Nobumasa Sugimoto

Krylov‐Bogoliubov‐Mitropolsky method for nonlinear wave modulation

The Krylov‐Bogoliubov‐Mitropolsky perturbation method is applied to systems of nonlinear dispersive waves including plasma waves such as ion‐acoustic, magneto‐acoustic, and electron plasma waves. It is found that long time slow modulation of the complex wave amplitude can be described by the nonlinear Schrodinger equation for a very wide class of nonlinear dispersive waves.

Burgers equation with a fractional derivative ; hereditary effects on nonlinear acoustic waves

This paper deals with initial-value problems for the Burgers equation with the inclusion of a hereditary integral known as the fractional derivative of order 4. Emphasis is placed on the difference between the local and global dissipation due to the second-order and the half-order derivatives, respectively. Exploiting the smallness of the coefficient of the second-order derivative, an asymptotic analysis is first developed. When a discontinuity appears, the matched-asymptotic expansion method is employed to derive a uniformly valid solution. If the coefficient of the halforder derivative is also small, as is usually the case, the evolution comprises three stages, namely a lossless near field, an intermediate Burgers region, and a hereditary far field. In view of these results, the equation is then solved numerically, under various initial conditions, by finite-difference and spectral methods. It is revealed that the effect of the fractional derivative accumulates slowly to give rise to a significant dissipation and distortion of the waveform globally, which is to be contrasted with the effect of the second-order derivative, significant only locally, in a thin ‘shock layer’.

Dispersion characteristics of sound waves in a tunnel with an array of Helmholtz resonators

This paper examines dispersion characteristics of sound waves propagating in a tunnel with an array of Helmholtz resonators connected axially. Assuming plane waves over the tunnel’s cross section except a thin boundary layer, weakly dissipative effects due to the wall friction and the thermoviscous diffusivity of sound are taken into account. Sound propagation in such a spatially periodic structure may be termed ‘‘acoustic Bloch waves.’’ The dispersion relation derived exhibits peculiar characteristics marked by emergence of ‘‘stopping bands’’ in the frequency domain. The stopping bands inhibit selectively propagation of sound waves even if no dissipative effects are taken into account, and enhance the damping pronouncedly even in a dissipative case. The stopping bands result from the resonance with the resonators as side branches and also from the Bragg reflection by their periodic arrangements. In the ‘‘passing bands’’ outside of the stopping bands, the sound waves exhibit dispersion, though subjected i...

Propagation of nonlinear acoustic waves in a tunnel with an array of Helmholtz resonators

It is proposed that an array of Helmholtz resonators connected to a tunnel in its axial direction will suppress the propagation of sound generated by a travelling train and especially the emergence of shock waves in the far field. Under the approximation that the resonators may be regarded as continuously distributed, quasi-onedimensional formulation is given for nonlinear acoustic waves by taking account of not only the resonators but also the wall friction due to the presence of a boundary layer and the diffusivity of sound. For a far-field propagation, the spatial evolution equation coupled with the equation for the response of the resonator is then derived. The linear dispersion relation suggests that the resonators, if appropriately designed, enhance the dissipation and give rise to the dispersion as well. By solving initialvalue problems for the evolution equation, the array of resonators is proved to be very effective in suppressing shock waves in the far field. The resonators themselves fail to counteract shock waves once formed, but rather prevent their emergency by rendering acoustic waves dispersive. By this dispersion, it becomes possible, in a special case, for an acoustic soliton to be propagated in place of a shock wave.

Note on Higher Order Terms in Reductive Perturbation Theory

It is remarked that the second order correction given by Ichikawa et al. for a single ion acoustic K-dV (Korteweg-de Vries) soliton contains a secular term. To eliminate the secularity, the method of multiple scales combined with the reductive perturbation method is proposed. It is then found that not only the second order correction is modified so as to be secular-free but also the phase factor of the lowest K-dV soliton suffers a modification proportional to its amplitude.

Acoustic solitary waves in a tunnel with an array of Helmholtz resonators

It is demonstrated theoretically that an acoustic solitary wave can be propagated in a tunnel with a periodic array of Helmholtz resonators, if the dissipative effects are made negligibly small. As wave propagation in such a periodic system is known as the Bloch waves, the array can give rise to the dispersion necessary to formation of the solitary wave. Explicit profiles of the solitary waves are obtained by solving the steady‐wave solutions to the nonlinear wave equations derived previously. It is found that the solitary wave is compressive and its propagation speed is slower than the usual sound speed a0, i.e., subsonic, but faster than a0(1−κ/2) in the linear long‐wave limit, κ being a small parameter representing the ratio of the cavity’s volume to the tunnel’s volume per axial spacing between the neighboring resonators. As the propagation speed approaches the upper bound, the height of the solitary wave increases to approach the limiting height, while as the speed approaches the lower bound, the sol...

Annihilation of shocks in forced oscillations of an air column in a closed tube

Effects of a periodic array of Helmholtz resonators on forced longitudinal oscillations of an air column in a closed tube are examined experimentally. The column is driven sinusoidally at a frequency near the lowest resonance frequency by oscillating bellows mounted on one end of the tube. Frequency responses are obtained for small and large amplitudes of the excitations. While the array lowers the resonance frequency and the peak value, its dispersive effect, i.e., the dependence of the sound speed on a frequency, can annihilate the shock effectively.

Marginal condition for the onset of thermoacoustic oscillations of a gas in a tube

This paper revisits a classical problem to derive a marginal condition for the onset of spontaneous thermoacoustic oscillations of a gas in a circular tube with one end open and the other closed by a flat wall, subjected to a temperature gradient along the side wall. Formulation is given in the framework of the linear theory and the first-order theory in the ratio of a boundary-layer thickness to the tube radius. An eigenvalue problem is posed on the second-order differential equation with variable coefficients of the axial coordinate for the excess pressure in the main-flow region outside of the boundary layer. A boundary layer on the end wall is taken into account in the form of an appropriate boundary condition. By using the idea of renormalization, the pressure is rescaled and a complex axial coordinate is introduced so that the pressure equation is transformed into a tractable form. It turns out that the equation includes a factor (frequency) determined by the product of the local sound speed and the...

Thermoacoustic-wave equations for gas in a channel and a tube subject to temperature gradient

This paper develops a general theory for linear propagation of acoustic waves in a gas enclosed in a two-dimensional channel and in a circular tube subject to temperature gradient axially and extending infinitely. A ‘narrow-tube approximation’ is employed by assuming that a typical axial length is much longer than a span length, but no restriction on a thickness of thermoviscous diffusion layer is made. For each case, basic equations in this approximation are reduced to a spatially one-dimensional equation in terms of an excess pressure by making use of a method of Fourier transform. This equation, called a thermoacoustic-wave equation, is given in the form of an integro-differential equation due to memory by thermoviscous effects. Approximations of the equations for a short-time and a long-time behaviour from an initial state are discussed based on the Deborah number and the Reynolds number. It is shown that the short-time behaviour is well approximated by the equation derived previously by the boundary-layer theory, while the long-time behaviour is described by new diffusion equations. It is revealed that if the diffusion layer is thicker than the span length, the thermoviscous effects give rise to not only diffusion but also wave propagation by combined action with temperature gradient, and that negative diffusion may occur if the gradient is steep.

Verification of acoustic solitary waves

Experiments and numerical simulations are carried out to verify the existence of the acoustic solitary wave in an air-filled tube with an array of Helmholtz resonators connected. Following up previous work (Sugimoto et al. 1999), the experiments are improved by using a newly designed piston driver to launch an initially plane pressure pulse and also by extending the tube length from 7.4 m to 10.6 m. To highlight the effect of the array of resonators, the case with no array is also examined in parallel. Direct and indirect checks are made to verify the existence of the solitary wave. The former compares the profiles and propagation speeds of pulses measured experimentally to the solitary-wave solution. The latter checks the validity of nonlinear wave equations in describing real wave evolution in the tube. Solving an initial-value problem numerically with weakly lossy effects of boundary layers and jet loss at the throat of the resonator, comparison is made between measured and simulated evolution. The validity of the equations in the lossy case is necessary to maintain the existence of the solitary wave in the lossless limit. It is revealed that nonlinear wave equations originally derived for unidirectional propagation in the tube can provide a good description of the real evolution, with some allowance for phase shifts on reflection at both ends of the tube. In particular, it turns out that the lossy effects are described quantitatively well. By establishing the validity of the equations, it is concluded that the acoustic solitary wave exists.