Abstract
A semi-infinite tube is joined to a semi-infinite cone. Waves propagating in the tube towards the join are partly reflected and partly radiated into the cone. The problem is to determine these wave fields. Two modal expansions are used, one in the tube and one in the cone. However, their regions of convergence do not overlap: there is a region \({\mathcal{D}}\) near the join where neither expansion converges. It is shown that the expansions can be connected by judicious applications of Green’s theorem in \({\mathcal{D}}\). The resulting equations are solved asymptotically, for long waves or for narrow cones. Related two-dimensional problems are also solved. Applications to acoustics, electromagnetics and hydrodynamics are considered.
References
Levine H, Schwinger J (1948) On the radiation of sound from an unflanged circular pipe. Phys Rev 73: 383–406
Noble B (1988) Methods based on the Wiener–Hopf technique. Chelsea, New York
Caussé R, Kergomard J, Lurton X (1984) Input impedance of brass musical instruments—comparison between experiment and numerical models. J Acoust Soc Am 75: 241–254
Risser JR (1949) Waveguide and horn feeds. In: Silver S (eds) Microwave antenna theory and design. McGraw-Hill, New York, pp 334–387
Love, AW (eds) (1976) Electromagnetic horn antennas. IEEE Press, New York
Olver AD, Clarricoats PJB, Kishk AA, Shafai L (1994) Microwave horns and feeds. IEE, London
Bird TS, Love AW (2007) Horn antennas. In: Volakis JL (eds) Antenna engineering handbook, 4th edn. McGraw-Hill, New York, pp 14-1–14-74
Green HE (2006) The radiation pattern of a conical horn. J Electromagn Waves Appl 20: 1149–1160
Jones DS (1986) Acoustic and electromagnetic waves. Oxford University Press, Oxford
Kaloshin VA (2009) Scattering matrix for a junction of two horns. Russ J Math Phys 16: 246–259
Borovikov VA, Kinber BY (1994) Geometrical theory of diffraction. Institution of Electrical Engineers, London
Webster AG (1919) Acoustical impedance, and the theory of horns and of the phonograph. Proc Natl Acad Sci USA 5: 275–282
Benade AH, Jansson EV (1974) On plane and spherical waves in horns with nonuniform flare I. Theory of radiation, resonance frequencies, and mode conversion. Acustica 31: 79–98
Pierce AD (1989) Acoustics. Acoustical Society of America, New York
Martin PA (2004) On Webster’s horn equation and some generalizations. J Acoust Soc Am 116: 1381–1388
Dean RG (1964) Long wave modification by linear transitions. J Waterw Harb Div Proc ASCE 90: 1–29
Lamb H (1932) Hydrodynamics, 6th edn. Cambridge University Press, Cambridge
LeBlond PH, Mysak LA (1978) Waves in the ocean. Elsevier, Amsterdam
Lewin L (1970) On the inadequacy of discrete mode-matching techniques in some waveguide discontinuity problems. IEEE Trans Microw Theory Tech MTT 18: 364–369
Chester W (1983) The acoustic impedance of a semi-infinite tube fitted with a conical flange. Z Angew Math Phys 34: 412–417
Rayleigh (1896) The theory of sound, vol 2. Reprinted, Dover, New York, 1945
Chester W (1987) The acoustic impedance of a semi-infinite tube fitted with a conical flange: Part II. J Sound Vib 116: 371–377
Dalrymple RA, Martin PA (1996) Water waves incident on an infinitely long rectangular inlet. Appl Ocean Res 18: 1–11
Chester W (1950) The propagation of sound waves in an open-ended channel. Phil Mag 41(7): 11–33
Rice SO (1949) A set of second-order differential equations associated with reflections in rectangular wave guides—application to guide connected to horn. Bell Syst Tech J 28: 136–156
Leonard DJ, Yen JL (1957) Junction of smooth flared wave guides. J Appl Phys 28: 1441–1448
Riblet HJ (1977) An alternate derivation of Lewin’s formula. IEEE Trans Microw Theory Tech MTT 25: 711–712
Bowman JJ, Senior TBA (1969) The wedge. In: Bowman JJ, Senior TBA, Uslenghi PLE (eds) Electromagnetic and acoustic scattering by simple shapes. North-Holland, Amsterdam, pp 252–283
Abramowitz, M, Stegun, IA (eds) (1965) Handbook of mathematical functions. Dover, New York
Bowman JJ (1969) The cone. In: Bowman JJ, Senior TBA, Uslenghi PLE (eds) Electromagnetic and acoustic scattering by simple shapes. North-Holland, Amsterdam, pp 637–701
Mittra R (1963) Relative convergence of the solution of a doubly infinite set of equations. J Res Natl Bur Stand 67D: 245–254
Mittra R, Lee SW (1971) Analytical techniques in the theory of guided waves. Macmillan, New York
Porter R, Porter D (2000) Water wave scattering by a step of arbitrary profile. J Fluid Mech 411: 131–164
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Martin, P.A. The horn-feed problem: sound waves in a tube joined to a cone, and related problems. J Eng Math 71, 291–304 (2011). https://doi.org/10.1007/s10665-011-9454-8
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DOI: https://doi.org/10.1007/s10665-011-9454-8