- Email: [email protected]
On wave propagation in waveguides
Physica D 162 (2002) 1–8
On wave propagation in waveguides P. Vainshtein∗ Faculty of Mechanical Engineering, Technion – Israel Institute of Technology, Haifa 32000, Israel Received 1 August 2001; received in revised form 7 November 2001; accepted 9 November 2001 Communicated by F.H. Busse
Abstract Cut-off frequencies are well known in waveguides of different nature to be the thresholds of propagation and evanescence. In the present paper, a theoretical study is made to demonstrate that a combined excitation of a plane wave and waveguide mode may lead to the propagation of a nonlinear wave pattern irrespective of whether the frequency of the waveguide mode is higher or below linear cut-off. A particular problem is examined when a piston performs sinusoidal rotational vibrations combined with translational displacements in a uniform semi-infinite two-dimensional fluid duct. The frequency of the vibrations is close to a cut-off frequency. The rotational vibrations are small compared to the translational displacements. An analytical solution of the problem is obtained for uniformly accelerated piston displacements. It is shown that longitudinal plane waves guide the duct modes along the duct, such that propagation takes place even if the forcing frequencies are below linear cut-off. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Cut-off frequency; Waveguides; Two-dimensional duct
1. Introduction Cut-off frequencies are commonly associated with waveguides in various physical contexts. It is well known that there exists a discrete set of frequencies, each of which is a threshold of a new propagating mode. According to the linearized theory, as the frequency decreases to the threshold below which the mode cannot propagate, the relevant group velocity tends to zero. Below the threshold, the mode changes drastically to an evanescent one. The familiar waveguides are acoustic ducts, water tanks, electromagnetic and dielectric optical waveguides (see [1,2]). As cut-off frequencies are of general interest in waveguides, an examination of problems is interesting to clarify the theoretical question of whether there are some ways of sound transmission if the driving frequency is below linear cut-off. In the present paper, an attempt to develop a relevant theory of nonlinear wave propagation where excitation frequencies are close to linear cut-off for the physically simple case of a slab acoustic duct with rigid walls is undertaken. ∗
Tel.: +972-4-82923266; fax: +972-4-8324533. E-mail address: [email protected] (P. Vainshtein). 0167-2789/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 8 9 ( 0 1 ) 0 0 3 7 6 - 1
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P. Vainshtein / Physica D 162 (2002) 1–8
Fig. 1. Schematic of a semi-infinite two-dimensional duct with a coordinate system. The thick lines show the piston performing combined translational displacements and sinusoidal vibrations.
2. Analysis and results The main subject of the study is the propagation of acoustic waves forced by piston motions combined translational displacements with rotational vibrations (see Fig. 1). Such piston motions are of a simple physical model of a source of sound generating two-dimensional acoustic waves in an acoustic duct. One may anticipate that due to nonlinear interactions of longitudinal waves and duct modes in the near field, the resulting wave motions in the far field will represent some two-dimensional flow pattern. In order to get an insight into this generally rather involved process we shall treat a particular problem where the rotational vibrations are much smaller than the translational displacements. Besides, a uniformly accelerated translational piston motion will be considered that will enable explicit solutions to be obtained. The exact nonlinear wave equation for an isentropic flow is (see [3]) ∂ 2φ ∂φ ∂φ 1 1 2 2 2 2 c0 ∇ φ − 2 = 2∇φ · ∇ (1) + (∇φ · ∇)(∇φ) + (γ − 1) + (∇φ) ∇ 2 φ, ∂t 2 ∂t 2 ∂t where φ is the velocity potential, c0 the initial speed of sound; x, y, z, t are Eulerian coordinates, ∇ and ∇ 2 are the gradient and Laplacian, γ the ratio of specific heat. Typical values of γ range from a maximum of 53 for monatomic gases through 75 for diatomic gases to values as low as 1.2 or even 1.1 for polyatomic gases at high temperature [1]. The condition γ → 1 is frequently used in asymptotic analysis (see for example [4]). It simplifies greatly the nonlinear analysis since the entropy waves are no longer coupled to the pressure field. We shall use this condition in the present paper in order to obtain an analytical solution of the formulated problem. The classical problem of the one-dimensional propagation of waves is specified by the motion of a piston with a characteristic time t0 , such that the distance traveled is small compared to c0 t0 (see [5]). This defines the small parameter of the problem, ε, and the piston curve xp (t) can be presented in terms of the dimensionless function f by t xp (t) = εc0 t0 f . (2) t0 The small rotational motion of a piston about the z-axis, such that the amplitude of the piston vibrations is small compared to the cross-sectional dimension of the duct h, can be presented by the piston curve as follows: πy xp (t) = δh sin Ωt cos , (3) h where Ω is the frequency of excitation and δ the small parameter. The problem of the wave propagation in a planar duct produced by the piston motion (3) has been investigated in [3]. An analytical theory based on an asymptotic approximation has been developed for the nonlinear response when the excitation frequency is close to the lowest cut-off frequency, ω0 = πc0 / h. The study has focused attention on the far fields of the steady-state flow. It has been shown that for the excitation frequencies below or above linear cut-off, the steady-state response in the far
P. Vainshtein / Physica D 162 (2002) 1–8
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field represents the evanescent and propagating duct modes, respectively. The main finding of [3] is that exactly at linear cut-off, the resonance transverse shock waves appear in the duct traveling between its side walls. It has been shown that this response decays algebraically in the far field. In the present paper, we consider the combined motion of a piston where the rotational vibrations are small compared to translational displacements πy t xp (t) = ε c0 t0 f + δh sin Ωt cos . (4) t0 h Under conditions (4) no flow parameters depend on z-variable. Thus, this variable and the corresponding velocity component are excluded from the analysis below. Let us introduce the following dimensionless variables: φ =
φ , c0 h
t =
tc0 , h
(x , y ) =
(x, y) . h
(5)
After dropping the primes, the dimensionless form of (1) is obtained simply by replacing c0 by unity. Eq. (4) takes the form Ω xp (t) = ε f (t) + δ sin π t cos πy . (6) ω0 The initial conditions are ∂φ = 0 (t = 0, 0 < x < ∞, 0 < y < 1). φ= ∂t
(7)
The fluid velocity components are u=
∂φ , ∂x
v=
∂φ . ∂y
(8)
The boundary condition on the piston surface is u(xp , y, t) −
∂xp ∂xp v(xp , y, t) = ∂y ∂y
(t > 0).
(9)
The no-flux requirement at the walls of the duct is v=0 We let Ω =ν ω0
(y = 0, 1).
(ν = O(1))
(10)
(11)
with ν being the parameter which is of the order of unity. Near field solution for the problem associated with the translational piston motion (2) may be found in [5]. It has been shown that the asymptotic expansion breaks down as x = O(ε −1 ). This result holds for the combined piston curve (6) as well since the rotational piston vibrations are considered much smaller than the translational displacements. In order to derive the effect of wave amplitude on wave development in the far field we use the method of multiple scales. We introduce the fast spatial variable, x, ˜ and the slow spatial variable X = εx associated with small parameter ε, and a corresponding differential operator ∂ ∂ ∂ = +ε . ∂x ∂ x˜ ∂X
(12)
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P. Vainshtein / Physica D 162 (2002) 1–8
Note that in the expansions below one may put x˜ = x that is correct to leading order (see [5]). Thus, we assume a far field expansion in the form φ = εφ1 (x, y, t, X) + ε 2 φ2 (x, y, t, X) + O(ε 3 ).
(13)
Substituting (13) in (1), one obtains, at O(ε), the linear wave equation, with the dependence on X not yet specified. At O(ε2 ) we find ∇ 2 φ2 −
∂ 2 φ2 ∂ 2 φ1 ∂φ1 ∂φ1 2 = −2 + 2∇φ1 · ∇ + (γ − 1) ∇ φ1 , 2 ∂x∂X ∂t ∂t ∂t
(14)
where the terms on the r.h.s. lead to secular terms. All terms on the right of (14) lead to particular solutions proportional to x, and these will render (13) invalid for large x unless we choose the right side of (14) to vanish. Thus, φ 1 is a uniform first approximation provided it satisfies the nonlinear equation ∂ 2 φ1 ∂φ1 γ − 1 ∂φ1 2 − ∇ φ1 = 0. − ∇φ1 · ∇ ∂t 2 ∂t ∂x∂X
(15)
The boundary condition for the far field is the same as that of the near field [5]. Thus, Eq. (15) is to be solved with u1 given at x = X = 0. Then this equation provides the uniformly valid first-order solution. For one-dimensional motions, (15) can be reduced to the simple wave or inviscid Burgers’ equation. First-order boundary conditions follow from (6), (9) and (10) u1 (0, y, t, 0) = f (t) + δπ ν cos πνt cos πy,
(16)
v1 = 0
(17)
(y = 0, 1).
We consider the distinguished limit γ −1 = µ = O(1), 2δ
δ → 0,
(18)
and seek a solution in the form of series with respect to small parameter δ φ1 (x, y, t, X) = φ (0) (x, t, X) + δφ (1) (x, y, t, X) + O(δ 2 ).
(19)
In the zeroth approximation, the solution is independent of the transverse variable y. Using (8) and (16) we easily obtain for this approximation the one-dimensional simple wave equation with γ = 1 ∂u(0) ∂u(0) − u(0) = 0, ∂X ∂θ
θ = t − x,
(20)
u(0) (θ, X = 0) = f (θ ).
(21)
Then the first-approximation equations are ∂u(1) ∂u(1) ∂u(0) ∂u(0) − u(0) − u(1) + µu(0) = 0, ∂X ∂θ ∂θ ∂θ
(22)
∂v1 ∂u(1) =− , ∂y ∂θ
(23)
u(1) (θ, X = 0) = πν cos πνt cos πy,
v (1) = 0
(y = 0, 1).
(24)
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The solution of the simple wave equation (20) under boundary condition (21) may be represented in the parametric form [5] u(0) (X, η) = f (η),
θ (X, η) = η − Xf (η),
(25)
where η is the parameter which is constant along the characteristics of Eq. (20). Explicit results are easily obtained for a uniformly accelerated piston motion where f (t) = ± 21 t 2 .
(26)
These piston curves bring about propagation of compression and expansion waves when sign plus and minus are taken, respectively. Elimination of the parameter η in Eq. (25) and using (26) enables the solution to be obtained in the form u(0) =
θ , 1−X
u(0) = −
θ , 1+X
θ > 0,
(27)
θ > 0, X 0, t > (−2X/ε)1/2 , X < 0,
(28)
where (27) and (28) describe the compression and expansion waves, respectively. Solution (27) is valid to the time of shock formation, that is for t < 1/ε. We seek a solution of Eqs. (22) and (23) in the form u(1) = U (X, θ ) cos πy,
v (1) = V (X, θ ) sin πy.
(29)
Substituting (27)–(29) in (22)–(24) yields the following equations for the compression and expansion waves, respectively, and the boundary condition θ ∂U 1 θ ∂U = 0, − − U +µ ∂X 1 − X ∂θ 1−X (1 − X)2
(30)
∂U θ ∂U 1 θ + + U +µ = 0, ∂X 1 + X ∂θ 1+X (1 + X)2
(31)
U (θ, 0) = πν cos πνθ.
(32)
We now define the new variables ζ = ln(1 − X),
(33)
ξ = ln(1 + X),
(34)
τ = ln θ.
(35)
Then Eqs. (30)–(32) take the form ∂U ∂U + + U − µ eτ −ζ = 0, ∂ζ ∂τ
(36)
∂U ∂U + + U + µ eτ −ζ = 0, ∂ξ ∂τ
(37)
U = πν cos(π ν eτ ) (ζ = ξ = 0).
(38)
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P. Vainshtein / Physica D 162 (2002) 1–8
Fig. 2. Propagation of the duct mode caused by the expansion wave. Dependence of the longitudinal velocity component u1 on time at y = 0 and different x.
Using (23), (29), (37) and (38), we obtain the following solution for the expansion waves: πν πνθ θ θ cos +µ , −µ 2 1+X 1+X 1+X (1 + X) µ θ2 µ θ2 πνθ − + . V = π sin 1+X 2 (1 + X)2 2 1+X
U=
(39) (40)
This solution shows that the plane expansion wave (28) causes the duct mode (29) to propagate along the duct. The propagation takes place for frequencies above and below linear cut-off. In Fig. 2, the dependence of the longitudinal velocity component u1 on time, calculated by (28), (29) and (39) at ε = 0.1, δ = 0.1, ν = 0.9, µ = 1.0, is presented at y = 0 and different x. Amplitude of the longitudinal velocity component at the leading front slightly decreases as it travels further along the duct. Note that the amplitude of the transverse velocity component vanishes at the leading front. The frequency of oscillations at x = 0 is the same as that of the related linear evanescent mode. It decreases as the distance from the original piston position increases. Using (23), (29), (36) and (38), we obtain the following solution for the compression waves: U=
πν πνθ θ θ cos −µ +µ , 1−X 1−X 1−X (1 − X)2
(41)
P. Vainshtein / Physica D 162 (2002) 1–8
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Fig. 3. Propagation of the duct mode caused by the compression wave. Dependence of the longitudinal velocity component u1 on time at y = 0 and different x.
πνθ µ θ2 µ θ2 V = π sin − . + 1−X 2 (1 − X)2 2 1−X
(42)
This solution shows that the plane compression wave (27) brings also about the propagation of the duct mode (29) for frequencies above and below linear cut-off. In Fig. 3, the dependence of the longitudinal velocity component u1 on time, calculated by (27), (29) and (41) at ε = 0.1, δ = 0.1, ν = 0.9, µ = 1.0, is presented at y = 0 and different x. In this case, the amplitude of the longitudinal velocity component at the leading front considerably increases as it travels further along the duct. A frequency of oscillations also increases as the distance from the original piston position increases. The obtained solutions describe two-dimensional wave patterns resulting from the combined excitation and nonlinear interactions. Indeed, while a front of a wave generated by the sinusoidal piston vibrations moves, reflecting from the duct walls, it interacts not only with other fronts of the same origin but also with a wave generated by the translational piston displacements. As a result, a two-dimensional wave pattern is formed which differs qualitatively from the linear acoustic modes as well as from nonlinear waves that could be generated by pure translational or rotational piston motions. One should not think, in particular, of the wave patterns propagating at frequencies below linear cut-off as evanescent modes changed into propagating modes. In reality we deal here with wave motions of a new kind produced by the combined piston motion.
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P. Vainshtein / Physica D 162 (2002) 1–8
It should be noted that combined piston motions are not the only way to produce sound including longitudinal plane waves and duct modes. Such two-dimensional sound patterns may also be generated with a compression driver mounted directly to the end of the duct to produce a plane wave, and drivers mounted to the top and bottom walls to generate duct modes. Hence, the conditions for the nonlinear interactions, investigated theoretically, are readily attainable in actual experiments. The obtained result may be of importance in applied research on transmission of sound in acoustic waveguides.
3. Conclusions Analytical calculations and approximations to the problem of wave propagation in two-dimensional acoustic ducts are presented. The main result is that for waves with wavelength larger than the cut-off wavelength propagation is achievable if the media is nonlinear. The importance of the phenomenon for sound transmission is discussed. Since electromagnetic media can generally be also nonlinear, the analysis of the possibility of such an effect in electromagnetic and optical waveguides might be of theoretical and applied interest.
Acknowledgements The author thanks the late I.M. Rutkevich for the many informal discussions. This work was financed by the TMR programme of the Commission of the European Union under grant code number R1097-2900 during a year’s visit by the author to the University College Dublin. References [1] [2] [3] [4] [5]
J. Lighthill, Waves in Fluids, Cambridge University Press, Cambridge, 1978. L.D. Landau, E.M. Lifshitz, Electrodynamics of Continuous Media, Pergamon Press, Oxford, 1984. P. Vainshtein, Transverse resonant oscillations in acoustic ducts, J. Comp. Acoust. 9 (2) (2001) 543. P.A. Blythe, D.G. Crighton, Shock-generated ignition: the induction zone, Proc. R. Soc. London Ser. A 426 (1989) 189. J.D. Cole, Perturbation Methods in Applied Mathematics, Springer, Berlin, 1968.
On wave propagation in waveguides P. Vainshtein∗ Faculty of Mechanical Engineering, Technion – Israel Institute of Technology, Haifa 32000, Israel Received 1 August 2001; received in revised form 7 November 2001; accepted 9 November 2001 Communicated by F.H. Busse
Abstract Cut-off frequencies are well known in waveguides of different nature to be the thresholds of propagation and evanescence. In the present paper, a theoretical study is made to demonstrate that a combined excitation of a plane wave and waveguide mode may lead to the propagation of a nonlinear wave pattern irrespective of whether the frequency of the waveguide mode is higher or below linear cut-off. A particular problem is examined when a piston performs sinusoidal rotational vibrations combined with translational displacements in a uniform semi-infinite two-dimensional fluid duct. The frequency of the vibrations is close to a cut-off frequency. The rotational vibrations are small compared to the translational displacements. An analytical solution of the problem is obtained for uniformly accelerated piston displacements. It is shown that longitudinal plane waves guide the duct modes along the duct, such that propagation takes place even if the forcing frequencies are below linear cut-off. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Cut-off frequency; Waveguides; Two-dimensional duct
1. Introduction Cut-off frequencies are commonly associated with waveguides in various physical contexts. It is well known that there exists a discrete set of frequencies, each of which is a threshold of a new propagating mode. According to the linearized theory, as the frequency decreases to the threshold below which the mode cannot propagate, the relevant group velocity tends to zero. Below the threshold, the mode changes drastically to an evanescent one. The familiar waveguides are acoustic ducts, water tanks, electromagnetic and dielectric optical waveguides (see [1,2]). As cut-off frequencies are of general interest in waveguides, an examination of problems is interesting to clarify the theoretical question of whether there are some ways of sound transmission if the driving frequency is below linear cut-off. In the present paper, an attempt to develop a relevant theory of nonlinear wave propagation where excitation frequencies are close to linear cut-off for the physically simple case of a slab acoustic duct with rigid walls is undertaken. ∗
Tel.: +972-4-82923266; fax: +972-4-8324533. E-mail address: [email protected] (P. Vainshtein). 0167-2789/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 8 9 ( 0 1 ) 0 0 3 7 6 - 1
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P. Vainshtein / Physica D 162 (2002) 1–8
Fig. 1. Schematic of a semi-infinite two-dimensional duct with a coordinate system. The thick lines show the piston performing combined translational displacements and sinusoidal vibrations.
2. Analysis and results The main subject of the study is the propagation of acoustic waves forced by piston motions combined translational displacements with rotational vibrations (see Fig. 1). Such piston motions are of a simple physical model of a source of sound generating two-dimensional acoustic waves in an acoustic duct. One may anticipate that due to nonlinear interactions of longitudinal waves and duct modes in the near field, the resulting wave motions in the far field will represent some two-dimensional flow pattern. In order to get an insight into this generally rather involved process we shall treat a particular problem where the rotational vibrations are much smaller than the translational displacements. Besides, a uniformly accelerated translational piston motion will be considered that will enable explicit solutions to be obtained. The exact nonlinear wave equation for an isentropic flow is (see [3]) ∂ 2φ ∂φ ∂φ 1 1 2 2 2 2 c0 ∇ φ − 2 = 2∇φ · ∇ (1) + (∇φ · ∇)(∇φ) + (γ − 1) + (∇φ) ∇ 2 φ, ∂t 2 ∂t 2 ∂t where φ is the velocity potential, c0 the initial speed of sound; x, y, z, t are Eulerian coordinates, ∇ and ∇ 2 are the gradient and Laplacian, γ the ratio of specific heat. Typical values of γ range from a maximum of 53 for monatomic gases through 75 for diatomic gases to values as low as 1.2 or even 1.1 for polyatomic gases at high temperature [1]. The condition γ → 1 is frequently used in asymptotic analysis (see for example [4]). It simplifies greatly the nonlinear analysis since the entropy waves are no longer coupled to the pressure field. We shall use this condition in the present paper in order to obtain an analytical solution of the formulated problem. The classical problem of the one-dimensional propagation of waves is specified by the motion of a piston with a characteristic time t0 , such that the distance traveled is small compared to c0 t0 (see [5]). This defines the small parameter of the problem, ε, and the piston curve xp (t) can be presented in terms of the dimensionless function f by t xp (t) = εc0 t0 f . (2) t0 The small rotational motion of a piston about the z-axis, such that the amplitude of the piston vibrations is small compared to the cross-sectional dimension of the duct h, can be presented by the piston curve as follows: πy xp (t) = δh sin Ωt cos , (3) h where Ω is the frequency of excitation and δ the small parameter. The problem of the wave propagation in a planar duct produced by the piston motion (3) has been investigated in [3]. An analytical theory based on an asymptotic approximation has been developed for the nonlinear response when the excitation frequency is close to the lowest cut-off frequency, ω0 = πc0 / h. The study has focused attention on the far fields of the steady-state flow. It has been shown that for the excitation frequencies below or above linear cut-off, the steady-state response in the far
P. Vainshtein / Physica D 162 (2002) 1–8
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field represents the evanescent and propagating duct modes, respectively. The main finding of [3] is that exactly at linear cut-off, the resonance transverse shock waves appear in the duct traveling between its side walls. It has been shown that this response decays algebraically in the far field. In the present paper, we consider the combined motion of a piston where the rotational vibrations are small compared to translational displacements πy t xp (t) = ε c0 t0 f + δh sin Ωt cos . (4) t0 h Under conditions (4) no flow parameters depend on z-variable. Thus, this variable and the corresponding velocity component are excluded from the analysis below. Let us introduce the following dimensionless variables: φ =
φ , c0 h
t =
tc0 , h
(x , y ) =
(x, y) . h
(5)
After dropping the primes, the dimensionless form of (1) is obtained simply by replacing c0 by unity. Eq. (4) takes the form Ω xp (t) = ε f (t) + δ sin π t cos πy . (6) ω0 The initial conditions are ∂φ = 0 (t = 0, 0 < x < ∞, 0 < y < 1). φ= ∂t
(7)
The fluid velocity components are u=
∂φ , ∂x
v=
∂φ . ∂y
(8)
The boundary condition on the piston surface is u(xp , y, t) −
∂xp ∂xp v(xp , y, t) = ∂y ∂y
(t > 0).
(9)
The no-flux requirement at the walls of the duct is v=0 We let Ω =ν ω0
(y = 0, 1).
(ν = O(1))
(10)
(11)
with ν being the parameter which is of the order of unity. Near field solution for the problem associated with the translational piston motion (2) may be found in [5]. It has been shown that the asymptotic expansion breaks down as x = O(ε −1 ). This result holds for the combined piston curve (6) as well since the rotational piston vibrations are considered much smaller than the translational displacements. In order to derive the effect of wave amplitude on wave development in the far field we use the method of multiple scales. We introduce the fast spatial variable, x, ˜ and the slow spatial variable X = εx associated with small parameter ε, and a corresponding differential operator ∂ ∂ ∂ = +ε . ∂x ∂ x˜ ∂X
(12)
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P. Vainshtein / Physica D 162 (2002) 1–8
Note that in the expansions below one may put x˜ = x that is correct to leading order (see [5]). Thus, we assume a far field expansion in the form φ = εφ1 (x, y, t, X) + ε 2 φ2 (x, y, t, X) + O(ε 3 ).
(13)
Substituting (13) in (1), one obtains, at O(ε), the linear wave equation, with the dependence on X not yet specified. At O(ε2 ) we find ∇ 2 φ2 −
∂ 2 φ2 ∂ 2 φ1 ∂φ1 ∂φ1 2 = −2 + 2∇φ1 · ∇ + (γ − 1) ∇ φ1 , 2 ∂x∂X ∂t ∂t ∂t
(14)
where the terms on the r.h.s. lead to secular terms. All terms on the right of (14) lead to particular solutions proportional to x, and these will render (13) invalid for large x unless we choose the right side of (14) to vanish. Thus, φ 1 is a uniform first approximation provided it satisfies the nonlinear equation ∂ 2 φ1 ∂φ1 γ − 1 ∂φ1 2 − ∇ φ1 = 0. − ∇φ1 · ∇ ∂t 2 ∂t ∂x∂X
(15)
The boundary condition for the far field is the same as that of the near field [5]. Thus, Eq. (15) is to be solved with u1 given at x = X = 0. Then this equation provides the uniformly valid first-order solution. For one-dimensional motions, (15) can be reduced to the simple wave or inviscid Burgers’ equation. First-order boundary conditions follow from (6), (9) and (10) u1 (0, y, t, 0) = f (t) + δπ ν cos πνt cos πy,
(16)
v1 = 0
(17)
(y = 0, 1).
We consider the distinguished limit γ −1 = µ = O(1), 2δ
δ → 0,
(18)
and seek a solution in the form of series with respect to small parameter δ φ1 (x, y, t, X) = φ (0) (x, t, X) + δφ (1) (x, y, t, X) + O(δ 2 ).
(19)
In the zeroth approximation, the solution is independent of the transverse variable y. Using (8) and (16) we easily obtain for this approximation the one-dimensional simple wave equation with γ = 1 ∂u(0) ∂u(0) − u(0) = 0, ∂X ∂θ
θ = t − x,
(20)
u(0) (θ, X = 0) = f (θ ).
(21)
Then the first-approximation equations are ∂u(1) ∂u(1) ∂u(0) ∂u(0) − u(0) − u(1) + µu(0) = 0, ∂X ∂θ ∂θ ∂θ
(22)
∂v1 ∂u(1) =− , ∂y ∂θ
(23)
u(1) (θ, X = 0) = πν cos πνt cos πy,
v (1) = 0
(y = 0, 1).
(24)
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The solution of the simple wave equation (20) under boundary condition (21) may be represented in the parametric form [5] u(0) (X, η) = f (η),
θ (X, η) = η − Xf (η),
(25)
where η is the parameter which is constant along the characteristics of Eq. (20). Explicit results are easily obtained for a uniformly accelerated piston motion where f (t) = ± 21 t 2 .
(26)
These piston curves bring about propagation of compression and expansion waves when sign plus and minus are taken, respectively. Elimination of the parameter η in Eq. (25) and using (26) enables the solution to be obtained in the form u(0) =
θ , 1−X
u(0) = −
θ , 1+X
θ > 0,
(27)
θ > 0, X 0, t > (−2X/ε)1/2 , X < 0,
(28)
where (27) and (28) describe the compression and expansion waves, respectively. Solution (27) is valid to the time of shock formation, that is for t < 1/ε. We seek a solution of Eqs. (22) and (23) in the form u(1) = U (X, θ ) cos πy,
v (1) = V (X, θ ) sin πy.
(29)
Substituting (27)–(29) in (22)–(24) yields the following equations for the compression and expansion waves, respectively, and the boundary condition θ ∂U 1 θ ∂U = 0, − − U +µ ∂X 1 − X ∂θ 1−X (1 − X)2
(30)
∂U θ ∂U 1 θ + + U +µ = 0, ∂X 1 + X ∂θ 1+X (1 + X)2
(31)
U (θ, 0) = πν cos πνθ.
(32)
We now define the new variables ζ = ln(1 − X),
(33)
ξ = ln(1 + X),
(34)
τ = ln θ.
(35)
Then Eqs. (30)–(32) take the form ∂U ∂U + + U − µ eτ −ζ = 0, ∂ζ ∂τ
(36)
∂U ∂U + + U + µ eτ −ζ = 0, ∂ξ ∂τ
(37)
U = πν cos(π ν eτ ) (ζ = ξ = 0).
(38)
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Fig. 2. Propagation of the duct mode caused by the expansion wave. Dependence of the longitudinal velocity component u1 on time at y = 0 and different x.
Using (23), (29), (37) and (38), we obtain the following solution for the expansion waves: πν πνθ θ θ cos +µ , −µ 2 1+X 1+X 1+X (1 + X) µ θ2 µ θ2 πνθ − + . V = π sin 1+X 2 (1 + X)2 2 1+X
U=
(39) (40)
This solution shows that the plane expansion wave (28) causes the duct mode (29) to propagate along the duct. The propagation takes place for frequencies above and below linear cut-off. In Fig. 2, the dependence of the longitudinal velocity component u1 on time, calculated by (28), (29) and (39) at ε = 0.1, δ = 0.1, ν = 0.9, µ = 1.0, is presented at y = 0 and different x. Amplitude of the longitudinal velocity component at the leading front slightly decreases as it travels further along the duct. Note that the amplitude of the transverse velocity component vanishes at the leading front. The frequency of oscillations at x = 0 is the same as that of the related linear evanescent mode. It decreases as the distance from the original piston position increases. Using (23), (29), (36) and (38), we obtain the following solution for the compression waves: U=
πν πνθ θ θ cos −µ +µ , 1−X 1−X 1−X (1 − X)2
(41)
P. Vainshtein / Physica D 162 (2002) 1–8
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Fig. 3. Propagation of the duct mode caused by the compression wave. Dependence of the longitudinal velocity component u1 on time at y = 0 and different x.
πνθ µ θ2 µ θ2 V = π sin − . + 1−X 2 (1 − X)2 2 1−X
(42)
This solution shows that the plane compression wave (27) brings also about the propagation of the duct mode (29) for frequencies above and below linear cut-off. In Fig. 3, the dependence of the longitudinal velocity component u1 on time, calculated by (27), (29) and (41) at ε = 0.1, δ = 0.1, ν = 0.9, µ = 1.0, is presented at y = 0 and different x. In this case, the amplitude of the longitudinal velocity component at the leading front considerably increases as it travels further along the duct. A frequency of oscillations also increases as the distance from the original piston position increases. The obtained solutions describe two-dimensional wave patterns resulting from the combined excitation and nonlinear interactions. Indeed, while a front of a wave generated by the sinusoidal piston vibrations moves, reflecting from the duct walls, it interacts not only with other fronts of the same origin but also with a wave generated by the translational piston displacements. As a result, a two-dimensional wave pattern is formed which differs qualitatively from the linear acoustic modes as well as from nonlinear waves that could be generated by pure translational or rotational piston motions. One should not think, in particular, of the wave patterns propagating at frequencies below linear cut-off as evanescent modes changed into propagating modes. In reality we deal here with wave motions of a new kind produced by the combined piston motion.
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It should be noted that combined piston motions are not the only way to produce sound including longitudinal plane waves and duct modes. Such two-dimensional sound patterns may also be generated with a compression driver mounted directly to the end of the duct to produce a plane wave, and drivers mounted to the top and bottom walls to generate duct modes. Hence, the conditions for the nonlinear interactions, investigated theoretically, are readily attainable in actual experiments. The obtained result may be of importance in applied research on transmission of sound in acoustic waveguides.
3. Conclusions Analytical calculations and approximations to the problem of wave propagation in two-dimensional acoustic ducts are presented. The main result is that for waves with wavelength larger than the cut-off wavelength propagation is achievable if the media is nonlinear. The importance of the phenomenon for sound transmission is discussed. Since electromagnetic media can generally be also nonlinear, the analysis of the possibility of such an effect in electromagnetic and optical waveguides might be of theoretical and applied interest.
Acknowledgements The author thanks the late I.M. Rutkevich for the many informal discussions. This work was financed by the TMR programme of the Commission of the European Union under grant code number R1097-2900 during a year’s visit by the author to the University College Dublin. References [1] [2] [3] [4] [5]
J. Lighthill, Waves in Fluids, Cambridge University Press, Cambridge, 1978. L.D. Landau, E.M. Lifshitz, Electrodynamics of Continuous Media, Pergamon Press, Oxford, 1984. P. Vainshtein, Transverse resonant oscillations in acoustic ducts, J. Comp. Acoust. 9 (2) (2001) 543. P.A. Blythe, D.G. Crighton, Shock-generated ignition: the induction zone, Proc. R. Soc. London Ser. A 426 (1989) 189. J.D. Cole, Perturbation Methods in Applied Mathematics, Springer, Berlin, 1968.