1 A. d dz I. INTRODUCTION. J. Acoust. Soc. Am. 116 (3), September /2004/116(3)/1381/8/$ Acoustical Society of America

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1 On Webster s horn equation and some generalizations P. A. Martin a) Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden, Colorado Received 7 October 23; revised 8 May 24; accepted 2 June 24 Sound waves along a rigid axisymmetric tube with a variable cross-section are considered. The governing Helmholtz equation is solved using power-series expansions in a stretched radial coordinate, leading to a hierarchy of one-dimensional ordinary differential equations in the longitudinal direction. The lowest approximation for axisymmetric motion turns out to be Webster s horn equation. Fourth-order differential equations are obtained at the next level of approximation. Comparisons with existing asymptotic theories for waves in slender tubes are made. 24 Acoustical Society of America. DOI:.2/ PACS numbers: 43.2.Mv LLT Pages: I. INTRODUCTION Webster s horn equation 99 gives a one-dimensional approximation for low-frequency sound waves along a rigid tube with a variable cross-sectional area A(z). The equation itself can be written as A d dz Az dp dzk 2 Pz, k/c, is the frequency, c is the constant speed of sound, and z is the coordinate along the tube. Equation can be derived by considering a thin layer of the fluid at z, perpendicular to the z-direction, with the assumption that the acoustic pressure is constant over this layer; this pressure is P(z). For a succinct derivation, see Pierce, 989, p. 36. Alternative derivations and extensions to other problems such as with fluid flow through the tube are available; see, for example, Reinstra, 22 and Hélie, 23. Exact solutions of Eq. are available for various specific A(z); for some recent results, see Kumar and Sujith, 997. In 967, Eisner published an excellent review of early work based on Eq.. On p. 27, we read: Eq. is usually called Webster s horn equation, but we see that there is little justification for this name. Daniel Bernoulli, Euler, and Lagrange all derived the equation and did most interesting work on its solution, more than 5 years before Webster. We are interested in giving a systematic derivation of Webster s equation, and of higher-order variants. We limit our analysis to axisymmetric tubes, and obtain Webster s equation when we seek axisymmetric solutions. We also consider nonaxisymmetric motions. To be specific, we consider a tube of finite length, and seek the frequencies of free vibration of the compressible fluid within the closed tube. In 96 three years before Webster s paper, Lord Rayleigh published a paper on axisymmetric motions in an axisymmetric tube. He began by writing the general solution of the axisymmetric Helmholtz equation in cylindrical polar coordinates (r,,z) as a Electronic mail: pamartin@mines.edu ur,,zj r d2 dz 2 k2 u z, u (z)u(,,z) is the value of u on the axis of the tube; as the Bessel function J (w) has a power-series expansion in integer powers of w 2, this provides meaning to the right-hand side of Eq. 2 see Eq. 26 below. Rayleigh then obtained an equation for u (z) by applying the boundary condition on the rigid wall of the tube; as a first approximation, he obtained Eq. see Eq. 8 in Rayleigh, 96. He also discussed some higher-order approximations. We shall adopt a similar approach, although we do not begin with an explicit representation such as Eq. 2: we shall use a Frobenius-type power-series expansion for the radial variation of u(r,,z). After formulating the problem in Sec. II, we examine the special case of circular cylinders in Secs. III and IV. The exact solution for the eigenfrequencies is recalled in Sec. III. Then, the approximate method is developed in Sec. IV. It is based on some observations of Boström 2 for the related axisymmetric problems of elastic waves in isotropic rods. In principle, we can obtain a heirarchy of approximations: We give explicit results for the first two members of this heirarchy. Apart from the merits of explaining the method for a simple case, we are also able to give a quantitative comparison with the exact solutions from Sec. III. In Sec. V, we consider axisymmetric, noncylindrical tubes. We change variables in the governing Helmholtz equation from r and z to and, is a scaled version of r chosen to that the lateral boundary is mapped to constant; this has the effect of complicating the partial differential equation via the chain rule but has the virtue that the lateral boundary condition is applied on a coordinate surface. The complications bring the shape of the boundary into the differential equation cf. Webster s equation but they can be dealt with readily because we then use a Frobenius-type expansion in the new radial variable. Again, we obtain a hierarchy of approximations. The first approximation gives a second-order ordinary differential equation; it reduces to Webster s equation for axisymmetric motions. The second approximation gives a fourth-order ordinary differential 2 J. Acoust. Soc. Am. 6 (3), September /24/6(3)/38/8/$2. 24 Acoustical Society of America 38

2 equation. In order to assess these approximations, we compare with some results of Ting and Miksis 983 and Geer and Keller 983. These authors began with the governing elliptic boundary-value problem for waves in slender tubes, and then obtained various asymptotic approximations. The comparisons are made in Sec. VI. Some concluding remarks can be found in Sec. VII. In summary, the approximate method described below has two virtues. First, the approximations can be improved. Second, the use of power series means that the basic method can be applied to much more complicated equations of motion, and to systems of such equations. For example, the propagation of elastic waves in nonuniform anisotropic rods can be studied; for some preliminary results in this direction, see Martin, 24. II. FORMULATION Consider a tube of circular cross-section and length L. Using cylindrical polar coordinates, (r,,z ), the interior of the tube is specified by r ar z, 2, z L, R (z ), so that 2a is the maximum diameter of the tube. It will be convenient to define dimensionless variables, using L as our length scale. Thus, we put rr /L, zz /L, R z Rz and a/l. Later, we shall regard as a small parameter. Hence, the tube becomes rrz, 2, z. Inside the tube, the acoustic potential U(r,,z,t) satisfies the wave equation r r r U r 2 U r 2 2 U L2 2 U 2 2 z c 2 t, 2 c is the speed of sound. On the lateral wall of the tube, the normal derivative of U vanishes U r 3 U Rz on rrz, z. 4 z We assume that R() and R() are both positive, and close the two ends of the tube with rigid circular discs, giving U/z at z and at z. 5 We seek free vibrations of the compressible fluid within the axisymmetric tube. Thus, we put U(r,,z,t) u(r,z)cos m cos t, m is a non-negative integer and is the frequency. The wave equation becomes r r r u r r 2 2 u z 2 k2 r 2 m 2 u, now kl/c is a dimensionless wave number. We are interested in determining eigenfrequencies so that there is a nontrivial u that satisfies Eq. 6 and the boundary conditions. 6 III. CIRCULAR TUBE: EXACT SOLUTION Consider a circular tube with R(z) for z. We can write down separated solutions of Eq. 6, u(r,z) J m (r)cos z, J m is a Bessel function, and and are arbitrary constants satisfying k The boundary conditions give j m, and N, N and are integers and j m, is the -th zero of J m :J m ( j m, ),, 2,...,. Hence, k 2 N 2 j m, 2 2. This is the exact solution of the problem. It can be found in textbooks for example, p. 22 of Kinsler et al., 982 and was given by Lord Rayleigh in the first edition of volume II of The Theory of Sound, published in 878 Rayleigh, 945. A. Axisymmetric modes mä The axisymmetric problem (m) is unusual, because we can take : both Eq. 6 and the lateral boundary condition on r are satisfied by u(r,z)cos z, so that one set of eigenfrequencies is given by kn for all. This set is elementary; it can be found in Sec. 62 of Lamb s book 96. The next set comes by using j, k 2 N , N,,2,...,. 9 B. Flexural modes mä We are also interested in nonaxisymmetric motions. As an example, we consider the case m flexural modes. As j,.84 and j,2 5.33, Eq. 7 gives the first two sets of eigenfrequencies as k 2 N and k 2 N for N,,2,..., later, we will compare the coefficients 3.39 and with those obtained by certain approximate theories. IV. CIRCULAR TUBE: APPROXIMATE METHOD If the tube is slender, a/l, we expect to be able to derive one-dimensional theories. We shall do this using power series in r. Note that we do not limit ourselves to polynomials in r, and so we are not limited, in principle, to very long waves. Nevertheless, it turns out that the low-order truncations obtained below work best for longer waves. We begin, as in the method of Frobenius, by writing ur,z r 2n u n z, n and u n (z) are to be found. Substitution in Eq. 6 gives 2 m 2 u z n r 2n2 n u n u n k 2 u n, n ()(2n2) 2 m 2. Just as in the method of Frobenius for ordinary differential equations, we require that 382 J. Acoust. Soc. Am., Vol. 6, No. 3, September 24 P. A. Martin: Webster s horn equation and some generalizations

3 every coefficient of r vanishes. For the first term to vanish, we obtain 2 m 2. As we want u to be bounded at r, we take m, and then we obtain u n zk 2 u n z4nnmu n z, n,,2,...,. Notice that this procedure does not determine u (z). However, following Boström 2, we note that u, u 2,..., are all determined by u ; for example, we have and u u k 2 u 4m u 2 u k 2 u 8m2 u iv 2k 2 u k 4 u 32mm2. 2 Then, regardless of the choice of u, the infinite series in Eq. will give an exact solution of Eq. 6, assuming that the series converges. The end boundary conditions, Eq. 5, give r 2n u n (z) at the two ends, whence n u n u n, n,,2,...,. 3 The lateral boundary condition reduces to u/r on r, and this gives n 2nm 2n u n z, z. Eliminating u n in favor of u, we obtain an equation for u (z) mu m2 2 u m4 4 u 2 mu m22 4m u k 2 u m4 4 32mm2 u iv 2k 2 u k 4 u. 4 At this stage, no approximations have been made. We obtain various approximations by truncating Eq. 4; this is done next. A. First approximation If we discard all terms with powers of greater than 2 in Eq. 4, we obtain u ze m u z, E m k,k 2 4mm/m From Eq. 3, we see that Eq. 5 is to be solved subject to u ()u (). If we look for solutions of Eq. 5 in the form u zcos z, 7 we find that 2 E m () (k,). Then, in order to satisfy the boundary conditions on the two ends of the tube, we obtain N, whence k 2 N 2 4mm/m2 2. This can be compared with the exact solution, Eq. 7. For m, we recover the exact set of solutions given by Eq. 8. For m, we can compare with Eq. ; when m, we have 4m(m)/(m2)8/32.67 which is in error by about 2%. B. Second approximation If we retain the terms in 4 in Eq. 4, we obtain u iv zc m u ze m 2 u z, C m k,2k 2 8m22 m4 2, E m 2 k,k 4 32mm2 m m m2k 2 4m The fourth-order Eq. 8 is to be solved subject to u u u u,. 2 2 using Eq. 2 to express u in terms of u and u, and Eq. 3. Substituting Eq. 7 in Eq. 8 gives 4 2 C m k,e m 2 k,, and then a routine calculation yields 2 k 2 m / 2 m 2 k,, say, m m 4(m2) 2 /(m4) and m 4 m m284m4m 2 m Finally, the end boundary conditions give k 2 N 2 m / 2.. Axisymmetric modes (mä) 25 When m, Eq. 25 reduces to Eq. 8 when we take the minus sign in. If we take the plus sign, we obtain k 2 (N) 2 8 2, the coefficient 8 can be compared with the exact 4.68 in Eq. 9. For this axisymmetric problem, we can compare with Rayleigh s approach. As J (w)/4w 2 /64w 4, Eq. 2 gives ur,zu z 4 r 2 u k 2 u 64 r 4 u iv 2k 2 u k 4 u. 26 If we discard the higher-order terms and apply the boundary condition, u/r onr, we obtain precisely Eq. 8 with m. This is reassuring but not surprising: The representations given by Eqs. 2 and are equivalent, although the latter can be used for nonaxisymmetric problems. J. Acoust. Soc. Am., Vol. 6, No. 3, September 24 P. A. Martin: Webster s horn equation and some generalizations 383

4 2. Flexural modes (mä) When m, we obtain k 2 N To obtain the lowest values, we take the minus sign, giving k 2 (N) ; the coefficient 3.53 differs from the exact 3.39 by about 4%. Thus, the fourth-order model gives good accuracy for the lowest set of eigenfrequencies. If we take the plus sign in Eq. 27, the coefficient multiplying 2 becomes.87, which can be compared with the exact value of given in the second of Eq.. Thus, as when m, the second set of eigenfrequencies is not approximated well by the fourth-order model. V. NONCYLINDRICAL TUBES Once we move away from cylinders, we no longer have exact solutions. Therefore, an approximate method will be required. For this reason, we choose to make a simple change of the independent variables, from (r,z) to,, so that the new geometry is a circular cylinder. The price of this change is that the new partial differential equation is more complicated. Thus, define new variables and by r/rz and z, 28 so that the tube is mapped onto the circular tube, given by,. Later, we will use z in place of, but it is clearer to distinguish the two variables at this stage. The chain rule gives u r R u z u, u R u R, 2 u r 2 u 2 R 2, 2 2 u z 2 u R 2 u u 2 22 R R R R R 2 u. Hence, Eq. 6 becomes 2 R 2 u 3 R 2 RR u 2 R 2 2 u 2 3 RR 2 u kr2 m 2 u, the lateral boundary condition, Eq. 4, becomes R 2 u u RR on,, 3 and the end boundary conditions, Eq. 5, become u R u at and at. 3 R To solve Eq. 29, we proceed as in Sec. IV, and write u, n 2n u n. Substitution in Eq. 29 gives 2 m 2 u n n ()(2n2) 2 m 2, and 2n2 n u n n ;, n z;r 2 u n 2RR2nu n k 2 R 2 2n 2 R 2 2nR 2 RRu n. As before, we take m and then n z;m4nnmu n z, n,,2,...,. In particular, we find that 4mu R 2 u 2mRRu k 2 R 2 mmr 2 mrru, 8m2u 2 R 2 u 2m2RRu k 2 R 2 m2m3r 2 RRu. Eliminating u from the last equation, using Eq. 33, gives 32mm2u 2 m u iv m u m u m u m u, m zr 4, m z4mr 3 R, m z2r 2 k 2 R 2 3mmR 2 RR, m z4mrk 2 R 2 RR 2 R mr3rrm2r 2, m zk 4 R 4 2mk 2 R 2 mr 2 RRmR 3 R iv mmr 2 4RR3R 2 m2r 2 6RRm3R When Eq. 32 is substituted in the lateral boundary condition Eq. 3, together with m, we obtain mu z n 2n2 2n2mu n z 2nmR 2 u n RRu n. Similarly, the end boundary conditions Eq. 3 give n 2n u n z R R 2nmu nz at 35 z and at z, 36 which immediately gives 384 J. Acoust. Soc. Am., Vol. 6, No. 3, September 24 P. A. Martin: Webster s horn equation and some generalizations

5 Rzu n zrz2nmu n z at z and at z, 37 for n,,2,...,. Below, we shall use the following shorthand notation: S zr/r, S 2 R/R, S 3 R/R, S 4 R iv /R. 38 A. First approximation If we retain only the terms up to 2 in Eq. 35, we find that mu 2 m2u mr 2 u RRu. Upon using Eq. 33, this gives u zd m zu ze m zu z, D m z2s 2m 2 /m2, E m ze m k,r mmm2 S 2 m2 ms 2, 39 and E m () are defined by Eq. 6. From Eq. 37, Eq. 39 is to be solved subject to Ru mru atz and at z. Equation 39 can be transformed so as to eliminate the first-derivative term. Thus, put u zr U z with m 2 2/m2. 4 Then, we find that U (z) solves U zk 2 KzU z, Kz 2m m2 m m2 S 2 S 2 2m 2 R 2, 4 subject to (m2)u 2(m)S U at z and at z. Notice that Eq. 4 has oscillatory solutions when k 2 K but exponential solutions when k 2 K. Equation 39 and its associated boundary conditions can also be written as a regular Sturm-Liouville problem. Thus, pzu qzwzu z, pwr 2, k 2, is defined by Eq. 4, and qzmr 2 m m2 m2s R 2S 2.. Axisymmetric modes (mä) When m, Eq. 39 reduces to Au k 2 Au, 42 A(z)a 2 R(z) 2 is the area of the circular crosssection at z; Eq. 42 is recognized as Webster s horn equation Eq.. The appropriate boundary conditions are u ()u (). 2. Flexural modes (mä) When m, Eq. 39 reduces to u z 2 3 S u z k S 2 S R 2u z, subject to Ru Ru atz and at z. B. Second approximation If we retain the terms up to 4 in Eq. 35, we obtain an equation containing u, u, u, u, and u 2. If we eliminate u, u, and u 2, using Eqs. 33 and 34, we obtain a fourthorder equation for u (z), u iv B m 2 zu C m 2 zu D m 2 zu E m 2 zu, 43 B m 2 4S 42mm 2 /m4, C m 2 C m k,r6ms 2 4mm 2 S 2 m4, 44 D m 2 D m k,r4ms 3 4mS m4 m4m2 S 2 3mm3S 2, E 2 m E 2 2 m k,rms 4 3mmS 2 4mm2 2mk2 m4 6m2 mm2 S 2 m4 S 2 S 2 S 2 4m2 m3 S m4 S 3 mmm2 m 2 4 m4s m4 8mm2 m4 2 R 2 mm2s 2 m2s 2, C m and E m (2) are defined by Eqs. 9 and 2, respectively, and D m k, 4S m4 42mm 2 k m2m Note that Eq. 43 reduces to Eq. 8 when R(z). Equation 43 is to be supplemented with boundary conditions. From Eq. 37, these are Ru mru and Ru m2ru 46 at z and at z. Eliminating u from the second of Eq. 46, using Eq. 33, and then using the first of Eq. 46, we obtain u 3mS u 2m 2 S 2 3S 2 u ms 3 u ; 47 J. Acoust. Soc. Am., Vol. 6, No. 3, September 24 P. A. Martin: Webster s horn equation and some generalizations 385

6 both this condition and u ms u are to be imposed at z and at z. Notice that the boundary conditions do not involve k 2.. Axisymmetric modes (mä) When m, Eq. 43 reduces to u iv 4R R u 2 k R 2 u 4R R k R 2 u k 4 8k2 2 R 2 u. 48 From Eq. 47, we see that Eq. 48 is to be solved subject to Eq. 2. We observe that Eq. 48 involves R and R, but not higher derivatives of R. Also, Eq. 48 can be derived directly by applying Eq. 4 to Eq Flexural modes (mä) When m, we obtain B 2 z 4 5 S, C 2 z2k S 2 6S R2, D 2 z 4 5 S k 2 6S 2 2S 2 2R 2 4S 3, E 2 zk 4 S 4 6S k2 2S 2 5S 2 36R S 8S 3 8S S 2 2S R 2 2S 2 3S R 2. Then, the differential equation Eq. 43 with m) is to be solved subject to u S u at z and at z. and u 3S u 3S 2 u S 3 u VI. COMPARISON WITH ASYMPTOTIC APPROXIMATIONS A number of formal asymptotic theories have been developed for waves in slender tubes. In the papers by Ting and Miksis 983 and by Geer and Keller 983, the tubes need not have circular cross-sections and they need not be straight. Here, we shall compare the results obtained from our one-dimensional theory with those obtained by specialising the analysis in Ting and Miksis, 983 and Geer and Keller, 983 to our axisymmetric geometries. Two asymptotic regimes are of interest to us. In one, k O() as, so that the wavelength is comparable to L, the length of the tube. All solutions of this kind are axisymmetric. In a second regime, ko( )as, so that the wavelength is comparable to al. Both kinds of solution are seen in the exact solutions for circular tubes Sec. III. A. Wavelength comparable to L For axisymmetric motion, we obtained the fourth-order Eq. 48, which we write here as R 2 u k 2 R 2 u 8 2 R 3 Ru iv 4Ru 2k 2 Ru 4k 2 Ru k 4 Ru. 49 In this equation, k and u (z) are unknown; they are to be determined subject to the four boundary conditions, Eq. 2. Let us look for solutions of the form k 2 k 2 2 k 2 2, 5 k and k 2 are to be found. Thus, we put u (z) v (z) 2 v (z) and Eq. 5 in Eq. 49. This is a classic singular perturbation, because Eq. 49 reduces to a second-order equation when, implying that the boundary conditions will have to be modified. Indeed, writing out the first two terms of the exact boundary condition, Eq. 36, we have u z 2 u 2R/Ru u z 4 2 R 2 u k 2 u, at each end of the tube, we have used Eq. 33. We will not be able to satisfy this condition for all allowable with, and so we integrate over each circular end of the tube, and impose u z 8 2 R 2 u k 2 u at z and at z; 5 this ensures that Eq. 36 is satisfied in an average sense. The terms in from Eqs. 49 and 5 give R 2 v k 2 R 2 v for z, with v v. 52 This is Webster s horn equation again, written as a regular 2 Sturm-Liouville problem: it is an eigenvalue problem for k and v (z); we normalize the solution using v zrz 2 dz. 53 The terms in 2 give R 2 v k 2 R 2 v k 2 2 R 2 v V, V 8 R 3 Rv iv 4Rv 2k 2 Rv 4k 2 Rv k 4 Rv 4 R 2 RRRRv 2 R 3 Rv, after using Eq. 52. Similarly, Eq. 5 gives v z 4 RRv 4 RRv at z and at z. 56 Thus v solves a forced version of Webster s horn equation with inhomogeneous boundary conditions. As the homogeneous form of Eq. 54 admits nontrivial solutions, Eq J. Acoust. Soc. Am., Vol. 6, No. 3, September 24 P. A. Martin: Webster s horn equation and some generalizations

7 will only have solutions if a consistency condition is satisfied. Thus, we multiply Eq. 54 by v (z) and Eq. 52 by v (z), subtract the two, and integrate over z. This gives 4 R2 v RRv z k 2 2 Vv dz, we have used Eqs. 52 2, 53, and 56. In order to cancel the left-hand side, write VV /4R 2 (RRv )/4R 2 (RRv ). Then, an integration by parts gives k R 2 RRv v dz Wv dz, WzV 4 R 2 RRv 57 4 R 3 Rv RR 2 v 2 3R 2 RRRR 3 v 58 and we have used Eq. 55. Integrating the first integral in Eq. 57 by parts, using Eq. 52 2, then gives k RRv 2 dzk, K (Wv /4R 3 Rv v )dz. We are going to show that K. Noting the first term in Eq. 58, and making use of Eq. 52, we have v v v v k 2 v 2R Rv v k 2 v 2R Rv v 2R Rv 2 2R 2 RRR 2 v RRv v. Hence, K/2 (RRv ) 2 (RR) 2 v v dz, and this is seen to vanish after another integration by parts. Thus, k 2 k RRv 2 dzo 4 as. 59 This elegant formula was derived in a different way by Geer and Keller 983; see their Eq. 2. B. Wavelength comparable to a In the differential equations, Eqs. 39 and 43, put k / and u zezwz with Ezexp i z z tdt, 6 (t) and w(z) are to be determined, and z is a constant. Suppose further that wzw zw z 2 w 2 z. Then, we obtain the following approximations: 6 u /Ew w O 2, u /E iw O, u /E 2 2 w 2iw iw 2 w O, u /E 3 i 3 w O 2, u iv /E 4 4 w w 2i3w 2w O 2, as. Let us begin with the second-order equation, Eq. 39. We note that D m () O() and E m () 2 E m () (,R)O() as. Then, the terms in 2 give z 2 E m,r. The terms in give 2w D m w, 62 which is a first-order differential equation for w. Rearranging gives w /2 w /2 w w 2 D m 2 q m R R, q m () (2m 2 )/(m2). An integration gives w 2 R 2q constant, 63 w,, and R are all functions of z only, and q q m (). Next, consider the fourth-order equation, Eq. 43. Substituting as before, we find that B m 2 O, D m 2 2 D m,ro C m 2 2 C m,ro, E m 2 4 E m 2,RO 2 and as. Then, we see that the terms in 4 give 4 2 C m,re m 2,R, which should be compared with Eq. 22. Thus, z 2 m 2,R, m is defined by Eq. 23. The terms in 3 give 4 2 C m,re m 2,Rw 2iw C m,r2 2 iw D m,r C m,r 3 B m The factor multiplying w vanishes, due to Eq. 64, leaving a first-order differential equation for w (z) w 2 D m,rc m,r 4 B 2 m 6 3. w 2 2 C m,r2 2 Rearranging this equation gives J. Acoust. Soc. Am., Vol. 6, No. 3, September 24 P. A. Martin: Webster s horn equation and some generalizations 387

8 w /2 w /2 4Dm,R2 2 B m q R 2 2C m,r2 2 m R, q m 2 4m2m 2 2mm3 m /m4 m. Here, we have used 2 2 m /R 2, and Eqs. 9, 23, 44, and 45. Hence, an integration gives Eq. 63 again, but with qq m (2). Let us compare our results obtained by solving ordinary differential equations asymptotically with those of Ting and Miksis 983 and Geer and Keller 983obtained by solving an elliptic boundary-value problem asymptotically. First, we obtained approximations for, given by Eqs. 62 and 65, as the exact formula is z 2 2 j m, /Rz The errors incurred here are exactly the same as those discussed in Sec. IV for a circular tube constant cross-section: We cannot expect to do any better for tubes of varying crosssection. Second, the analysis of Ting and Miksis 983 and Geer and Keller 983 leads to Eq. 63 with q for all m. If we take the minus sign in Eq. 24, we find that q (2) and q (2).43; it is not clear why this last number is not closer to, given that the corresponding values of are close. VII. CONCLUDING REMARKS The method of Sec. V for waves in axisymmetric tubes leads to eigenvalue problems for ordinary differential equations. The simplest first approximation leads to a regular Sturm Liouville problem. This is convenient because efficient software is readily available for solving such problems numerically Pruess and Fulton, 993. The next second approximation is expected to be more accurate, and, indeed, we have shown this in some asymptotic regimes. The second approximation leads to an eigenvalue problem for a fourthorder differential equation. However, it does not fall into the class of regular fourth-order Sturm-Liouville problems; see, for example, the review by Greenberg and Marletta 2. In fact, we can say little about the theoretical properties of the simplest equation, namely Eq. 48, which models axisymmetric motions. Evidently, higher-order approximations could be developed, leading to ordinary differential equations of order 2n with n3,4,...; the necessary derivations would be expedited using software for symbolic manipulations. The method of Sec. V could also be extended to other cross-sections, using a scaling that depends on the angle as well as on the longitudinal coordinate z. Another possibility is to abandon power series in favor of Neumann series, which are series of Bessel functions of various orders. This would permit better representation of u(r,z) for fixed z, but at the expense of additional complication. Finally, the basic power-series method can be extended to various elastodynamic problems, generalizing the work of Boström 2 on rods and of Boström, Johansson, and Olsson 2 on plates; for axisymmetric motions in nonuniform anisotropic rods, see Martin, 24. Indeed, the fact that the power-series method is relatively insensitive to complications in the governing partial differential equations means that it may be worth developing further. Boström, A. 2. On wave equations for elastic rods, Z. Angew. Math. Mech. 8, Boström, A., Johansson, G., and Olsson, P. 2. On the rational derivation of a hierarchy of dynamic equations for a homogeneous, isotropic, elastic plate, Int. J. Solids Struct. 38, Eisner, E Complete solutions of the Webster horn equation, J. Acoust. Soc. Am. 4, Geer, J. F., and Keller, J. B Eigenvalues of slender cavities and waves in slender tubes, J. Acoust. Soc. Am. 74, Greenberg, L., and Marletta, M. 2. Numerical methods for higher order Sturm Liouville problems, J. Comput. Appl. Math. 25, Hélie, T. 23. Unidimensional models of acoustic propagation in axisymmetric wave-guides, J. Acoust. Soc. Am. 4, Kinsler, L. E., Frey, A. R., Coppens, A. B., and Sanders, J. V Fundamentals of Acoustics, 3rd ed. Wiley, New York. Kumar, B. M., and Sujith, R. I Exact solutions for the longitudinal vibration of non-uniform rods, J. Sound Vib. 27, Lamb, H. 96. The Dynamical Theory of Sound, 2nd ed. Dover, New York. Martin, P. A. 24. Waves in wood: axisymmetric waves in slender solids of revolution, Wave Motion to be published. Pierce, A. D Acoustics Acoustical Society of America, New York. Pruess, S., and Fulton, C. T Mathematical software for Sturm- Liouville problems, ACM Trans. Math. Softw. 9, Rayleigh, Lord 96. On the propagation of sound in narrow tubes of variable section, Philos. Mag., Series 6 3, Rayleigh, Lord 945. The Theory of Sound Dover, New York. Reinstra, S. W. 22. The Webster equation revisited, paper AIAA , 8th AIAA/CEAS Aeroacoustics Conference, Breckenridge, Colorado, June 22. Ting, L., and Miksis, M. J Wave propagation through a slender curved tube, J. Acoust. Soc. Am. 74, Webster, A. G. 99. Acoustical impedance, and the theory of horns and of the phonograph, Proc. Natl. Acad. Sci. U.S.A. 5, J. Acoust. Soc. Am., Vol. 6, No. 3, September 24 P. A. Martin: Webster s horn equation and some generalizations