Asymptotic Behavior of Waves in a Nonuniform Medium

Transcription

1 Available at Appl Appl Math ISSN: Vol 12, Issue 1 June 217, pp Applications Applied Mathematics: An International Journal AAM Asymptotic Behavior of Waves in a Nonuniform Medium 1 Nezam Iraniparast, 2 Lan Nguyen 3 Mikhail Khenner Department of Mathematics Western Kentucky University Bowling Green, KY NezamIraniparast@wkuedu; 2 LanNguyen@wkuedu; 3 MikhailKhenner@wkuedu Received: June 13, 216; Accepted: April 2, 217 Abstract An incoming wave on an infinite string, that has uniform density except for one or two jump discontinuities, splits into transmitted reflected waves These waves can explicitly be described in terms of the incoming wave with changes in the amplitude speed But when a string or membrane has continuous inhomogeneity in a finite region the waves can only be approximated or described asymptotically Here, we study the cases of monochromatic waves along a nonuniform density string plane waves along a membrane with nonuniform density In both cases the speed of the physical system is assumed to tend to a constant when the spatial variable gets very large In the case of a string with local inhomogeneity it is possible to find solutions that are asymptotic to sinusoidal waves involving the limiting speed of the string On the other h when the coefficients in the equation of the vibrating string are small deviations from a constant, we use a special Green s function to approximate the solution We also find a finite series of sinusoidal waves with constant speeds, that are asymptotic to the solutions of a vibrating membrane problem The number of waves in the series depends on the width of the membrane, the limiting speed of the waves the time frequency of the sinusoidal waves The technique we use to show asymptotic approximations involves reducing the pde equations of the physical models to a first-order system of ode s Keywords: Scattering; Asymptotic convergence; Wave; String; Membrane; Eigenvalue; Eigenvector; Boundary conditions MSC 21 No: 74J2, 35B4 217

2 218 N Iraniparast et al 1 Introduction Consider the equation of the waves along an infinite vibrating string, u tt c 2 xu xx =, x, t R,, where, cx = { c1, x, c 2, x > The solution of this equation is well known to be Strauss 28, { fx c1 t + c 2 c 1 c ux, t = 2 +c 1 f x c 1 t, x <, 2c 2 c 2 +c 1 f c 1 c 2 x c 2 t, x > Here the incident wave function f is defined so that fx = for x <, cx is the speed of the waves c 1, c 2 are constants The solution above shows that a single inhomogeneity in the density of the string creates a reflection a transmission in the propagation of the waves In the region x <, the solution is the sum of an outgoing a reflected wave both traveling at the speed c 1 In the region x >, the solution is a transmitted wave traveling at the speed c 2 In both regions, the amplitudes of the waves have changed according to the expressions given in terms of c 1 c 2 Since the speed c is the ratio tension/mass, the discontinuity is as a result of the change in the mass density of the string In Iraniparast 211, the number of sudden changes in the mass density of the string has been increased to two This means the speed c has two jump discontinuities along the string In this case, there is a solution ux, t with a formulation similar to the one shown above in terms of a single function f However, the expressions for the solution are much longer more complicated, because the waves that are reflected at each interface interact with the incoming outgoing waves to produce new waves If the inhomogeneity of the string is made up of more than two single impurities in the string, then the computation to find an exact solution will be much more complicated, if not impossible But if we assume a continuous change in the mass density, there are approximations to the solutions that could help determine the behavior of the waves in the far-field regions of the string The purpose of this article is to study the wave equation for the string when there is a restoring/repelling force qu, with the mass density S, tension T the coefficient q all continuous functions of the spatial variable x Furthermore, we assume that lim x /Sx = c 2 where c is a constant, for example, the speed of the waves reaches a constant far away from the source of the inhomogeneity of the string In Section 2, we show that under certain conditions on the coefficients, there are two monochromatic wave solutions One is asymptotically approximated by an outgoing sinusoidal wave,

3 AAM: Intern J, Vol 12, Issue 1 June the other by an estimated incoming sinusoidal wave In Section 3, we give an example numerically show that the waves undergo a change in their behavior when they travel through the inhomogeneity, but later resume a sinusoidal propagation with a different amplitude speed In Section 4, we show that if the mass, tension, the coefficient of the restoring/repelling force have small deviations of oɛ from a constant, a solution u up to oɛ can be calculated In Section 5, we apply the method of Section 2 to the problem of the vibration of a membrane We consider a vibrating membrane with no restoring/repelling force, an infinite length in the x direction a finite width in the y direction The mass τ tension T will both be continuous functions of x We assume a Dirichlet condition on the boundary impose other conditions to show that there are solutions w n x, y, t that are asymptotic to the sum of n sinusoidal waves We should add here that there is voluminous amount of work regarding the phenomenon of the waves subject to changes in the media In the context of scattering, we mention a few here In Elschner Hu 212, time-harmonic two-dimensional elastic scattering for unbounded surfaces is studied The scattering is due to a source term whose support is located within a distance above the surface In Martin 23, the existence uniqueness of time harmonic acoustic pressure waves electromagnetic fields scattered by an inhomogeneous medium is shown For a discontinuous media, Sabatier 1989 studied the scattering of waves in a frequency domain where the impedance factor has discontinuities Montroll Hart 1951 use scalar timeindependent plane-wave functions satisfying the Laplace equation to compute the scattered field when the scatterers are homogeneous isotropic with well-defined boundaries For far-field patterns, Colton Paivarinta 199 studied the time-harmonic waves satisfying Maxwell s equations for electric magnetic fields The sources of scattering are regions in the space where the conductivity permittivity are functions of the space variables Finally, in Feuer Akeley 1947, the effect of a resonant ring in a circular wave guide on transverse electric waves incident on the ring has been studied The waves with frequency other than the resonant frequency get scattered 2 Asymptotic Behavior Consider the more general case where the mass density, tension restoring/repelling force of the string depend on the spatial variable x Sxu tt x u x + qxu =, 1 where we assume that far away, the string is uniformly dense in the sense that, lim x Sx = c2, Sx, > Suppose the wave motion 1 has the frequency ω, that is, it is a monochromatic wave of the form,

4 22 N Iraniparast et al Then v must satisfy, ux, t = vxe iωt Let, v + T xv + ω 2 Sx qxv = 2 Then, equation 2 reads, T x = px, Sx = mx, qx = nx v + pxv + ω 2 mx nxv = 3 If we transform equation 2 to a first-order system with v x = zx, we have, d dx v z = 1 n ω 2 m p v z 4 Reformulating, equation 4, d dx v z = [ 1 ω2 c 2 + ω 2 1 m c 2 + n p ] v z Let, ũ = v z, A = 1 ω 2 c 2, W x = ω 2 1 m c 2, Rx = n p Then, the equation 4 becomes, The eigensystem of A consists of, d = A + W x + Rxũ 5 dxũ

5 AAM: Intern J, Vol 12, Issue 1 June µ 1 = iω c, v 1 = 1 iω c µ 2 = iω c, v 2 = 1 iiω c The eigenvalues of A + W are, λ j = ±i ω 2 m, j = 1, 2 Furthermore we have, lim λ j = ±µ j x The Theorem 81 in Codington Levinson 1955 on asymptotic convergence of the solutions of the system 5 requires the extra assumptions that, lim W x =, x W x dx <, Rx dx <, that for D ij = Rλ i λ j, we either have, for a given k j = 1, 2, or, x D kj dσ, x, x2 x 1 D kj dσ >, x 2 > x 1, x2 x 1 D kj dσ < L, x 2 > x 1, where L is a constant Since in our case D ij =, the second condition is automatically satisfied Then, there is a solution φ k x of the system 5 such that for some x, x <, Denoting, x lim φ kx exp iω mσdσ = v k x x φ 1 = v1 z 1, φ 2 = v2 z 2,

6 222 N Iraniparast et al we have, v1 z 1 1 iω c x exp iω mσdσ, x, x v2 z 2 1 iω c x exp iω mσdσ, x x Therefore, equation 1 has solutions u 1 u 2 that satisfy, We have the following theorem Theorem 1 u 1 x, t e iω x x mσdσ t, x, u 2 x, t e iω x x mσdσ+t, x Consider the equation of the vibrations of an infinite string with mass density Sx tension, given by, Sxu tt x u x + qxu =, x < Assume qx CR,, Sx C 1 R Let, px = T x, Sx mx =, qx nx = Assume that, lim x Sx = c2, m x dx =, nx + px dx < Then for the waves of the form ux, t = vxe iωt there are solutions u 1 x, t, u 2 x, t, x such that, u 1 x, t e iω x x mσdσ t, x, u 2 x, t e iω x x mσdσ+t, x

7 AAM: Intern J, Vol 12, Issue 1 June An Example Let, Sx = exp a 1 + x2, = Sx, qx = 2ax 1 + x2 2 Then, Sx mx = = 1, a qx 2ax exp, nx = = 1 + x x2 px = T x 2ax = 1 + x2 2

8 224 N Iraniparast et al Then, Rx dx = W x dx = lim x Sx = 1, ω 2 m x dx =, nx + px dx = 1 + a e a, a > The conditions of the Theorem 1 are all satisfied For the choices of ω = 1 a = 3 a pair of the solutions of the equation 3 are depicted in Figures 1 2 The asymptotic solutions of equation 1 would be these waves moving along the string with the passage of time 4 Approximating Suppose, Sx = ρ + ɛs 1 x, = D + ɛt 1 x, qx = q + ɛq 1 x, where ρ, D q are constants Let, S 1 x, T 1 x, q 1 x, x vx = v x + ɛv 1 x + Substituting these in equation 2 for v, the equations up to o1, oɛ terms, respectively, will be, Dv x + ω 2 ρ q v =, Dv 1x + ω 2 ρ q v = q 1 ω 2 S 1 v T 1 v The solution v, assuming q < ω 2 ρ, is of the form, ω2 ρ q o v x = A exp i D x, where A is a constant The solution for the second equation using the fundamental solution,

9 AAM: Intern J, Vol 12, Issue 1 June where, Gx, x = 1 2ik expik x x, is, k = ω2 ρ q D, v 1 x = A x ] [e ikx q 1 ω 2 S 1 + k 2 T 1 dx + e ikx q 1 ω 2 S 1 k 2 T 1 e 2ikx dx 2ikD x Then, νx = v x + ɛv 1 x, will be an approximation of v up to oɛ Higher-order approximations can be obtained the same way, if necessary 5 Guided Waves Many guided wave propagations, in optics, electromagnetics acoustics, are described by waves propagating on the surface of a membrane Lamb 1995 Consider a membrane occupying the region, Ω = {x, y x <, y d} Let wx, y, t represent the vibrations in the absence of the external forces Then, it satisfies, τx, y 2 w t 2 = T x, y w + T x, y w, x, y, t Ω [,, 6 x x y y with the boundary conditions, wx,, t = = wx, d, t, 7 where τx, y is the density T x, y the tension in the membrane Letting τ T depend on x only assuming separation of variables, wx, y, t = XxY ygt, equation 6 reduces to the following three ODE s with separation constant γ 2 as follows,

10 226 N Iraniparast et al g t + γ 2 gt = The solution to this is gt = RBe iγt Letting wx, y, t = XxY ye iγt, the boundary value problem for Y y is, Y + k 2 1Y =, Y = = Y d with eigenvalues, k1,j 2 = j2 π 2, j = 1, 2, 3, d2 eigenfunctions, And finally the equations for X j are, jπy Y j y = a j sin, j = 1, 2, 3, d X j + T γ 2 τ T X j + T j2 π 2 X d 2 j =, j = 1, 2, 3, As in Section 2, we transform the second order ODE above to a first order system with X jx = Z j x We have, d dx Xj Z j = 1 j 2 π 2 d 2 τγ2 T T T Xj Z j Suppose, Rewrite the above equation in the form, lim x τx = c2 d dx Xj Z j = [ 1 + j 2 π 2 d 2 γ2 c 2 T T + ] Xj Z j γ 2 1 τ c 2 T This time,

11 AAM: Intern J, Vol 12, Issue 1 June A = 1 j 2 π 2 γ2 d 2 c 2, W = γ 2 1 τ c 2 T, R = T T The eigenvalues of A are, µ 1 j = j2 π 2 d 2 γ2 c, j2 π 2 µ2 j = 2 γ2 d 2 c 2 The corresponding eigenvectors are, p 1 j = 1 1 j 2 π 2 d 2 γ2 c 2, p 2 j = 1 1 j 2 π 2 d 2 γ2 c 2, respectively The eigenvalues of A + W are, If the conditions, λ k j2 π j = ± 2 γ2 τ, k = 1, 2 d 2 T j 2 π 2 d 2 γ2 <, j = 1, 2,, c2 are satisfied, then µ k j are imaginary according to the Theorem 81 in Codington Levinson 1955 there are solutions Xj k, x such that, Xj 1 e i x γ 2 τ x T j2 π 2 d 2 dσ, x, Xj 2 e i x γ 2 τ x T j2 π 2 d 2 dσ, x, for some x This implies that there is a solution w n x, y, t to the membrane problem 6, 7 x, such that,

12 228 N Iraniparast et al w n x, y, t n< γd πc j=1 [a j sin jπy x γ 2 τ d e i x T j2 π 2 d 2 dσ 8 +b j sin jπy x γ 2 τ d ei x T j2 π 2 d 2 dσ ]e iγt, x Based on the above argument the theorem of Codington Levinson 1955, we can state the following Theorem 2 Consider the membrane problem, τx 2 w t 2 = w + w, x, y, t Ω [, 9 x x y y wx,, t = = wx, d, t, x, t [, [,, 1 where τx is the area mass density is the tension Assume, lim x τx = c2, τx dx <, T x dx < Then, among the solutions of the type, of 9, 1, there are solutions, wx, y, t = XxY ye iγt, w n x, y, t, n 1, such that w n x, y, t, is given as in 8 x, 6 Conclusions Waves incident on a string with a jump discontinuity in its mass density will scatter In such cases the exact behavior of the reflected transmitted waves can be determined If the mass density of the string has a local continuous inhomogeneity, it can be shown that all time harmonic waves

13 AAM: Intern J, Vol 12, Issue 1 June far away from such interference are asymptotic to sinusoidal vibrations The asymptotic behavior of the transmitted reflected waves away from this non-uniformity can be determined The speed of the asymptotic waves depends on the ratio mx = Sx, that is, the ratio of mass density tension In the case that mass density, tension the coefficient of the restoring force are all small perturbations from constants, the time-harmonic solutions up to oɛ is approximated The approximated solutions have constant wave speeds, but variable amplitudes The amplitudes depend on the oɛ parts of expansions for the coefficients of the equation In two-dimensions the problem of the waves along a guide in the form of a long insulated strip is considered Far away, the waves become asymptotic to a finite sum of forward backward moving waves that depend on the ratio of the mass density tension of the guide its width Acknowledgments The authors would like to thank our colleague for his time reading the manuscript making valuable editing remarks REFERENCES Codington, E Levinson, N 1955 Theory of Ordinary Differential Equations, McGraw- Hill Colton, D Paivarinta, L 199 Far-Field Patterns for Electromagnetic Waves in an Inhomogeneous Medium, SIAM J MATH ANAL, Vol 21, No 6, pp Elschner, J Hu, G 212 Elastic Scattering by Unbounded Rough Surfaces, SIAM J MATH ANAL, Vol 44, No 6, pp Feuer, P Akeley, E S 1947 Scattering of Electromagnetic Radiation by a Thin Circular Wave Guide, Department of Physics, Purdue University, Lafayette, Indiana Iraniparast, N 211 Wave Scattering in In-homogeneous Strings, ISRN Mathematical Analysis, Volume 211 Lamb, G L 1995 Introductory Applications of Partial Differential Equations, Wiley Martin, P A 23 Acoustic Scattering by Inhomogeneous Obstacles, SIAM J APPL MATH, Volume 64, No 1, pp Montroll, E W Hart, R W 1951 Scattering of Plane Waves by Soft Obstacles II Scattering by Cylinders, Spheroids, Disks, J App Physics, Vol 22, No 1 Sabatier, P C 1989 On the scattering by Discontinuous Media, Inverse Problems in Prtial Differentail Equations, pp 85-1 Strauss, W A 28 Partial Differential Equations, Wiley