WKB solution of the wave equation for a plane angular sector

Transcription

1 PHYSICAL REVIEW E VOLUME 58, NUMBER JULY 998 WKB solution of the wave equation for a plane angular sector Ahmad T. Abawi* Roger F. Dashen Physics Department, University of California, San Diego, La Jolla, California 993 Received 3 December 997; revised manuscript received March 998 The two angular Lamé differential equations that satisfy boundary conditions on a plane angular sector PAS are solved by the Wentzel-Kramers-Brillouin WKB method. The WKB phase constants are derived by matching the WKB solution with the asymptotic solution of the Weber equation. The WKB eigenvalues eigenfunctions show excellent agreement with the exact eigenvalues eigenfunctions. It is shown that those WKB eigenvalues eigenfunctions that contribute substantially to the scattering amplitude from a PAS can be computed in a rather simple way. An approximate formula for the WKB normalization constant, which is consistent with the WKB assumptions, is derived compared with the exact normalization constant. S63-65X PACS numbers: 4.5.p I. INTRODUCTION In a previous paper, formulas for the Wentzel- Kramers-Brillouin WKB eigenvalues satisfying Dirichlet or Neumann boundary condition on a plane angular sector PAS were reported the WKB eigenvalues,,, were compared with the exact eigenvalues for PAS s of different corner angles 6, 9,. A historical review of the solution of the wave equation for a PAS was also given in the above paper. It suffices to say that, to our knowledge, no approximate solution of this problem has been reported in the literature. In this paper, the two coupled Lamé equations are solved by the WKB method. The WKB analysis in this paper is valid for large values of. Depending on the sign of, one of the two Lamé equations has turning points. When is small the turning points occur the angles are small. In this region it proves more accurate to obtain the WKB phase constants by matching it with the asymptotic solution of the Weber equation. For large values of, it is shown that this phase constant reduces to the phase constant obtained when the solution is matched with the asymptotic solution of the Airy s equation as is commonly done in quantum mechanics. This paper is organized in the following way: In the second section the WKB solution is formulated the WKB phase constants are derived. In the third section formulas for the WKB eigenvalues for Dirichlet Neumann boundary conditions are derived a comparison between the WKB exact eigenfunctions is presented. The fourth section contains a derivation of approximate WKB solutions which are valid for small values of /. Finally, an approximate formula for the WKB normalization constant consistent with WKB assumptions is derived in section five. II. THE GENERAL SOLUTION The angular part of the wave equation in the sphero-conal coordinate system are expressed by the two angular Lamé differential equations,: *Present address: Propagation Division, SPAWAR Systems Center, D88, 5356 Hull Street, San Diego, CA 95-5l. cos d d cos d d sin cos d d cos d d sin, the sphero-conal coordinate system variables are related to the Cartesian coordinate variables, x, y, z by xr cos cos, yr sin sin, zr cos cos, cos, is the corner angle. The range of the variables is,, r. The geometry of the sphero-conal coordinate system is shown in Fig.. The construction of this coordinate system is described, its orthogonality proved, in. The WKB solution of Eqs. are respectively given by X/98/58/53/$5. PRE The American Physical Society

2 5 AHMAD T. ABAWI AND ROGER F. DASHEN PRE 58 FIG.. The geometry of the sphero-conal coordinate system. r is the distance from the origin to the point p. 4 sin cos sin t cos 4 sin cos sin t cos / d, / d. Notice that the solution can be obtained from the solution by replacing with, with, with. This is also evident in the differential equations, Eqs.. For, the turning point for the equation, t, the turning point for the equation is t t (,,). For, t t t (,,), / t,,arcsin. The above solutions are valid for t t t t. For the regions t t t t the solutions are given by 4 sin sin exp. cos d /, A similar solution for the equation which is valid for can be obtained from the above equation by replacing,, with,,, respectively. The solutions given by Eqs. 4 6 are separated by the turning points they are both singular. To match these solutions at the turning points, the Lamé equation is approximated by another differential equation at the turning point then the asymptotic solution of this differential equation is matched with the WKB solution of the Lamé equation. By this process the phase constants are obtained. Although the WKB solutions obtained in this paper are general, we are particulary interested in those solutions that correspond to the small absolute values of the eigenvalue. The reason for this is that the expression for the scattering amplitude for a PAS, derived in a separate paper 3, contains either the eigenfunctions or their derivatives evaluated at the surface of the PAS or its edges. Observe that the surface of the PAS in the sphero-conal coordinate system is its edges are. Since the WKB eigenfunctions are decaying exponentials to the left of the turning point near zero to the right of the turning point near, significant contribution to the solution comes from those eigenfunctions that have a turning point near the surface or near the edges. They correspond to eigenfunctions with small absolute values of the eigenvalue. To derive a differential equation that approximates the Lamé equation at the turning points, we use the following transformation: 4 cos. For this transformation converts Eq. to d d p, with p sin cos. By using the Liouville transformation 4 yx dx Eq. 7 can be transformed to d y dx dx d / d, p dx d / d dx d / dx y. The first term in the curly brackets can be set equal to any smooth function of x 4, for small / the second term can be ignored. Near the turning points we set dx p d x 4 a, 8 resulting in 7 d y dx x 4 a y, 9

3 PRE 58 WKB SOLUTION OF THE WAVE EQUATION FOR A which is the Weber equation. In the above equation the plus sign is applicable to the equation the minus sign is applicable to the equation. The parameter a is determined from Eq. 8 by requiring that the turning points of the Lamé equation the Weber equation occur at the same time, thus ensuring the regularity of d/dx at the turning points: t sin /d cos a x / 4 a dx. A similar equation is obtained for the equation by replacing t by t, by, by, a by a. Once a is determined from the above equation, can be related to x by t sin /d cos x x / 4 a dx. sin cos t a is found to be a /, x d a,e Ke, e /,, x / 4 a dx. 3 is the elliptic integral of the third kind. For a can be obtained from the above equation by replacing by interchanging. Equation 3 has a power series expansion given by a O 4, If only the first term in the above expansion is retained,. a. 4 If this value of a is used in Eqs., near the turning points small the above equations respectively yield x/, x/. 5 Equation 9 is the desired differential equation. The asymptotic solution of this equation will be used to determine. A. The WKB phase constants The Weber equation has solutions 5 d y dx x 4 a y 6 Wa,x 3/4 G /G 3 y e G 3 /G y o, y e x,aa x! a x4 4! 7 a 3 7 a x6, 8 6! y o x,axa x3 3! a 3 x5 5! a 3 3 a x7, 9 7! G a 4 ia, G 3 a 3 4 ia. For xa the Weber equation has asymptotic solutions 5 Wa,x a / x cos 4 x a ln x 4 a / Wa,x ax sin 4 x a ln x 4 a, ae a e a, aarg ia. From these two solutions we construct an even solution

4 54 AHMAD T. ABAWI AND ROGER F. DASHEN PRE 58 ỹ e x,a 4 G 3a G a / / a / Comparing the phase amplitude of this equation with a x that of ỹ e (x,a) we find an odd solution ỹ o x,a 4 G a cos x 4 a ln x 4 aa, / a / G 3 a / a x cos x 4 a ln x 4 aa, tana a ea e a. In the region is small so that the Lamé equation can be approximated by the Weber equation, yet is large so that the WKB solution for the Lamé equation is valid, x / / a, 4 aa a lna a A 4 G 3a G a It can be shown that / a a /. a 3 4 arctan tanh a DaDa. In terms of this new quantity we have 8 a Da a lna a. 3 The phase constant for the odd solution is obtained by comparing I with the phase of Eq.. Itis 5 8 a Daa lna a. A similar analysis gives both Eq. 5 Eqs. or are solutions of the same differential equation. The phase constant can therefore be determined by matching the phases of the two solutions. Depending on the prescribed boundary conditions, either ỹ e or ỹ o will be employed to obtain these phase constants. First consider the even solution for the equation. From Eq. we have sin I t x a cos x 4a dx / d x 4 x 4aalnxx 4aa lna. Exping the last expression in powers of a keeping only terms of first order gives I x 4 a ln x a lna a. The WKB solution, Eq. 5, thus becomes x A 4 4 x cos x 4 a lnx a lna a. 8 a Da a lna a A 4 G 3a G a And for the odd solution we find / a a 5 8 a Da a lna a. 4 /. 5 For the role of the the equations are interchanged. In other words, in this case the equation is the one with the turning points. However, the expressions obtained for the phase constants still remain valid. In summary we have 8 a Da a lna a, 8 a Da a lna a, 5 8 a Da a lna a, 5 8 a Da a lna a.

5 PRE 58 WKB SOLUTION OF THE WAVE EQUATION FOR A The above phase constants were derived for small values of a when the turning points lie close to or. For large positive values of a, D(a) /4 (a) a lnaa. In this limit /4, 3/,, /. For large negative values of a, D(a) /4 (a) alnaa. In this limit /, /4, 3/. These are the phase constants that would have been obtained had the problem been treated by the regular WKB method often employed in solving the Schrödinger equation in quantum mechanics. In solving the Schrödinger equation by the WKB method, the phase constants are obtained by matching the phase of the WKB solution with the phase of the asymptotic solution of the Airy s equation. III. THE WKB SOLUTIONS By determining the phase terms,,,, we now have the complete WKB solutions of the the equations. In this section we derive the WKB eigenvalue equations by applying the boundary conditions imposing the requirement that the solutions for ; ( t ; t ) those for ; ( t ; t ) join each other smoothly in their common region of validity. This results in relationships for the WKB eigenvalues. Both Dirichlet Neumann boundary conditions will be considered. A. Dirichlet boundary condition For the Dirichlet boundary condition we have,,,. That is, the solution must be even at both, the solution must be even at odd at. The WKB solution of the equation valid in the region t is A hcos t sin cos / d, t, h. 4 sin Since for the Dirichlet boundary condition is even at both, the WKB solution for t is given by Ahcos t sin / cos d, t. The two solutions, () (), must join smoothly in the region t t. Let us define J t sin cos / sin t t cos / / d d, 6 sin cos t t t t J b. / d sin cos sin In terms of these quantities we have cos / d / d b t sin cos / d, A hcosj b A hcos b. then Equating gives

6 56 AHMAD T. ABAWI AND ROGER F. DASHEN PRE 58 A cosj cos bsinj sin ba cos b. The solution to this equation, which gives A, constant independent of the parameter b, is obtained by setting J m, m,,... Substituting for, we find J adaa lnaam 4, m,,... For Dirichlet boundary condition is even around odd around. Then A hcos t sin cos / d,, h is h with the appropriate change of variables. The solution for t is A hcos t sin / cos d, t. Since this solution is odd with respect to, the solution for t is given by Ahcos t sin / cos d, t. Requiring that () () join each other smoothly results in J n, n,,... Note that the reason n starts from is to guarantee that n, since J is positive. Substituting for we find J aa lnaan, n,,... For Dirichlet boundary condition we therefore have the set of eigenvalue equations J adaa lnaam 4, J aa lnaan, m,,..., n,,... 7 The integrals defined by J J can be expressed in terms of elliptic integrals 6 J J,,,r,,r Kr for J J,,r Kr, for. In the above,,r r /, r, / is the elliptic integral of the third kind, K is the elliptic integral of the first kind. We also have the following identity for J J : J J. 8

7 PRE 58 WKB SOLUTION OF THE WAVE EQUATION FOR A FIG.. The top two panels show the the eigenfunctions for the exact solid lines, the WKB dashed lines the power series solution to the Weber equation dotted lines of the Lamé equations, subject to Dirichlet boundary condition. The corresponding exact eigenvalues are,6.8 85,.44 73, the corresponding WKB eigenvalues are,6.8 34, The bottom two panels show the same eigenfunctions for Neumann boundary condition with exact eigenvalues, , WKB eigenvalues, , Note how the power series solutions overlay the exact solutions for close to zero connect smoothly with the WKB solutions. B. Neumann boundary condition For the Neumann boundary condition we have,,,. The solution is thus odd with respect to the solution is odd with respect to even with respect to. A similar approach yields the eigenvalue equations: J adaa lnaam 3 4, m,,... J aa lnaan, n,, Note that Eqs. 7 9 are two parameter eigenvalue equations. This means that for a given value of n m each set of equations must be solved simultaneously. We used the Newton-Raphson method in two dimensions to compute the eigenvalues, for each boundary condition. The WKB eigenvalues agree remarkably well with the exact eigenvalues. C. The WKB eigenfunctions Once the eigenvalues are computed, the WKB eigenfunctions can be obtained from Eqs It is obvious that either of these solutions multiplied by a constant is still a solution. However, the amplitudes A A were determined in such a way that the WKB solutions can be compared with the exact solution with the solution of the Weber equation without the need to multiply them by a constant. This proves convenient when comparing individual eigenfunctions. The complete normal mode solution of the wave equation for this problem involves a sum over the product of all eigenfunctions divided by the normalization constant that removes any amplitude ambiguities 3. The WKB normalization constant is derived later in this paper. For even solutions the amplitudes A A given by Eqs.

8 58 AHMAD T. ABAWI AND ROGER F. DASHEN PRE 58 sin / cos 5 are determined such that z() z(), z can be either or. For odd solutions, on the other h, the initial conditions are z() z(). Because the derivative of the solution is nonzero at the initial point, A A must be multiplied by the scale factors d/dx d/dx, respectively. To see this, take, for example,, which near reduces to the solution of the Weber equation, y(x). Then d dyx dyx d d dx x dx d. J / sin sin d, a,. Since dy/dx x is chosen to be unity, multiplied by d/dx satisfies the above equation. The same is true for. d/dx d/dx may be obtained from Eq. 5. As was pointed out before, we are particularly interested in those solutions for which the turning points lie close to or. In this case near the solution can be obtained from the power series solution of the Weber equation given by Eqs For a given set of eigenvalues,, a is determined from Eq.. Then the power series solution are obtained by choosing or as independent variables determining x from Eqs. or, which is then used in Eqs The results are shown in Fig.. By only using a few terms, the power series solution overlays the exact solution for close to zero smoothly connects with the WKB solution in each case. In this way an excellent approximation to the exact solution can be obtained for the entire interval between. IV. SOLUTION FOR LARGE AND SMALL It was pointed out earlier that the main contribution to the scattering amplitude comes from those eigenfunctions that correspond to small values of. It will be shown in this section that when /, the expressions for the eigenvalues the integrals appearing in the phases of the WKB solutions can be approximated by simple algebraic functions. First, consider the integral For small values of this integral is approximated by 7 Similarly, J arctan a ln8 a lnaao. J arctan a ln8 a lnaao. 3 3 By adding the eigenvalue equations for the Dirichlet boundary condition, Eq. 7, using Eq. 8 we find mn Da 4. 3 For Neumann boundary condition we similarly find mn 3 Da By subtracting the eigenvalue equations for the Dirichlet boundary condition using Eqs. 3, 3, 3 we find mn 3 4 Da a ln 8mn 3 Da 4 adanm In the same way we find for the Neumann boundary condition mn 5 Da 4 a ln 8mn 5 Da 4 adanm In the above equations use has been made of the relationships arctan, arctan. Equations have the advantage that they only depend on the parameter a. This allows one to solve these equations for a by performing a search in one dimension as opposed to two dimensions when the equations depend on as well then use Eqs to determine. The eigenvalues obtained in this way are still in excellent agreement with the exact eigenvalues. The phase of the WKB solution for the equation, Eq. 4 is

9 PRE 58 WKB SOLUTION OF THE WAVE EQUATION FOR A sin / V. THE WKB NORMALIZATION cos d. f t This can be written as ff/u, f t / sin cos / d, 36 In this section we derive an approximate formula for the WKB normalization coefficient that is valid for small values of large values of. The normalization coefficent for the solution of the wave equation in the sphero-conal coordinate system is given by N Lamé Lamé d d, 38 u / sin cos d / sin cos d, sin sin cos cos. Let us define Lamé Lamé Lamé Lamé,. Using the WKB eigenvalue equations the expressions for, we find for both Dirichlet Neumann boundary conditions f n 4 Da. Similarly the phase of the equation can be written as gg u. For the Dirichlet boundary condition we find g m, for the Neumann boundary condition we find g m Da. In the above g is f with replaced by, by, by. For small it may be shown that 7 u arcsin cos a ln cos cos sin, 37 a/( ). A similar solution can be obtained for u by replacing with a with a. then by submitting these quantities in Eq. 38, wefind N sin Lamé d cos Lamé d cos sin Lamé d cos Lamé d. cos The above integrals are of the general form Lamé d, 39 can be either or represents the other functions appearing in Eq. 39. Now let us write Lamé ()()d as Lamé d Lamé WKB Lamé d WKB Lamé d, 4 WKB Lamé is the WKB solution of the Lamé equation. The first integral can be written as

10 6 AHMAD T. ABAWI AND ROGER F. DASHEN PRE 58 Lamé WKB Lamé d Lamé WKB Lamé d Lamé WKB Lamé d Lamé WKB Lamé d. Weber WKB Weber d Weber WKB Weber d Weber WKB Weber d. For, Weber WKB Weber thus the second integral in the above equation is zero Since for close to or the Lamé equation reduces to the Weber equation, is chosen to be small enough such that for, the solution of the Lamé equation can be represented by the solution of the Weber equation, denoted by Weber, the WKB solution of the Lamé equation can be represented by the WKB solution of the Weber equation, denoted by WKB Weber, given by Eqs. or. In the region, both the solution of the Lamé equation the solution of the Weber equation can be represented by their WKB solutions: Lamé WKB Lamé, Weber WKB Weber. Based on this argument the middle integral in the above equation is zero in the other integrals Laḿe can be replaced by Weber, resulting in Thus Weber WKB Weber d Weber WKB Weber d. Lamé d Weber WKB Weber d Weber WKB Weber d Lamé WKB Lamé d WKB Lamé d. 4 Weber WKB Weber d Weber WKB Weber d. Note that in the first integral the functions in the square brackets are valid near in the second integral they are valid near. In the equation for the normalization we either have Substituting these in Eq. 4 we find Lamé d or sin cos Weber WKB Weber d Weber WKB Weber d WKB Lamé d. In the second integral in the above equation let, so Weber WKB Weber d Weber WKB Weber d. Next, let us write In the first case we have cos. sin Lamé cos d Weber WKB sin Weber cos d Weber WKB sin Weber cos d WKB sin Lamé cos d.

11 PRE 58 WKB SOLUTION OF THE WAVE EQUATION FOR A... 6 There is negligible contribution from the first two integrals on the right h side because for small values of, sin for large values of, Weber () WKB Weber (). Therefore, we find sin Lamé cos d WKB sin Lamé cos d. In the second case Lamé cos d Weber WKB Weber cos d Weber WKB Weber cos d WKB Lamé cos d. Here again there is negligible contribution from the first two integrals for large values of since Weber () WKB Weber (). For small valuers of thus we have Let us define cos, Lamé cos d Weber WKB Weber d WKB Lamé Weber WKB Weber d cos d. Weber WKB Weber d, TABLE I. Comparison between the exact WKB normalization coefficients. (n,m) exact WKB N exact N WKB, , , , , , , , , , , , , , Lamé cos d WKB Lamé cos d. Following a similar procedure using integrals involving the variable substituting in Eq. 38 yields N sin WKB Lamé d cos C WKB Lamé cos d sin WKB Lamé d cos C WKB Lamé cos d, Weber WKB Weber d, Weber WKB Weber d, C C are proportionality constants given by 4 Weber WKB Weber d, Lamé dc Weber xdx, 43 then

12 6 AHMAD T. ABAWI AND ROGER F. DASHEN PRE 58 Lamé dc Weber xdx. 44 accurate, small, the approximation leading to Eq. 45 is valid, N WKB matches N exact very closely. Expressions for C C are derived in the Appendix. The evaluation of the integrals appearing in the expressions for s is quite lengthy. In this paper we only report the results. The interested reader may refer to Ref. 7 for a detailed derivation. We find the WKB normalization coefficient N A A ln 8 Re ia arcsin e a e a, 45 is the digamma function defined by (z) d(z)/dz In the above equation the plus sign is used for the Dirichlet boundary condition the minus sign is used for the Neumann boundary condition. Table I shows a comparison between the exact the WKB normalization coefficients for a 9 PAS subject to Dirichlet boundary condition. In Table I exact exact exact, WKB WKB WKB, n m are defined in Eq. 7. As can be seen from Table I, for large values of n, the WKB solution is APPENDIX: THE DERIVATION OF C C We use the fact that for small angles the Laḿe the Weber equations have the same solution up to a multiplicative constant. First, note that the WKB solution of the Weber equation using the amplitude factors in Eqs. in Eq. 7 is given by Then Weber x / a a 4 x 4a cos x x 4a dx. Weber x dx a a x 4a On the other h we have cos x x 4adxdx. A Lamé da cos sin cos sin d / d. The arguments of the cosines in the above two equations are the same according to Eq.. For small, we replace sin by use to get Lamé d / x, A cos x x 4adxdx. x 4a A Comparing Eqs. A A with Eq. 43 yields C A a a, a/( ). In a similar manner we find A C a a.

13 PRE 58 WKB SOLUTION OF THE WAVE EQUATION FOR A Ahmad T. Abawi, Roger F. Dashen, Herbert Levine, J. Math. Phys. 38, L. Kraus L. Levine, Commun. Pure Appl. Math. 9, Ahmad T. Abawi Roger F. Dashen, Phys. Rev. E 56, F. W. J. Olver, Asymptotics Special Functions Academic, New York, Hbook of Mathematical Functions, edited by M. Abramowitz I. A. Stegun Dover, New York, I. S. Gradshteyn I. M. Ryzhik, Tables of Integrals, Series, Products, nd ed. Academic, New York, Ahmad T. Abawi, Ph.D. thesis, University of California, San Diego, 993.