5603-1-F Copy THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL ENGINEERING Radiation Laboratory FINAL REPORT ON AFOSR GRANT 289-63 by Frederick B. Sleator June 1964. ^ ' AFOSR Grant No. 289-63 5603-1-F = RL-2133 Contract With: United States Air Force Air Force Offioe of Scientific Research Washinton 25, D.C. Admimistered through: OFFICE OF RESEARCH ADMINISTRATION * ANN ARBOR

THE UNIVERSITY OF MICHIGAN 5603-1-F FINAL REPORT ON AFOSR GRANT 289-63 I. Introduction The class of diffraction and radiation problems to which the spheroidal geometry is appropriate is extensive and of considerable interest from many standpoints, and considerable effort has been expended in this field over the past quarter century or more. Generally speaking, these problems yield readily to the standard methods of attack, to the extent that their solutions are expressible in terms of the appropriate sets of eigenfunctions of the separated wave equation. Unfortunately these spheroidal functions, though much more extensively tabulated now than a few years ago, still represent a serious obstacle in the attainment of quantitative results. The dependence of each function on four independent parameters or arguments indicates the immensity of the number of values which may be required, and the complete lack of recurrence relations and scarcity of other computation aids makes the calculation of these values extremely tedious for anything less than a large-scale facility. It has been the objective of this investigation, as well as an earlier one under AFOSR Grant 62-265, to develop and exploit new relations between the spheroidal functions and other better known functions which might reduce the labor and complication involved in obtaining new numerical values of the spheroidal functions and thus facilitate the quantitative solutions of new spheroidal diffraction problems or the extension of existing solutions into ranges not previously covered. II. Discussion In the earlier investigation it was intended to develop certain relations between spheroidal and elliptic functions with a view toward making use of the fairly comprehensive tables which exist for the latter. Although the results of this effort were largely negative, in that the relations found did not seem to be useful in general, the original ideas have yielded a relationship involving certain integrals of exponential type, which is apparently new and merits further consideration. This is derived briefly as follows. 1

THE UNIVERSITY OF MICHIGAN 5603-1-F In the standard literature on spheroidal functions(, there appears the following expansion of the product of a radial and an angular function: OD I S (c,rl) Rm(cn, ) = im dmn(c)Pm (cos9) (kr) r function in terms of a series of spherical ones, as in the expansion mn I Smn(, ) = dmn(c) Pm+r (2) mn r=O 1 r m+r(2) If eq. (2) is substituted in (1) and both sides are then multiplied by P m (r)) drc m+3 and integrated between -1 and +1, the orthogonality of the Legendre functions leads to the form 2(2m+ d) d n( R(l ) = > r+m-ndmn() m (s ) (2m + 2(c i d (c) m P +((cos 8)) rO(2m+2,+l)B 1 mn = r m+r m+r(kr()) Pm+(n) d (3) or, setting 2m+2s1! *irm Pm+r(cos(r)) Pm+s(r) m+r(kr(r7))d7 Cm 2 (2m+s)t M+r mrs (-1 ()e.g. Flammer, Spheroidal Wave Functions, Stanford University Press, 1957. 2

THE UNIVERSITY OF MICHIGAN 5603-1-F ao inR() (C, )m dmn d (c) Cm (c, ) mn S r r rs r=0, 1 Since the integration over r t with e held constant is actually a line integral over an elliptic contour, the arguments of the spherical functions in the integrand are found from the geometry of the spherical and spheroidal coordinate systems to be gn / a r -1 cos6l8) = (Y) = Examination of the integrals which remain reveals that these are not elliptic but are reducible to combinations of elementary and exponential (i. e. sine or cosine) integrals. If the index s is allowed to range from 0 to oo, there results an infinite system of linear homogeneous equations in the "unknowns" d (c), which r can be written r=o, 1 det (c) [C (c, ) 6 iR(1) (c,( = 0, Existence of a solution to this system requires the vanishing of the determinant, i.e. rs rs mn and the quantity inR(1) (c, ) is thus cast in the role of an eigenvalue of the infinite mn matrix rs(c, )j}, which could be determined by standard methods if the elements of the matrix were known. Once this eigenvalue and the requisite number of elements are known, the system (4) can presumably be solved (approximately) for some finite number of coefficients dmn(c). r 3

THE UNIVERSITY OF MICHIGAN 5603-1-F Considering the intractable nature of the integrals in (3) and the difficulties inherent in the manipulation of infinite matrices, it is not clear whether or not the above development is of any immediate practical value. It appears, however, on the basis of the known properties of the spheroidal coefficients that it should be possible to construct a Hulbert space, the elements of which are vectors d with components d (c), and if this is the case the considerable body of general theory r on linear transformations in Hilbert space would be applicable to the matrix tC m rs and might yield vmults if interest. The advantages to be gained by this line of attack appeared uncertain, and the task seemed large, and consequently it was not pursued further. Instead, the principal effort has been expended on a simpler and more direct miod of obtaining approximations to the spheroidal functions, which we proceed to outline in general terms before giving the specific forms of interest. Given a linear ordinary differential equation of the form L U = [X+ p(] U (5) whose solutions U(9, X) are to be determined, suppose there is a simpler equation, which can be written M V = AV, (6) x whose solutions V(x,j) are well known, and further, that we can find a function 9(x) such that L(x) M lim Now for some particular value e let + (+p()) 5 P. Substituting this and the function 9(x) in eq. (5) then gives an equation of the form 4

THE UNIVERSITY OF MICHIGAN 5603-1-F M U = I U + e(x)U (7) x o where, under reasonable assumptions of regularity, e(x) is small in some neighborhood of x = x(9 ), and in this neighborhood a solution U of eq. (7) must thus be approximated by a solution V(x(f), j) of eq. (6) and in general an iteration scheme can be developed to extend the range or improve the approximation. The accuracy of the result of course depends intimately on the natures of the operators and functions involved and the region in which the forms are applied. Furthermore the utility is limited in that no information on the normalization is involved, i. e. the value of the unknown function is determinable only in the neighborhood of a point at which it is already known. Nonetheless the procedure seems to offer advantagee in certain applications, some of which are described briefly here in terms of the spheroidal functions. For the sake of definiteness we consider first the radial functions, for which the operator Lg in (5) has the form L d9 L = d - - 1) d 2_1 2 2 and p(g) = -c. If for the approximating equation (6) we take the ordinary Bessel equation, whose solutions are perhaps the most convenient set of functions we can hope to use, then the requisite transformation g(x) is easily found to be 2+x2 2 2 or = a2 -x where a is an arbitrary constant, and the transformed equation can be written explicitly as 5

THE UNIVERSITY OF MICHIGAN 5603-1-F dn 412X n m d2 dU 22 2) 2 2 2 2 2 4 2 mn + mn + 4 a x ) m mn 2 x dx 2 24 2 2 222 mn dx- L -x) x (aL -x ) - For our immediate purposes we can abandon the constant a, which seems to be immaterial, and consider the case where e i 1. The quantity AO in eq. (7) is 2 0 0 then simply 4(c -X ), and for the sake of clarity in the argument which follows, rn 2 we incorporate this in the independent variable, i.e. set ca 2x c -X and mn write the transformed equation finally as d 2U dU 2 mn +1 mn + - + E(a) = a du 2 JUmn = da a acia with 2 ) 6(c -mn 2 -2 21 X 22)Ii (a) a= E 2 2(c2_X )+aX2 2]- 1. l(cc -X ) - 2 1 n J n m If e(a) is small, the function U (a) should be approximated by a cylindrical function, mn i.e. U (a) a Z (a) (8) mn m where a is independent of a, but a now depends on c and X as well as x. To mn this approximation, then, the only dependence of U on n is through a. The mn accuracy of the approximation however depends on the magnitude of e(a), which of course depends on n, and this dependence must be examined. In terms of the variable x, it develops that 2 r (ox))- X2 +(2-x2)(1 -x2. (l-x), mn 2 C 6

THE UNIVERSITY OF MICHIGAN 5603-1-F In general, if c is fixed, the eigenvalue Xn increases with n, so that for fixed mn c and x << 1, there will be an infinite range of values of n over which E is uni2 formly bounded and of order x. Once the constant a and the eigenvalues X mn are determined, the approximation (8) should thus be useful for all n above a certain value. Another possible application of the above forms is in the determination of the set of eigenvalues which make the radial functions vanish on a certain spheroidal surface, as required in the Sommerfeld-type solution of a scattering problem. Suppose that for given m and some x within the range of applicability of (8) one of the requisite eigenvalues, say X is known. Then if a = 2x c - X, we can mo o mo write U - aJ (a )+bN (a) = 0 mo m o m o in which a and b, being independent of a, cannot depend on n. Then N (ar ) a _ m o b J (a ) m o and for any other n, we can write aJ (a)+bN (a) = b (a)N ( J (a)N ( - m n m n J (a) m n m n m o m o or J (a ) N (a ) - J (a ) N (a) - 0 m o m n m n m o which is a familiar equation, some roots of which are tabulated in the standard literature on Bessel functions. The nth eigenvalue X is thus given approximately mn 7

THE UNIVERSITY OF MICHIGAN 5603-1-F in terms of the nth root of this equation by the form 2 2 a x N, 2 n X - -. mn 4 It appears that it should be possible to obtain analogous approximations to the ordinary eigenvalues, i.e. the values of X for which the angular functions are mn finite at rl = +1. To date, however, this objective has not been attained, despite a fairly intensive effort. The analysis is complicated by the facts that at the finite singular points the functions vanish to order m/2, and at the origin the dependence on the parameter c disappears. Perhaps the most useful application of the above forms will prove to be in the determination of values of the functions in neighborhoods of points at which they are already known, i.e. in the development of refined methods of interpolation and extrapolation. To illustrate some of these, we now consider the angular functions, whose transformed equation is written dS dS 2 2 2 mn 1 mn 4 2 4(p- 1) m + - (p. 2 4S =0, 2 p dp mn 2 2+ 2 (p2+1)4 2 mn dp L (p + 1) (p +1) p -1 with p = In some neighborhood of an arbitrary point ri, the angular function S (c, n) can be expressed approximately by the form nmn nn S (cn) a J (Y p)+ B N ( P'Yo 9) mn mn m o mn m o mn\ 22 where 7 (1-i ) mn-c r and an, mn are independent of rl. If two values are given in the allowable range, mng. values of at two poits or values of n and its derivative at one pointmn e.g. values of S at two points or values of S and its derivative at one point, Inn mn 8

THE UNIVERSITY OF MICHIGAN 5603-1-F the constants a and f3 can be determined and the expression (9) used for any mn mn ri within a certain neighborhood of the reference point or points, whose extent' depends on these points and on the accuracy required. As an illustration of this procedure we compute an approximate value of So (c, r) for c = 3.0, ri = 0.5, - 0 0 given the values of S at 1= cos 50 =.6428, 72= cos 70.3420. On the basis of (9), we can write in general S (, ) ~1 [nSmn(C r l)02 mn (c, 2 L n mn 12 where Inn v mn mn -na ij Jmo Pi) m(o Pj) mo j m o i Using the tabulated value of X (3.0), S (3.0,. 6428) and S (3.0,. 3420) and the 00 oo oo oo expression (10) for 7o, we obtain the value S (3.0,.5) =.7575, which compares 00 reasonably well with the tabulated value S (3.0,.5) =.7571. oo In the neighborhood of the origin, i. e. In << 1, the forms simplify somewhat. Since the parity of the angle functions is that of the quantity m+no,either the function or its derivative will vanish at r} = 0, according as m+n is odd or even, and we can thus write almost immediately for Ir1 << 1 S (c, r1) = 2 S n(C, o) ki f N m ~n) 2mn mn mn L1 - 1-,/ m mn - J' (\ m N ( i)] m+n even m v mn In mn rl- j -S (C, O) J(N (t ) 2 mn m\ mn 7l-n/m mn -J (V \ ) XN (' _)] 1 m+n odd. m mn - m mn 1 - 9

THE UNIVERSITY OF MICHIGAN 5603-1-F Various other particular applications of the above forms suggest themselves, some of which involve the zeros of functions such as the A... A careful examination 1] of the accuracy obtainable and the range of applicability of the methods should accompany investigation of these applications. m. Conclusions An adequate assessment of the forms and nn thods described in the preceding discussion will depend, of course, on some fairly voluminous computations, as wel as some more detailed analytical work. Preferably these two lines of advance should be closely coordinated and the overall direction should be determined in accordance with the requirements of the unsolved or only partly solved physical problems. Although the present forms should give values sufficiently accurate for practical applications, it is clear that they are more suitable for filling in gaps in existing tables as the need arises than for large-scale production of new ones. One of the principal features of the Bessel function approximation is that it exhibits the dependence of the spheroidal functions on the wavelength-eccentricity parameter c and on the eigenvalue XAn more explicitly than the usual representations. Further exploitation of this feature should be of value in a number of circumstances. Respectfully submitted rederick B. Sleator 10