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- Written by Bjørn Kolbrek

# Higher Order Modes

Higher order modes describes the deviation from simple wave propagation in horns and sound fields in general. If the wave front in the horn is not plane, cylindrical or spherical, it can be described as a sum of mode functions. In one of my earlier blog posts, and also in the reports available under Horn Theory, I have discussed this in some more detail.

Since it appears that many believe that the existence of higher order modes in horns (often just called HOM) is a quite recent discovery, I decided to put together a compilation of references on the topic. This list will probably be updated contiously as I find more references, but here are the first ones. Some of them have URLs, but you may not have permission to download the papers from the journals. If you have access to a university library, you may be able to download them there. I will not download them for you...

V. A. Hoersch: Non-Radial Harmonic Vibrations within a Conical Horn, Phys. Rev. 25, 218–224 (1925)

Perhaps the first analysis of modes in conical horns. The boundary condition at the horn mouth is not realistic (zero pressure condition), but the analysis is valid. It illustrates that the existence of HOMs were known at a very early stage in electroacoustics.

K. Sato: On the Sound Field Due to a Conical Horn with a Source at the Vertex, Japanese Journal of Physics 5, no 3, 103-109, 1929

The sound field internal and external to a conical horn is given as an expansion in spherical harmonics (modes in spherical coordinates). Directivity plots for nearfield and farfield are given.

R. J. Alfredson: The propagation of sound in a circular duct of continuously varying cross-sectional area, Journal of Sound and Vibration, Volume 23, Issue 4, 22 August 1972, Pages 433-442

The horn is divided into short cylindrical segments, where a modal description is used to compute the sound field. Modes are matched at the discontinuities, the number of modes determined iteratively.

A. Cummings: Sound transmission in curved duct bends, J. Sound Vibr. 35, no 4, 451-477 (1974)

A modal description of the sound field in a rectangular bend. Mode functions are cosine and Bessel functions.

A. H. Benade and E. V. Jansson: On plane and spherical waves in horns with nonuniform flare. I- Theory of radiation, resonance frequencies, and mode conversion, Acoustica 31, 79-98 (1974)

The existence of HOMs and their effect on the resonance frequencies of musical instruments are discussed in general terms. Some simplified calculations. Discussion of mode conversion.

E. R. Geddes: Acoustic Waveguide Theory Revisited, J. Audio Eng. Soc. 41, no 6, 452-461 (1993)

A more detailed analysis of the Oblate Spheroidal waveguide than the original 1989 paper. The existence of HOMs in the waveguide is taken into account and analyzed.

V. Pagneux, N. Amir and J. Kergomard: A study of wave propagation in varying cross‐section waveguides by modal decomposition. Part I. Theory and validation, J. Acoust. Soc. Am. 100, 2034 (1996)

V. Pagneux, N. Amir and J. Kergomard: A study of wave propagation in varying cross‐section waveguides by modal decomposition. Part II. Results, J. Acoust. Soc. Am. 101, 2504 (1997)

V. Pagneux, N. Amir and J. Kergomard: Wave propagation in acoustic horns through modal decomposition, Transactions of the Wessex Institute, 1997

The papers by Pagneux et al descibe computational methods for the calculation of horn performance based on mode matching of the sound field. The focus is on musical instruments, but the method is of course also applicable to other horns. Axisymmetric and 2D horns are described.

S. Félix, V. Pagneux: Sound propagation in rigid bends: A multimodal approach, J. Acoust. Soc. Am. 110, 1329 (2001)

Multimodal description of sound propagation through bends in tubes.

J. A. Kemp: Theoretical and experimental study of wave propagation in brass musical instruments, PhD dissertation, University of Edinburgh, 2002

A multimodal method for computing the throat impedance and mouth volume velocity of horns. Partly based on the work of Pagneux, Amir and Kergomard.

E. Geddes: Audio Transducers, 2002

Describes mode based methods for several geometries, including the oblate spheroidal coordinate system. The measurement chapter partly describes how to measure the modal amplitudes at the mouth of a horn or other transducers.

Behler, Gottfried K.; Makarski, Michael: On the Velocity Distribution at the Interface of Horn Driver and Horn, AES Convention:116 (May 2004) Paper Number:6097

Measurement of the sound field at the interface surface between a compression driver and horn, and the decomposition of this sound field into modes described in cylindrical coordinates.

Makarski, Michael: Do Higher Order Modes at the Horn Driver's Mouth Contribute to the Sound Field of a Horn Loudspeaker?, AES Convention:117 (October 2004) Paper Number:6188

Description of how a non-planar velocity wave front at the horn throat influences the sound field at the horn mouth, and the radiated sound field.

Makarski, Michael: Tools for the Professional Development of Horn Loudspeakers, PhD thesis, RWTH Aachen, 2006

The PhD thesis contains the information from the above two articles, and more. Modal decomposition of sound fields, and modal exitation of horns are described un detail. If you want to measure HOMs, this is a good place to start.

Leo L. Beranek and Tim J. Mellow: Acoustics: Sound Fields and Transducers, Academic Press, 2012

This is an update of the classical book Acoustics by Beranek from 1954, and it's the most useful book on acoustics that I have. If you plan to buy only one book on loudspeakers and sound radiation, this is the one. (For room acoustics, noise control etc, there other books that have more information, but for a loudspeaker designer, this one is a must). It covers many analytical and semianalytical methods to calculate sound fields, many of them are based on eigenmode or eigenfunction expansions.