Higher Order Modes
Higher order modes in 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.
Horn Research Update
Since I have followed the Integrated PhD programme at NTNU, I did not finish my MSc until this summer (2013), even though I started om my PhD in the summer of 2011. Anyway, my MSc is now completed, and I'm working full time on my PhD.
My MSc thesis was of course also on horns, where I extended the Modal Propagation Method to rectangular horns with asymmetries. I also did an attempt on curved horns, but that was a very complex subject, as it turned out. The modes in the bend is described by Bessel functions of non-integer and imaginary order, and I also needed to find the zeros of such functions. Not that easy... You can see the result of my work here: MSc Thesis.
This fall I started working on the modal radiation impedance of rectangular ducts. I implemented some of this in my MSc thesis, so the basic functions for computing the radiation impedance for a horn mounted in an infinite baffle are available. However, most horns are not mounted in infinite baffles. They are mounted in finite baffles, unbaffled, or placed on the floor, near walls, in real rooms. Therefore I started looking into the influence of mutual impedance.
If a surface vibrates with a certain velocity, there will be a force pressing back on this surface due to the reaction of the air the surface moves. The force will be frequency dependent, and it will be more or less out of phase with the velocity. At low frequencies, there is nearly a 90 degree phase difference between the two, and little power is radiated. At higher frequencies they fall into phase, and the source radiates with maximum efficiency. The ratio of this force to the velocity is called the radiation impedance.
If two or more surfaces are vibrating, or if a source vibrates near a hard wall, so that an image source is formed due to the "mirroring effect" of the wall, each source will generate a pressure on the other. The ratio of the pressure of one source to the velocity of the other is called the mutual radiation impedance. It has to be taken into account whenever sources are near walls, or when there are multiple sources.
In addition to mutual impedance, diffraction from the baffle edges for horns in finite baffles also creates pressure on the radiating surface. To quantify this effect is what I'm currently working on.
All these simulation models do of course have to be checked against reality. The workshop at the university has therefore built me some pretty horns to measure. They have an aluminium inner skin, and 4 layers of 3mm MDF laminated on top of that, finished with some bracing, as shown on the picture below:
Having this horn ready, I mounted it in a fairly large baffle, and measured in the anechoic chamber at the University. The test setup is shown below:
I measured both the throat impedance using the two-microphone method, and the pressure at various points.
Below is a graph of the measured throat impedance, compared to the impedance simulated with the Modal Propagation Method, using 16 modes in each direction. Seems like I'm on the right track with this method....
The deviations from the simulated impedance above 2.5kHz is first the cross-modes in the measurement tube, and then, above 4kHz, we can see the effect of the microphone spacing becoming less than half a wavelength.
This horn is intended as a 1:4 scale model of a 50Hz midbass horn. An even shorter version of it, designed to be mounted near a floor/wall intersection, is also under way. This is quite common for midbass horns, and the idea is to try out this horn at various positions in a scale model of a room.
The general rule of thumb has been that by placing a horn near a wall, one can get away with a smaller mouth. This is not necessarily the case. Below is an example. The pale (overlay) lines are the throat impedance for the horn mounted in an infinite baffle, the same as above. The darker lines shows what happens when a wall is placed near the horn (30cm from the center of the mouth, which is 35x35cm). It is actually worse than the horn in the baffle alone.
Placing the horn nearer the wall (20cm from the center of the mouth) improves the situation. And the nearer it gets, the better. Conversely, at 40cm the impedance is much more peaky than at 30cm.
But note also that the higher frequency ripple increases.
More data will be published on this later. But for the time being, it is worth keeping in mind that things are not as simple as rules of thumb may lead us to believe. Especially with horns.
Modes in horns
Not only plane or spherical waves propagate in horns. Above a certain frequency, the wave front becomes much more complex, and this complex wave front can be described as a weighted sum of mode functions.
You may be familiar with standing waves in rooms. These standing waves can only exist at certain frequencies, and their shape is given by the so-called boundary conditions. For a room with hard walls, the boundary condition is that the velocity perpendicular to the wall should be zero. That means that the pressure has its maximum values at the walls. A room will have modes in all three directions, and the pressure at any point will be a sum of the contributions from all excited modes.
Another example of modes is the vibration of a string. A string is fixed at the ends, so the boundary condition is that the displacement should be zero at the ends. The mode functions for displacement are sine functions. Then the string can vibrate with shapes that are sums of sine functions that can fulfil the condition of zero displacement at the ends.
In horns, the pressure distribution across a cross-section can also be viewed as a sum of mode functions that meet the boundary conditions. For an axisymmetric horn, using a plane cross section, these functions can be cylindrical Bessel functions. These are the mode functions for a cylindrical tube of constant cross section. A horn can be approximated by a series of short, cylindrical tubes. At each change of cross section, the mode functions in the two tubes must be matched. The mode functions will not be the same in the two tubes. This means that a single mode coming to a change in cross section will generate a series of other modes in the following duct. This is known as mode coupling.
This fact can be used to simulate horns, by letting the wave at the throat propagate through the horn, generating new modes at each change in cross section, and letting them propagate or decay through the next part of the horn, and so on. A method to do this is described by Kemp in his thesis. During my project in Numerical Acoustics this fall (2011), I implemented this method and compared it to BEM. The report can be found here. The end result was in very good agreement with BERIM (a boundary element method for horns in infinite baffles), and much faster. I plan to make the method available, hopefully both as a little Windows executable, and as a Matlab toolbox. Update: The Matlab toolbox is available here.
Update: Spring 2012 I undertook a more detailed evaluation of the modal propagation method, to check the accuracy and computation times etc. The report can be downloaded here.
Today I added the option to donate to my horn research. Since my stipend is small, I have to raise funds in other places. So I thought I should try adding a PayPal donate button. Hopefully that will help a little. And it will also motivate me to put up more material on these pages, maybe also some SW now and then.
This is my first blog post here. I'm not sure how much time I will devote to writing blog posts, but I hope to put up something here every now and then. Anyway, this is my
Horn Research Blog.
This summer (2011) I was accepted for the integrated PhD programme at NTNU, the Norwegian University of Science and Technology. My thesis will be on horns for domestic use. All this means that I'm officially a horn researcher!
The integrated PhD means that I will be doing my Masters in parallel with the first year of the PhD. So far I have not had much time to do proper horn research. But I have been working on an article on finite horns, and a simulation method for horns that includes higher order modes.
That concludes my first posting.