We demonstrate the ability of a simple algorithm based on the venerable method of images (MOIs), to accurately model the detailed frequency response of a multidimensional, rectangular, lossy resonant cavity. The convergence properties of the model's infinite series solution are shown to be determined by the cavity's quality factor *Q*. A 1D example demonstrates that the MOI series converges to the exact solution. Next, a comparison to precisely measure 2D cavity data confirms that a straightforward extension of the 1D algorithm to multiple dimensions provides accurate results. The algorithm is short, easily understandable by undergraduate students and relatively undemanding to code. An example using ^{®}mathematica is provided.

## References

1.

The standard deviation of the observed peak frequencies away from the least-squares fit to $\omega (k)=v\varphi \u2009k$ was only $2.1\u2009Hz$ for frequencies below $3.5\u2009kHz$. The phase velocity $v\varphi $ of sound was determined from the fit to be $344.79\u2009\xb1\u20090.24\u2009m/s$ in excellent agreement with that calculated using an ideal gas expression for dry air at the measured lab temperature of $22.4\u2009C$: $344.72\u2009m/s$. See, for example, the

*HyperPhysics Concepts*website of Georgia State University,*Speed of Sound*: <http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/souspe3.html>.2.

By “phasor” we mean, not the science fiction weapon, but the signal's complex-valued amplitude and phase at time

*t*= 0. We then get the phase at any other time by multiplying the phasor by $exp\u2009(\u2212i\omega t)$. Note that we use the “physics phase convention” for a traveling wave: $exp\u2009(i\u2009k\xb7r\u2212i\omega t)$, rather than the “engineering convention” $exp\u2009(j\omega t\u2212j\u2009k\xb7r)$. Of course,*i*and*j*each denote $\u22121$, but the differing symbols might provide the reader with a hint as to which convention is being used (and*j*avoids possible confusion with a symbol for electrical current).3.

Those “in the know” will recognize this discussion as providing a somewhat simplified version of a wave representation of the signals on the transmission line, where we have taken the line's characteristic wave impedance $\u22611$. The current phasor on the line at position

*x*is then $I(x)=V+(x)\u2212V\u2212(x)$, and a free boundary for the voltage becomes a fixed boundary for the current, e.g., $I(L)=V+(L)\u2212V\u2212(L)=0$.4.

Wikipedia describes several of these methods, e.g., <https://en.wikipedia.org/wiki/Finite_element_method>, <https://en.wikipedia.org/wiki/Method_of_moments_(electromagnetics)_>.

5.

In general, the method of images (MOI) enjoys a long history in the computer-aided engineering analysis of room and structural acoustics. Pioneered in a now-classic paper by Allen and Berkley (Ref. 15), it used image sources to estimate the time-domain, reverberation response of a room to a short sound impulse. The resulting reverberation time envelope characterizes the suitability of a space as, for example, a lecture hall or recording studio, a diagnostic introduced in 1895 by W. C. Sabine (Ref. 16). The method does not, however, employ coherent superposition of waves from the source and its reflections, and it is, therefore, unsuitable for frequency response calculations of a cavity whose dimensions are of the order of a wavelength, as in our case. The use of MOI-based calculations of a space's acoustic reverberation response has become a standard of the industry. The Wayverb website, Ref. 17, includes a summary of current techniques employed by several popular acoustic engineering tools: <https://reuk.github.io/wayverb/context.html#existing-software>. All of these tools, however, are inappropriate for the task at hand.

6.

George

Green

, An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism

(T. Wheelhouse

, Nottingham, England

, 1828

). Available at Google Books <https://books.google.com/books?id=GwYXAAAAYAAJ>.Also, definitions and examples are available in:

John D.

Jackson

, Classical Electrodynamics

, 3rd ed. (John Wiley and Sons, Inc., New York

, 1998

), ISBN 978-0471309321.7.

We consider only specular reflections for our model, so that angles of incidence and reflection are equal. The assumed ideal, rigid boundaries have reflection coefficients equal to unity, independent of the angle of incidence. These assumptions are quite accurate for the thick, rigid walls of the small resonant cavity under consideration. For large cavities (such as offices, classrooms, or auditoriums), these assumptions are inadequate, because diffuse reflections and wall losses (attenuation) are nontrivial and can be strongly dependent on the angle of incidence.

8.

The requirement that multiple mirror images of the cavity generated as described completely tile the plane and have disjoint interior regions greatly restricts the geometries of candidate cavities for this method. Rectangles and equilateral triangles are among the very few available. More complicated algorithms can be used to generate the set of image sources for a more general range of polygonal cavity shapes, but such considerations are beyond the scope of this paper. For examples see the quite sophisticated approach presented by Cuenca

*et al.*, in their 2009 paper, Ref. 18. It employs coherent MOI superposion to analyze the frequency responses of elastic vibrations in thin, convex, solid plates.9.

Of course, the RHS of Eq. (10) does not evaluate to a pressure amplitude. It should be scaled by a factor representing the pressure phasor amplitude at some standard distance

*δ*from the source, e.g., $p(\delta )\delta 1/2$. As will be seen, this factor is unimportant for our analysis, so it may be taken to be $\u22611$.10.

This result leads to a conclusion analogous to the venerable “Olbers paradox” of cosmology (H. R. Olbers, 1823). See, for example,

Edward R.

Harrison

, Darkness at Night: A Riddle of the Universe

(Harvard U. P.

, Cambridge

, 1987

), ISBN 978–0674192706. See also: <https://en.wikipedia.org/wiki/Olbers_paradox>11.

The algorithm as implemented could perform approximately $1.8\xd7106$ image

*p*(*r*) calculations per second of CPU time on a fairly modest desktop: 3.1 GHz, 4-core**Intel i5 with 16 GB RAM. Two versions of the notebook are provided: MOIannotated.nb has heavily commented**^{®}**Mathematica code explaining its details and including example function calls, whereas MOI.nb is much more terse and streamlined. The website also holds other files related to acoustic cavity resonance investigations. <http://sophphx.caltech.edu/MOI>.**^{®}12.

At the “standard” threshold of human hearing, $0\u2009dB\u2009SPL$ (sound pressure level), a sound wave's RMS pressure amplitude is approximately $2\xd710\u221210$ standard atmospheres, or about 0.15

*μ*torr. At $121\u2009dB\u2009SPL$, over a million times the threshold amplitude, measured vibration amplitude of the human eardrum in the area where it is coupled to the inner ear is 350 nm; at $0\u2009dB\u2009SPL$, this would correspond to an amplitude a million times smaller, or only $\u223c1/300$ atomic radius. See:Juergen

Tonndorf

and Shyam M.

Khanna

, “Tympanic-membrane vibrations in human cadaver ears studied by time-averaged holography

,” J. Acous. Soc. Am.

52

, 1221

–1233

(1972

).13.

E. A.

Dean

, “Atmospheric effects on the speed of sound

,” Technical Report No. AD-A076060

, Defense Technical Information Center

(1979

) <http://sophphx.caltech.edu/MOI/Dean_1979.pdf≥.14.

Carl Gottfried Neumann (1832–1925), German (Prussian) mathematician

. He made several important contributions to applied mathematics and mathematical physics

. See the MacTutor History of Mathematics Archive website, <https://mathshistory.st-andrews.ac.uk/Biographies/Neumann_Carl/>.15.

Jont B.

Allen

and David A.

Berkley

, “Image method for efficiently simulating small-room acoustics

,” J. Acoust. Soc. Am.

65

, 943–950

(1979

).16.

Harvard professor Wallace Clement Sabine (1868–1919) was arguably the “father” of architectural acoutics and its use of

*sound reverberation*time as a primary indicator of the suitability of a space's acoustics for concert music or as a lecture hall. Modern computational methods for architectural acoustic design are clearly directly descended from his successful 1895 effort to improve the acoustics of a lecture hall on the Harvard campus. SeeWallace Clement

Sabine

, Collected Papers on Acoustics

(Harvard U. P

., Cambridge

, 1922

). Available at *Internet Archive's Open Library*: <https://openlibrary.org/works/OL160998W>.17.

The

*Wayverb*website ([copyright]Reuben

Thomas

, 2016

) provides open-source software for acoustic modeling of large rooms and structures. The software, which has not been used or endorsed by the authors, is described as employing a variety of modeling techniques. Available at: <https://reuk.github.io/wayverb/>.18.

Jacques

Cuenca

, François

Gautier

, and Laurent

Simon

, “The image source method for calculating the vibrations of simply supported convex polygonal plates

,” J. Sound Vib.

322

, 1048

–1069

(2009

).© 2023 Author(s). Published under an exclusive license by American Association of Physics Teachers.

2023

Author(s)

AAPT members receive access to the

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