Hostname: page-component-8448b6f56d-cfpbc Total loading time: 0 Render date: 2024-04-20T00:12:20.962Z Has data issue: false hasContentIssue false
  • English
  • Français

Trapped modes in two-dimensional waveguides

Published online by Cambridge University Press:  26 April 2006

M. Callan ,
C. M. Linton  and
D. V. Evans
M. Callan
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK
C. M. Linton
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK
D. V. Evans
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

A two-dimensional acoustical waveguide described by two infinite parallel lines a distance 2d apart has a circle of radius a < d positioned symmetrically between them. The potential satisfies the two-dimensional Helmholtz equation in the fluid region between the circle and the lines, and the normal gradient of the potential vanishes on both. For motions which are antisymmetric about the centreline of the guide there exists a cutoff frequency below which no propagation down the guide is possible. It is proved that for a circle of sufficiently small radius there exists a trapped mode, having a frequency close to the cutoff frequency, which is antisymmetric about the centreline of the guide and symmetric about a line through the centre of the circle perpendicular to the centreline. The method used is due to Ursell (1951) who established the existence of a trapped surface wave mode in the vicinity of a long totally submerged horizontal circular cylinder of small radius in deep water. Numerical computations in the present work reveal that a single trapped mode appears to exist for all values of a ≤ d and not just when the circle is small. The present method, when used to attempt to construct a solution antisymmetric about both the centreline and a line perpendicular to it through the centre of the circle does not lead to a trapped mode. The trapped modes can equally well be regarded as surface-wave modes, as in an infinitely long tank of water with a free surface, into which has been placed symmetrically, a vertical rigid circular cylinder extending throughout the depth. Numerical evidence for the existence of such trapped modes when the cylinder is of rectangular cross-section was presented in Evans & Linton (1991).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 229 , August 1991 , pp. 51 - 64
Copyright
© 1991 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bogdanoff, D. W. 1983 Compressible effects in turbulent shear layers. AIAA J. 21, 926–927.Google Scholar
Chinzei, N., Masuya, G., Komuro, T., Murakami, A. & Kuden, K. 1986 Spreading of two-stream supersonic turbulent mixing layers. Phys. Fluids 29, 1345–1347.Google Scholar
Greenough, J. A., Riley, J. J., Soetrisbmno, M. & Eberhardt, D. S. 1989 The effects of walls on a compressible mixing layer. AIAA Paper 89–0372.Google Scholar
Ikawa, H. & Kubota, T. 1975 Investigation of supersonic turbulent mixing with zero pressure gradient. AIAA J. 13, 566–572.Google Scholar
Jackson, T. L. & Grosch, C. E. 1989 Absolute/convective instabilities and the convective Mach number in a compressible mixing layer. ICASE Rep. 89–38.Google Scholar
Mei, C. C. 1985 Resonant reflection of surface water waves by periodic sandbars. J. Fluid Mech. 152, 315–335.Google Scholar
Mei, C. C., Hara, T. & Naciri, M. 1988 Note on Bragg scattering of water waves by parallel bars on the seabed. J. Fluid Mech. 186, 147–162.Google Scholar
Mack, L. M. 1989 On the inviscid acoustic-mode instability of supersonic shear flows. Proc. Fourth Symp. on Numer. and Phys. Aspects of Aerodyn. Flows.
Miles, J. W. 1958 On the disturbed motion of a plane vortex sheet. J. Fluid Mech. 4, 538–552.Google Scholar
Nayfeh, A. H. 1973 Perturbation Methods. Wiley-Interscience.
Papamoschou, D. 1986 Experimental investigation of heterogeneous compressible shear layers. Ph.D. thesis, California Institute of Technology.
Papamoschou, D. & Roshko, A. 1986 Observations of supersonic free shear layers. AIAA Paper 86–0162.Google Scholar
Papamoschou, D. & Roshko, A. 1988 The compressible turbulent shear layers: an experimental study. J. Fluid Mech. 197, 453–477.Google Scholar
Ragab, S. A. & Wu, J. L. 1989 Linear instabilities in two-dimensional compressible mixing layers. Phys. Fluids A1, 957–966.Google Scholar
Reshotko, E. 1976 Boundary-layer stability and transition. Ann. Rev. Fluid Mech. 8, 311–349.Google Scholar
Smith, A. M. O. & Gamberoni, N. 1956 Transition, pressure gradient and stability theory. Douglas Aircraft Company (El Segundo) Rep. ES 26388.Google Scholar
Tam, C. K. W. & Hu, F. Q. 1989 The instability and acoustic wave modes of supersonic mixing layers inside a rectangular channel. J. Fluid Mech. 203, 51–76.Google Scholar
Zhuang, M., Kubota, T. & Dimotakis, P. E. 1988 On the instability of inviscid compressible free shear layers. AIAA Paper 88–3538.Google Scholar
Zhuang, M., Kubota, T. & Dimotakis, P. E. 1990 The effect of walls on a spatially growing supersonic shear layer.. Phys. Fluids A 2, 599–604.Google Scholar