Skip to main content Accessibility help
×
Home

Trapped modes in two-dimensional waveguides

  • M. Callan (a1), C. M. Linton (a1) and D. V. Evans (a1)

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 ad 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).

Copyright

References

Hide All
Bogdanoff, D. W. 1983 Compressible effects in turbulent shear layers. AIAA J. 21, 926927.
Chinzei, N., Masuya, G., Komuro, T., Murakami, A. & Kuden, K. 1986 Spreading of two-stream supersonic turbulent mixing layers. Phys. Fluids 29, 13451347.
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.
Ikawa, H. & Kubota, T. 1975 Investigation of supersonic turbulent mixing with zero pressure gradient. AIAA J. 13, 566572.
Jackson, T. L. & Grosch, C. E. 1989 Absolute/convective instabilities and the convective Mach number in a compressible mixing layer. ICASE Rep. 89–38.
Mei, C. C. 1985 Resonant reflection of surface water waves by periodic sandbars. J. Fluid Mech. 152, 315335.
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, 147162.
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, 538552.
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.
Papamoschou, D. & Roshko, A. 1988 The compressible turbulent shear layers: an experimental study. J. Fluid Mech. 197, 453477.
Ragab, S. A. & Wu, J. L. 1989 Linear instabilities in two-dimensional compressible mixing layers. Phys. Fluids A1, 957966.
Reshotko, E. 1976 Boundary-layer stability and transition. Ann. Rev. Fluid Mech. 8, 311349.
Smith, A. M. O. & Gamberoni, N. 1956 Transition, pressure gradient and stability theory. Douglas Aircraft Company (El Segundo) Rep. ES 26388.
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, 5176.
Zhuang, M., Kubota, T. & Dimotakis, P. E. 1988 On the instability of inviscid compressible free shear layers. AIAA Paper 88–3538.
Zhuang, M., Kubota, T. & Dimotakis, P. E. 1990 The effect of walls on a spatially growing supersonic shear layer.. Phys. Fluids A 2, 599604.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Trapped modes in two-dimensional waveguides

  • M. Callan (a1), C. M. Linton (a1) and D. V. Evans (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed