vols. 7-73, pp 38-33 oi:.48/www.scientific.net/kem.7-73.38 4 rans ech ublications, witzerlan Citation & Copyright (to be inserte by the publisher imulation

More information

3-D FEM Modeling of fiber/matrix interface debonding in UD composites including surface effects

3-D FEM Modeling of fiber/matrix interface debonding in UD composites including surface effects IOP Conference Series: Materials Science an Engineering 3-D FEM Moeling of fiber/matrix interface eboning in UD composites incluing surface effects To cite this article: A Pupurs an J Varna 2012 IOP Conf.

More information

5-4 Electrostatic Boundary Value Problems

5-4 Electrostatic Boundary Value Problems 11/8/4 Section 54 Electrostatic Bounary Value Problems blank 1/ 5-4 Electrostatic Bounary Value Problems Reaing Assignment: pp. 149-157 Q: A: We must solve ifferential equations, an apply bounary conitions

More information

1. The electron volt is a measure of (A) charge (B) energy (C) impulse (D) momentum (E) velocity

1. The electron volt is a measure of (A) charge (B) energy (C) impulse (D) momentum (E) velocity AP Physics Multiple Choice Practice Electrostatics 1. The electron volt is a measure of (A) charge (B) energy (C) impulse (D) momentum (E) velocity. A soli conucting sphere is given a positive charge Q.

More information

Sensors & Transducers 2015 by IFSA Publishing, S. L.

Sensors & Transducers 2015 by IFSA Publishing, S. L. Sensors & Transucers, Vol. 184, Issue 1, January 15, pp. 53-59 Sensors & Transucers 15 by IFSA Publishing, S. L. http://www.sensorsportal.com Non-invasive an Locally Resolve Measurement of Soun Velocity

More information

12.11 Laplace s Equation in Cylindrical and

12.11 Laplace s Equation in Cylindrical and SEC. 2. Laplace s Equation in Cylinrical an Spherical Coorinates. Potential 593 2. Laplace s Equation in Cylinrical an Spherical Coorinates. Potential One of the most important PDEs in physics an engineering

More information

Thermal conductivity of graded composites: Numerical simulations and an effective medium approximation

Thermal conductivity of graded composites: Numerical simulations and an effective medium approximation JOURNAL OF MATERIALS SCIENCE 34 (999)5497 5503 Thermal conuctivity of grae composites: Numerical simulations an an effective meium approximation P. M. HUI Department of Physics, The Chinese University

More information

Homework 7 Due 18 November at 6:00 pm

Homework 7 Due 18 November at 6:00 pm Homework 7 Due 18 November at 6:00 pm 1. Maxwell s Equations Quasi-statics o a An air core, N turn, cylinrical solenoi of length an raius a, carries a current I Io cos t. a. Using Ampere s Law, etermine

More information

Implicit Differentiation

Implicit Differentiation Implicit Differentiation Thus far, the functions we have been concerne with have been efine explicitly. A function is efine explicitly if the output is given irectly in terms of the input. For instance,

More information

Characterization of lead zirconate titanate piezoceramic using high frequency ultrasonic spectroscopy

Characterization of lead zirconate titanate piezoceramic using high frequency ultrasonic spectroscopy JOURNAL OF APPLIED PHYSICS VOLUME 85, NUMBER 1 15 JUNE 1999 Characterization of lea zirconate titanate piezoceramic using high frequency ultrasonic spectroscopy Haifeng Wang, Wenhua Jiang, a) an Wenwu

More information

'HVLJQ &RQVLGHUDWLRQ LQ 0DWHULDO 6HOHFWLRQ 'HVLJQ 6HQVLWLYLW\,1752'8&7,21

'HVLJQ &RQVLGHUDWLRQ LQ 0DWHULDO 6HOHFWLRQ 'HVLJQ 6HQVLWLYLW\,1752'8&7,21 Large amping in a structural material may be either esirable or unesirable, epening on the engineering application at han. For example, amping is a esirable property to the esigner concerne with limiting

More information

Chapter 2 Governing Equations

Chapter 2 Governing Equations Chapter 2 Governing Equations In the present an the subsequent chapters, we shall, either irectly or inirectly, be concerne with the bounary-layer flow of an incompressible viscous flui without any involvement

More information

Separation of Variables

Separation of Variables Physics 342 Lecture 1 Separation of Variables Lecture 1 Physics 342 Quantum Mechanics I Monay, January 25th, 2010 There are three basic mathematical tools we nee, an then we can begin working on the physical

More information

The effect of nonvertical shear on turbulence in a stably stratified medium

The effect of nonvertical shear on turbulence in a stably stratified medium The effect of nonvertical shear on turbulence in a stably stratifie meium Frank G. Jacobitz an Sutanu Sarkar Citation: Physics of Fluis (1994-present) 10, 1158 (1998); oi: 10.1063/1.869640 View online:

More information

Qubit channels that achieve capacity with two states

Qubit channels that achieve capacity with two states Qubit channels that achieve capacity with two states Dominic W. Berry Department of Physics, The University of Queenslan, Brisbane, Queenslan 4072, Australia Receive 22 December 2004; publishe 22 March

More information

Math 342 Partial Differential Equations «Viktor Grigoryan

Math 342 Partial Differential Equations «Viktor Grigoryan Math 342 Partial Differential Equations «Viktor Grigoryan 6 Wave equation: solution In this lecture we will solve the wave equation on the entire real line x R. This correspons to a string of infinite

More information

19 Eigenvalues, Eigenvectors, Ordinary Differential Equations, and Control

19 Eigenvalues, Eigenvectors, Ordinary Differential Equations, and Control 19 Eigenvalues, Eigenvectors, Orinary Differential Equations, an Control This section introuces eigenvalues an eigenvectors of a matrix, an iscusses the role of the eigenvalues in etermining the behavior

More information

Optimization of Geometries by Energy Minimization

Optimization of Geometries by Energy Minimization Optimization of Geometries by Energy Minimization by Tracy P. Hamilton Department of Chemistry University of Alabama at Birmingham Birmingham, AL 3594-140 hamilton@uab.eu Copyright Tracy P. Hamilton, 1997.

More information

Analytic Scaling Formulas for Crossed Laser Acceleration in Vacuum

Analytic Scaling Formulas for Crossed Laser Acceleration in Vacuum October 6, 4 ARDB Note Analytic Scaling Formulas for Crosse Laser Acceleration in Vacuum Robert J. Noble Stanfor Linear Accelerator Center, Stanfor University 575 San Hill Roa, Menlo Park, California 945

More information

The Principle of Least Action

The Principle of Least Action Chapter 7. The Principle of Least Action 7.1 Force Methos vs. Energy Methos We have so far stuie two istinct ways of analyzing physics problems: force methos, basically consisting of the application of

More information

Calculus of Variations

Calculus of Variations Calculus of Variations Lagrangian formalism is the main tool of theoretical classical mechanics. Calculus of Variations is a part of Mathematics which Lagrangian formalism is base on. In this section,

More information

water adding dye partial mixing homogenization time

water adding dye partial mixing homogenization time iffusion iffusion is a process of mass transport that involves the movement of one atomic species into another. It occurs by ranom atomic jumps from one position to another an takes place in the gaseous,

More information

Math Notes on differentials, the Chain Rule, gradients, directional derivative, and normal vectors

Math Notes on differentials, the Chain Rule, gradients, directional derivative, and normal vectors Math 18.02 Notes on ifferentials, the Chain Rule, graients, irectional erivative, an normal vectors Tangent plane an linear approximation We efine the partial erivatives of f( xy, ) as follows: f f( x+

More information

The derivative of a function f(x) is another function, defined in terms of a limiting expression: f(x + δx) f(x)

The derivative of a function f(x) is another function, defined in terms of a limiting expression: f(x + δx) f(x) Y. D. Chong (2016) MH2801: Complex Methos for the Sciences 1. Derivatives The erivative of a function f(x) is another function, efine in terms of a limiting expression: f (x) f (x) lim x δx 0 f(x + δx)

More information

We G Model Reduction Approaches for Solution of Wave Equations for Multiple Frequencies

We G Model Reduction Approaches for Solution of Wave Equations for Multiple Frequencies We G15 5 Moel Reuction Approaches for Solution of Wave Equations for Multiple Frequencies M.Y. Zaslavsky (Schlumberger-Doll Research Center), R.F. Remis* (Delft University) & V.L. Druskin (Schlumberger-Doll

More information

Chapter 4. Electrostatics of Macroscopic Media

Chapter 4. Electrostatics of Macroscopic Media Chapter 4. Electrostatics of Macroscopic Meia 4.1 Multipole Expansion Approximate potentials at large istances 3 x' x' (x') x x' x x Fig 4.1 We consier the potential in the far-fiel region (see Fig. 4.1

More information

Crack onset assessment near the sharp material inclusion tip by means of modified maximum tangential stress criterion

Crack onset assessment near the sharp material inclusion tip by means of modified maximum tangential stress criterion Focuse on Mechanical Fatigue of Metals Crack onset assessment near the sharp material inclusion tip by means of moifie maximum tangential stress criterion Onřej Krepl, Jan Klusák CEITEC IPM, Institute

More information

Linear First-Order Equations

Linear First-Order Equations 5 Linear First-Orer Equations Linear first-orer ifferential equations make up another important class of ifferential equations that commonly arise in applications an are relatively easy to solve (in theory)

More information

Assignment 1. g i (x 1,..., x n ) dx i = 0. i=1

Assignment 1. g i (x 1,..., x n ) dx i = 0. i=1 Assignment 1 Golstein 1.4 The equations of motion for the rolling isk are special cases of general linear ifferential equations of constraint of the form g i (x 1,..., x n x i = 0. i=1 A constraint conition

More information

Quantum Mechanics in Three Dimensions

Quantum Mechanics in Three Dimensions Physics 342 Lecture 20 Quantum Mechanics in Three Dimensions Lecture 20 Physics 342 Quantum Mechanics I Monay, March 24th, 2008 We begin our spherical solutions with the simplest possible case zero potential.

More information

The total derivative. Chapter Lagrangian and Eulerian approaches

The total derivative. Chapter Lagrangian and Eulerian approaches Chapter 5 The total erivative 51 Lagrangian an Eulerian approaches The representation of a flui through scalar or vector fiels means that each physical quantity uner consieration is escribe as a function

More information

Asymptotics of a Small Liquid Drop on a Cone and Plate Rheometer

Asymptotics of a Small Liquid Drop on a Cone and Plate Rheometer Asymptotics of a Small Liqui Drop on a Cone an Plate Rheometer Vincent Cregan, Stephen B.G. O Brien, an Sean McKee Abstract A cone an a plate rheometer is a laboratory apparatus use to measure the viscosity

More information

Lectures - Week 10 Introduction to Ordinary Differential Equations (ODES) First Order Linear ODEs

Lectures - Week 10 Introduction to Ordinary Differential Equations (ODES) First Order Linear ODEs Lectures - Week 10 Introuction to Orinary Differential Equations (ODES) First Orer Linear ODEs When stuying ODEs we are consiering functions of one inepenent variable, e.g., f(x), where x is the inepenent

More information

Negative-Index Refraction in a Lamellar Composite with Alternating. Single Negative Layers

Negative-Index Refraction in a Lamellar Composite with Alternating. Single Negative Layers Negative-Inex Refraction in a Lamellar Composite with Alternating Single Negative Layers Z. G. Dong, S. N. Zhu, an H. Liu National Laboratory of Soli State Microstructures, Nanjing University, Nanjing

More information

TMA 4195 Matematisk modellering Exam Tuesday December 16, :00 13:00 Problems and solution with additional comments

TMA 4195 Matematisk modellering Exam Tuesday December 16, :00 13:00 Problems and solution with additional comments Problem F U L W D g m 3 2 s 2 0 0 0 0 2 kg 0 0 0 0 0 0 Table : Dimension matrix TMA 495 Matematisk moellering Exam Tuesay December 6, 2008 09:00 3:00 Problems an solution with aitional comments The necessary

More information

How the potentials in different gauges yield the same retarded electric and magnetic fields

How the potentials in different gauges yield the same retarded electric and magnetic fields How the potentials in ifferent gauges yiel the same retare electric an magnetic fiels José A. Heras a Departamento e Física, E. S. F. M., Instituto Politécnico Nacional, México D. F. México an Department

More information

23 Implicit differentiation

23 Implicit differentiation 23 Implicit ifferentiation 23.1 Statement The equation y = x 2 + 3x + 1 expresses a relationship between the quantities x an y. If a value of x is given, then a corresponing value of y is etermine. For

More information

Lie symmetry and Mei conservation law of continuum system

Lie symmetry and Mei conservation law of continuum system Chin. Phys. B Vol. 20, No. 2 20 020 Lie symmetry an Mei conservation law of continuum system Shi Shen-Yang an Fu Jing-Li Department of Physics, Zhejiang Sci-Tech University, Hangzhou 3008, China Receive

More information

Chapter 6: Energy-Momentum Tensors

Chapter 6: Energy-Momentum Tensors 49 Chapter 6: Energy-Momentum Tensors This chapter outlines the general theory of energy an momentum conservation in terms of energy-momentum tensors, then applies these ieas to the case of Bohm's moel.

More information

Time-of-Arrival Estimation in Non-Line-Of-Sight Environments

Time-of-Arrival Estimation in Non-Line-Of-Sight Environments 2 Conference on Information Sciences an Systems, The Johns Hopkins University, March 2, 2 Time-of-Arrival Estimation in Non-Line-Of-Sight Environments Sinan Gezici, Hisashi Kobayashi an H. Vincent Poor

More information

EXPONENTIAL FOURIER INTEGRAL TRANSFORM METHOD FOR STRESS ANALYSIS OF BOUNDARY LOAD ON SOIL

EXPONENTIAL FOURIER INTEGRAL TRANSFORM METHOD FOR STRESS ANALYSIS OF BOUNDARY LOAD ON SOIL Tome XVI [18] Fascicule 3 [August] 1. Charles Chinwuba IKE EXPONENTIAL FOURIER INTEGRAL TRANSFORM METHOD FOR STRESS ANALYSIS OF BOUNDARY LOAD ON SOIL 1. Department of Civil Engineering, Enugu State University

More information

Introduction to the Vlasov-Poisson system

Introduction to the Vlasov-Poisson system Introuction to the Vlasov-Poisson system Simone Calogero 1 The Vlasov equation Consier a particle with mass m > 0. Let x(t) R 3 enote the position of the particle at time t R an v(t) = ẋ(t) = x(t)/t its

More information

Conservation Laws. Chapter Conservation of Energy

Conservation Laws. Chapter Conservation of Energy 20 Chapter 3 Conservation Laws In orer to check the physical consistency of the above set of equations governing Maxwell-Lorentz electroynamics [(2.10) an (2.12) or (1.65) an (1.68)], we examine the action

More information

Calculus of Variations

Calculus of Variations 16.323 Lecture 5 Calculus of Variations Calculus of Variations Most books cover this material well, but Kirk Chapter 4 oes a particularly nice job. x(t) x* x*+ αδx (1) x*- αδx (1) αδx (1) αδx (1) t f t

More information

Conservation laws a simple application to the telegraph equation

Conservation laws a simple application to the telegraph equation J Comput Electron 2008 7: 47 51 DOI 10.1007/s10825-008-0250-2 Conservation laws a simple application to the telegraph equation Uwe Norbrock Reinhol Kienzler Publishe online: 1 May 2008 Springer Scienceusiness

More information

Efficient Macro-Micro Scale Coupled Modeling of Batteries

Efficient Macro-Micro Scale Coupled Modeling of Batteries A00 Journal of The Electrochemical Society, 15 10 A00-A008 005 0013-651/005/1510/A00/7/$7.00 The Electrochemical Society, Inc. Efficient Macro-Micro Scale Couple Moeling of Batteries Venkat. Subramanian,*,z

More information

Crack-tip stress evaluation of multi-scale Griffith crack subjected to

Crack-tip stress evaluation of multi-scale Griffith crack subjected to Crack-tip stress evaluation of multi-scale Griffith crack subjecte to tensile loaing by using periynamics Xiao-Wei Jiang, Hai Wang* School of Aeronautics an Astronautics, Shanghai Jiao Tong University,

More information

A simple model for the small-strain behaviour of soils

A simple model for the small-strain behaviour of soils A simple moel for the small-strain behaviour of soils José Jorge Naer Department of Structural an Geotechnical ngineering, Polytechnic School, University of São Paulo 05508-900, São Paulo, Brazil, e-mail:

More information

ELECTRON DIFFRACTION

ELECTRON DIFFRACTION ELECTRON DIFFRACTION Electrons : wave or quanta? Measurement of wavelength an momentum of electrons. Introuction Electrons isplay both wave an particle properties. What is the relationship between the

More information

Electromagnet Gripping in Iron Foundry Automation Part II: Simulation

Electromagnet Gripping in Iron Foundry Automation Part II: Simulation www.ijcsi.org 238 Electromagnet Gripping in Iron Founry Automation Part II: Simulation Rhythm-Suren Wahwa Department of Prouction an Quality Engineering, NTNU Tronheim, 7051, Norway Abstract This paper

More information

A Note on Exact Solutions to Linear Differential Equations by the Matrix Exponential

A Note on Exact Solutions to Linear Differential Equations by the Matrix Exponential Avances in Applie Mathematics an Mechanics Av. Appl. Math. Mech. Vol. 1 No. 4 pp. 573-580 DOI: 10.4208/aamm.09-m0946 August 2009 A Note on Exact Solutions to Linear Differential Equations by the Matrix

More information

Lecture XII. where Φ is called the potential function. Let us introduce spherical coordinates defined through the relations

Lecture XII. where Φ is called the potential function. Let us introduce spherical coordinates defined through the relations Lecture XII Abstract We introuce the Laplace equation in spherical coorinates an apply the metho of separation of variables to solve it. This will generate three linear orinary secon orer ifferential equations:

More information

Microwave Reflection from the Region of Electron Cyclotron Resonance Heating in the L-2M Stellarator )

Microwave Reflection from the Region of Electron Cyclotron Resonance Heating in the L-2M Stellarator ) Microwave Reflection from the Region of Electron Cyclotron Resonance Heating in the L-2M Stellarator German M. BATANOV, Valentin D. BORZOSEKOV, Nikolay K. KHARCHEV, Leoni V. KOLIK, Eugeny M. KONCHEKOV,

More information

arxiv: v1 [cond-mat.stat-mech] 9 Jan 2012

arxiv: v1 [cond-mat.stat-mech] 9 Jan 2012 arxiv:1201.1836v1 [con-mat.stat-mech] 9 Jan 2012 Externally riven one-imensional Ising moel Amir Aghamohammai a 1, Cina Aghamohammai b 2, & Mohamma Khorrami a 3 a Department of Physics, Alzahra University,

More information

fv = ikφ n (11.1) + fu n = y v n iσ iku n + gh n. (11.3) n

fv = ikφ n (11.1) + fu n = y v n iσ iku n + gh n. (11.3) n Chapter 11 Rossby waves Supplemental reaing: Pelosky 1 (1979), sections 3.1 3 11.1 Shallow water equations When consiering the general problem of linearize oscillations in a static, arbitrarily stratifie

More information

Wave Propagation in Grounded Dielectric Slabs with Double Negative Metamaterials

Wave Propagation in Grounded Dielectric Slabs with Double Negative Metamaterials 6 Progress In Electromagnetics Research Symposium 6, Cambrige, US, March 6-9 Wave Propagation in Groune Dielectric Slabs with Double Negative Metamaterials W. Shu an J. M. Song Iowa State University, US

More information

RETROGRADE WAVES IN THE COCHLEA

RETROGRADE WAVES IN THE COCHLEA August 7, 28 18:2 WSPC - Proceeings Trim Size: 9.75in x 6.5in retro wave 1 RETROGRADE WAVES IN THE COCHLEA S. T. NEELY Boys Town National Research Hospital, Omaha, Nebraska 68131, USA E-mail: neely@boystown.org

More information

TOWARDS THERMOELASTICITY OF FRACTAL MEDIA

TOWARDS THERMOELASTICITY OF FRACTAL MEDIA ownloae By: [University of Illinois] At: 21:04 17 August 2007 Journal of Thermal Stresses, 30: 889 896, 2007 Copyright Taylor & Francis Group, LLC ISSN: 0149-5739 print/1521-074x online OI: 10.1080/01495730701495618

More information

A new proof of the sharpness of the phase transition for Bernoulli percolation on Z d

A new proof of the sharpness of the phase transition for Bernoulli percolation on Z d A new proof of the sharpness of the phase transition for Bernoulli percolation on Z Hugo Duminil-Copin an Vincent Tassion October 8, 205 Abstract We provie a new proof of the sharpness of the phase transition

More information

arxiv: v1 [physics.class-ph] 20 Dec 2017

arxiv: v1 [physics.class-ph] 20 Dec 2017 arxiv:1712.07328v1 [physics.class-ph] 20 Dec 2017 Demystifying the constancy of the Ermakov-Lewis invariant for a time epenent oscillator T. Pamanabhan IUCAA, Post Bag 4, Ganeshkhin, Pune - 411 007, Inia.

More information

A Model of Electron-Positron Pair Formation

A Model of Electron-Positron Pair Formation Volume PROGRESS IN PHYSICS January, 8 A Moel of Electron-Positron Pair Formation Bo Lehnert Alfvén Laboratory, Royal Institute of Technology, S-44 Stockholm, Sween E-mail: Bo.Lehnert@ee.kth.se The elementary

More information

Lecture 2 Lagrangian formulation of classical mechanics Mechanics

Lecture 2 Lagrangian formulation of classical mechanics Mechanics Lecture Lagrangian formulation of classical mechanics 70.00 Mechanics Principle of stationary action MATH-GA To specify a motion uniquely in classical mechanics, it suffices to give, at some time t 0,

More information

1 Heisenberg Representation

1 Heisenberg Representation 1 Heisenberg Representation What we have been ealing with so far is calle the Schröinger representation. In this representation, operators are constants an all the time epenence is carrie by the states.

More information

Diagonalization of Matrices Dr. E. Jacobs

Diagonalization of Matrices Dr. E. Jacobs Diagonalization of Matrices Dr. E. Jacobs One of the very interesting lessons in this course is how certain algebraic techniques can be use to solve ifferential equations. The purpose of these notes is

More information

The Three-dimensional Schödinger Equation

The Three-dimensional Schödinger Equation The Three-imensional Schöinger Equation R. L. Herman November 7, 016 Schröinger Equation in Spherical Coorinates We seek to solve the Schröinger equation with spherical symmetry using the metho of separation

More information

Semiclassical analysis of long-wavelength multiphoton processes: The Rydberg atom

Semiclassical analysis of long-wavelength multiphoton processes: The Rydberg atom PHYSICAL REVIEW A 69, 063409 (2004) Semiclassical analysis of long-wavelength multiphoton processes: The Ryberg atom Luz V. Vela-Arevalo* an Ronal F. Fox Center for Nonlinear Sciences an School of Physics,

More information

Adhesive Wear Theory of Micromechanical Surface Contact

Adhesive Wear Theory of Micromechanical Surface Contact International Journal Of Computational Engineering esearch ijceronline.com Vol. Issue. hesive Wear Theory of Micromechanical Surface Contact iswajit era Department of Mechanical Engineering National Institute

More information

Optimized Schwarz Methods with the Yin-Yang Grid for Shallow Water Equations

Optimized Schwarz Methods with the Yin-Yang Grid for Shallow Water Equations Optimize Schwarz Methos with the Yin-Yang Gri for Shallow Water Equations Abessama Qaouri Recherche en prévision numérique, Atmospheric Science an Technology Directorate, Environment Canaa, Dorval, Québec,

More information

inflow outflow Part I. Regular tasks for MAE598/494 Task 1

inflow outflow Part I. Regular tasks for MAE598/494 Task 1 MAE 494/598, Fall 2016 Project #1 (Regular tasks = 20 points) Har copy of report is ue at the start of class on the ue ate. The rules on collaboration will be release separately. Please always follow the

More information

ON THE OPTIMALITY SYSTEM FOR A 1 D EULER FLOW PROBLEM

ON THE OPTIMALITY SYSTEM FOR A 1 D EULER FLOW PROBLEM ON THE OPTIMALITY SYSTEM FOR A D EULER FLOW PROBLEM Eugene M. Cliff Matthias Heinkenschloss y Ajit R. Shenoy z Interisciplinary Center for Applie Mathematics Virginia Tech Blacksburg, Virginia 46 Abstract

More information

APPROXIMATE SOLUTION FOR TRANSIENT HEAT TRANSFER IN STATIC TURBULENT HE II. B. Baudouy. CEA/Saclay, DSM/DAPNIA/STCM Gif-sur-Yvette Cedex, France

APPROXIMATE SOLUTION FOR TRANSIENT HEAT TRANSFER IN STATIC TURBULENT HE II. B. Baudouy. CEA/Saclay, DSM/DAPNIA/STCM Gif-sur-Yvette Cedex, France APPROXIMAE SOLUION FOR RANSIEN HEA RANSFER IN SAIC URBULEN HE II B. Bauouy CEA/Saclay, DSM/DAPNIA/SCM 91191 Gif-sur-Yvette Ceex, France ABSRAC Analytical solution in one imension of the heat iffusion equation

More information

Applications of First Order Equations

Applications of First Order Equations Applications of First Orer Equations Viscous Friction Consier a small mass that has been roppe into a thin vertical tube of viscous flui lie oil. The mass falls, ue to the force of gravity, but falls more

More information

Vectors in two dimensions

Vectors in two dimensions Vectors in two imensions Until now, we have been working in one imension only The main reason for this is to become familiar with the main physical ieas like Newton s secon law, without the aitional complication

More information

Solution to the exam in TFY4230 STATISTICAL PHYSICS Wednesday december 1, 2010

Solution to the exam in TFY4230 STATISTICAL PHYSICS Wednesday december 1, 2010 NTNU Page of 6 Institutt for fysikk Fakultet for fysikk, informatikk og matematikk This solution consists of 6 pages. Solution to the exam in TFY423 STATISTICAL PHYSICS Wenesay ecember, 2 Problem. Particles

More information

(3-3) = (Gauss s law) (3-6)

(3-3) = (Gauss s law) (3-6) tatic Electric Fiels Electrostatics is the stuy of the effects of electric charges at rest, an the static electric fiels, which are cause by stationary electric charges. In the euctive approach, few funamental

More information

Solutions to Math 41 Second Exam November 4, 2010

Solutions to Math 41 Second Exam November 4, 2010 Solutions to Math 41 Secon Exam November 4, 2010 1. (13 points) Differentiate, using the metho of your choice. (a) p(t) = ln(sec t + tan t) + log 2 (2 + t) (4 points) Using the rule for the erivative of

More information

Damage detection of shear building structure based on FRF response variation

Damage detection of shear building structure based on FRF response variation , pp.18-5 http://x.oi.org/10.1457/astl.013.3.05 Damage etection of shear builing structure base on FRF response variation Hee-Chang Eun 1,*, Su-Yong Park 1, Rae-Jung im 1 1 Dept. of Architectural Engineering,

More information

Lecture 1b. Differential operators and orthogonal coordinates. Partial derivatives. Divergence and divergence theorem. Gradient. A y. + A y y dy. 1b.

Lecture 1b. Differential operators and orthogonal coordinates. Partial derivatives. Divergence and divergence theorem. Gradient. A y. + A y y dy. 1b. b. Partial erivatives Lecture b Differential operators an orthogonal coorinates Recall from our calculus courses that the erivative of a function can be efine as f ()=lim 0 or using the central ifference

More information

PH 132 Exam 1 Spring Student Name. Student Number. Lab/Recitation Section Number (11,,36)

PH 132 Exam 1 Spring Student Name. Student Number. Lab/Recitation Section Number (11,,36) PH 13 Exam 1 Spring 010 Stuent Name Stuent Number ab/ecitation Section Number (11,,36) Instructions: 1. Fill out all of the information requeste above. Write your name on each page.. Clearly inicate your

More information

Lagrangian and Hamiltonian Mechanics

Lagrangian and Hamiltonian Mechanics Lagrangian an Hamiltonian Mechanics.G. Simpson, Ph.. epartment of Physical Sciences an Engineering Prince George s Community College ecember 5, 007 Introuction In this course we have been stuying classical

More information

qq 1 1 q (a) -q (b) -2q (c)

qq 1 1 q (a) -q (b) -2q (c) 1... Multiple Choice uestions with One Correct Choice A hollow metal sphere of raius 5 cm is charge such that the potential on its surface to 1 V. The potential at the centre of the sphere is (a) zero

More information

Least-Squares Regression on Sparse Spaces

Least-Squares Regression on Sparse Spaces Least-Squares Regression on Sparse Spaces Yuri Grinberg, Mahi Milani Far, Joelle Pineau School of Computer Science McGill University Montreal, Canaa {ygrinb,mmilan1,jpineau}@cs.mcgill.ca 1 Introuction

More information

Switching Time Optimization in Discretized Hybrid Dynamical Systems

Switching Time Optimization in Discretized Hybrid Dynamical Systems Switching Time Optimization in Discretize Hybri Dynamical Systems Kathrin Flaßkamp, To Murphey, an Sina Ober-Blöbaum Abstract Switching time optimization (STO) arises in systems that have a finite set

More information

Further Differentiation and Applications

Further Differentiation and Applications Avance Higher Notes (Unit ) Prerequisites: Inverse function property; prouct, quotient an chain rules; inflexion points. Maths Applications: Concavity; ifferentiability. Real-Worl Applications: Particle

More information

Stable and compact finite difference schemes

Stable and compact finite difference schemes Center for Turbulence Research Annual Research Briefs 2006 2 Stable an compact finite ifference schemes By K. Mattsson, M. Svär AND M. Shoeybi. Motivation an objectives Compact secon erivatives have long

More information

Application of the homotopy perturbation method to a magneto-elastico-viscous fluid along a semi-infinite plate

Application of the homotopy perturbation method to a magneto-elastico-viscous fluid along a semi-infinite plate Freun Publishing House Lt., International Journal of Nonlinear Sciences & Numerical Simulation, (9), -, 9 Application of the homotopy perturbation metho to a magneto-elastico-viscous flui along a semi-infinite

More information

12.5. Differentiation of vectors. Introduction. Prerequisites. Learning Outcomes

12.5. Differentiation of vectors. Introduction. Prerequisites. Learning Outcomes Differentiation of vectors 12.5 Introuction The area known as vector calculus is use to moel mathematically a vast range of engineering phenomena incluing electrostatics, electromagnetic fiels, air flow

More information

Physics 505 Electricity and Magnetism Fall 2003 Prof. G. Raithel. Problem Set 3. 2 (x x ) 2 + (y y ) 2 + (z + z ) 2

Physics 505 Electricity and Magnetism Fall 2003 Prof. G. Raithel. Problem Set 3. 2 (x x ) 2 + (y y ) 2 + (z + z ) 2 Physics 505 Electricity an Magnetism Fall 003 Prof. G. Raithel Problem Set 3 Problem.7 5 Points a): Green s function: Using cartesian coorinates x = (x, y, z), it is G(x, x ) = 1 (x x ) + (y y ) + (z z

More information

PHY 114 Summer 2009 Final Exam Solutions

PHY 114 Summer 2009 Final Exam Solutions PHY 4 Summer 009 Final Exam Solutions Conceptual Question : A spherical rubber balloon has a charge uniformly istribute over its surface As the balloon is inflate, how oes the electric fiel E vary (a)

More information

II. First variation of functionals

II. First variation of functionals II. First variation of functionals The erivative of a function being zero is a necessary conition for the etremum of that function in orinary calculus. Let us now tackle the question of the equivalent

More information

Nested Saturation with Guaranteed Real Poles 1

Nested Saturation with Guaranteed Real Poles 1 Neste Saturation with Guarantee Real Poles Eric N Johnson 2 an Suresh K Kannan 3 School of Aerospace Engineering Georgia Institute of Technology, Atlanta, GA 3332 Abstract The global stabilization of asymptotically

More information

Physics 2212 K Quiz #2 Solutions Summer 2016

Physics 2212 K Quiz #2 Solutions Summer 2016 Physics 1 K Quiz # Solutions Summer 016 I. (18 points) A positron has the same mass as an electron, but has opposite charge. Consier a positron an an electron at rest, separate by a istance = 1.0 nm. What

More information

MULTISCALE FRICTION MODELING FOR SHEET METAL FORMING

MULTISCALE FRICTION MODELING FOR SHEET METAL FORMING MULTISCALE FRICTION MODELING FOR SHEET METAL FORMING Authors J. HOL 1, M.V. CID ALFARO 2, M.B. DE ROOIJ 3 AND T. MEINDERS 4 1 Materials innovation institute (M2i) 2 Corus Research Centre 3 University of

More information

Gravitation as the result of the reintegration of migrated electrons and positrons to their atomic nuclei. Osvaldo Domann

Gravitation as the result of the reintegration of migrated electrons and positrons to their atomic nuclei. Osvaldo Domann Gravitation as the result of the reintegration of migrate electrons an positrons to their atomic nuclei. Osvalo Domann oomann@yahoo.com (This paper is an extract of [6] liste in section Bibliography.)

More information

Extinction, σ/area. Energy (ev) D = 20 nm. t = 1.5 t 0. t = t 0

Extinction, σ/area. Energy (ev) D = 20 nm. t = 1.5 t 0. t = t 0 Extinction, σ/area 1.5 1.0 t = t 0 t = 0.7 t 0 t = t 0 t = 1.3 t 0 t = 1.5 t 0 0.7 0.9 1.1 Energy (ev) = 20 nm t 1.3 Supplementary Figure 1: Plasmon epenence on isk thickness. We show classical calculations

More information

Chapter 2 Lagrangian Modeling

Chapter 2 Lagrangian Modeling Chapter 2 Lagrangian Moeling The basic laws of physics are use to moel every system whether it is electrical, mechanical, hyraulic, or any other energy omain. In mechanics, Newton s laws of motion provie

More information

Shape Effect on Blind Frequency for Depth Inversion in Pulsed Thermography

Shape Effect on Blind Frequency for Depth Inversion in Pulsed Thermography Shape Effect on Blin Frequency for Depth Inversion in Pulse Thermography M. Genest 1, E. Grinzato 2, P. Bison 2, S. Marinetti 2 C. Ibarra-Castaneo 1, X. Malague 1 1 Electrical an Computing Eng. Dept.,

More information