Acoustics, Sound Fields and Transducers



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The story behind “Acoustics, Sound Fields

and Transducers”. Leo Beranek and Tim

Mellow. Academic Press, 600 pp. 2012.
If you ask any

engineer or academic

working in

the field of elect

r o a c o u s t i c s

which is their

favorite text book,

the reply will

nearly always be

Acoustics by Leo

Beranek [1].

Why is a book published in 1954

without revisions still so popular? There are

many reasons: It deals mainly with fundamental

principles which have not changed. It

is well structured. The author’s passion for

the subject is infectious – the opening sentence

is “Acoustics is a most fascinating subject”.

Wave propagation is explained pictorially

before diving into mathematics.

Electrical circuit analogies are used to provide

insight into the operation of transducers.

Formulas are given to help readers to

work out their own designs. It is hardly surprising

that Acoustics has become one of the

most cited books on the subject.

My background was in electrical engineering,

so naturally Leo’s circuit analogies

sparked my interest in acoustics and his book

was my bible for many years. I was always fascinated by the plots of directivity patters and

radiation impedances for pistons with and

without baffles. In fact Lord Rayleigh (John

W. Strutt) had derived an elegant closed-form

formula for the radiation impedance of a piston

in an infinite baffle in 1894 [2], long

before the direct radiator loudspeaker had

been invented.

However, I had much more difficulty in

reproducing the impedance of a piston without

a baffle. In 2002, I looked up the paper

that Leo cited in the footnotes [3] expecting

to find an equation that I could simply enter

into my computer and use to plot the result.

It wasn’t there. I had to go right back to

Christoffel Bouwkamp’s 1941 PhD thesis [4],

which is 25 pages long, but contains no final

equation that can be used. Every step is

needed. At the end, the author states “The

time-consuming computation of eigenvalues,

eigenfunctions, and other quantities important

for the physical interpretation was done

on a small hand-driven Brunsviga desk calculator”.

This was certainly a heroic achievement

at the time and I imagine that he burnt

much midnight oil.

At about the same time, I discovered a

paper [5] by another Dutch researcher, Hans

Streng, which used more powerful boundary

integral methods [6] to determine the sound

radiated from a circular electrostatic loudspeaker

membrane. I then met a Finnish colleague

(coincidentally with the same forename

as the author of my favorite book), Leo

Kärkkäinen, who was working at Nokia

Research Center in Helsinki. When I showed

him Streng’s paper, he was quite excited

about it because he could see that it could be

used to describe any surface velocity distribution

such as that of a piston in a circular baffle.

This led to our first published paper [7].

With his physics background, Leo became

my mentor in wave theory and mathematical

methods. He even modified the Streng

method in order to avoid the use of collocation

and thus calculate everything directly. In

my second paper [8] I applied this method to

the Bouwkamp problem of waves diffracted

through a circular aperture. Meanwhile, I

had been busy creating my own document of

everything I had learnt about wave theory

starting from first principles until one day,

when I showed it to Leo Kärkkäinen, he said

“This would make a nice book. Have you

thought of publishing it?”

A few years later, on 23rd April 2007 to be

exact, I was thinking that what the world of

engineering acoustics needed was a text book

that covered everything from lumped-element

theory using circuit analogies, as covered

in Leo Beranek’s book, to wave theory

and sound radiation/scattering problems.

However, I could not cover the fundamental

principles any better than Leo had already

done and I did not want to plagiarize anything.

The idea of writing an updated version

of Acoustics suddenly hit me like a thunderbolt.

It seemed like completely the right

thing to do even if it would involve a huge

amount of work. The reason I know the date

so exactly is that in my excitement I immediately

fired off an email to Leo Kärkkäinen!

I wrote to Leo Beranek who amazingly

wrote back informing me that he held the

copyright and that I could use anything I

liked. However, he was not keen on being a

coauthor because he was too busy contributing

to the current literature on concert hall

acoustics, even if I wrote all the new material

and he simply reviewed it. This was hardly

surprising since at the time since I was completely unknown with only a handful of published papers to my name [7, 8, 9]. However,

as time went on, Leo became more enthusiastic

about the project and contributed much

new material, including two whole chapters

on sound in enclosures and rooms for loudspeaker listening. I was certainly delighted

when he eventually decided he would like to

have his name on it as that was the best possible

endorsement of all the hard work

involved.


The first task was to obtain an optical scan

of the original book as a Word document.

Unfortunately it then took me a year to correct

it because, in addition to numerous

scanning errors, the software did not recognize

any mathematical symbols or Greek letters.

I also had the formidable task of reproducing

the figures by drawing over PDFs of

them in Word, rather like virtual tracing

paper. I first met Leo Beranek face to face at

the 2007 ICA meeting in Madrid. We also met

a few times in London, Boston and at the

2008 ASA meeting in Miami but, because of

the distance between Surrey and Boston,

most of our collaboration was done via email.

While working on the new text, we were

answering all of the questions that I had collected

in my mind over the years. For example:

What are the independent constituent

variables that determine the efficiency of a

loudspeaker? How is the radiated sound pressure of an unbaffled loudspeaker determined

from its equivalent circuit? How does a finite

open or closed baffle affect the frequency

response? Can we design a simple crossover

which does not produce time-delay waveform

distortion? What are the 2-port networks for

horns of different profiles? How much radiated

sound power is needed to reproduce

music or speech in an auditorium of a given

size? How does the shape of the radiator

affect the response? Is there a difference

between flat, convex or concave radiators?

Where does the Kirchhoff-Helmholtz boundary

integral come from and what does it

mean? Can we have a unified approach to

sound radiation/scattering instead of the current

patchwork of different methods? What is

the equivalent circuit of a very narrow tube

with viscous and thermal losses and a slip

boundary condition?

Leo paid particular attention to the ordering

of contents and specifically asked me to

include a new section on transmission-line

loudspeakers because so many had asked him

about how the Bose Wave system worked. He

also requested a new chapter on cellphone

acoustics because he was curious as to how

so much sound could be produced by something

the size of a deck of cards and felt that

it would bring the book right up to date.

Because I had worked for Nokia for so many

years, I was too close to the subject to explain

it clearly to the lay person and Leo’s input

here proved invaluable. I am also indebted to

my colleague Enrico Pascucci for his marvelous

photographs and many useful suggestions

for the chapter.
Leo wanted a new title which indicated

direct lineage with the original while reflecting

the change in emphasis and suggested

“Acoustics: Transducers and airborne

acoustics”. Although “airborne” was intended

to indicate “non-structural”, I suggested

replacing it with “sound fields” in order to

sound less “outdoors”. Also, the term sound

field is fairly general as it can mean either a

free field or sound in an enclosure. He then

suggested putting “sound fields” before

“transducers” because that is the order in

which the two subjects are introduced and

we settled for that.

A new version of Acoustics had to include

the work of Neville Thiele [10] and Richard

Small [11]. They proposed just six parameters

to completely describe the low-frequency

behavior of a loudspeaker, which are now

commonly known as the Thiele-Small

parameters, and Small showed how to obtain

them from the input impedance [12]. Also,

they produced tables/charts which enable

anybody to choose a frequency response

shape for a given drive unit and engineer the

cabinet and bass-reflex port accordingly. Leo

is rightly proud of the fact that these authors

used his book as their starting point and that

it led to the development of smaller loudspeakers

using the acoustic suspension principle

[13].
During my previous experience as an analogue

filter designer, the Thiele/Small

approach was standard. One would never

design a filter by messing around with component

values until it “looked about right”. If

standard tables/charts were not available for

the element values of a particular circuit, I

had to derive the transfer function by hand

and solve for the polynomial coefficients in

terms of the circuit element labels. This was

a very laborious process which often involved

many pages of algebra, so I fully understood

the significance of Thiele and Small’s work.

Virtually no circuit simulation tools use

the transfer function. Instead they calculate

all the node voltages at every frequency step.

I had been interested in deriving transfer

functions back in the 1980s for studying the

transient responses and stability of amplifiers.

Back then, computers could only do

such calculations numerically. In order to

solve for circuit element values, symbolic

computation was needed, but at the time,

Maple was the only software which did this

and it was only available in universities. It

was shortly after joining Nokia in 1999 that

Noel Lobo introduced me to the numerical

and symbolic power of Mathematica and

taught me how to use it effectively.

Another Nokia colleague, Andrew Bright,

had the foresight to ask me to write a program

that could derive a polynomial transfer

function from the net list of a circuit. I

started with the method described in a

recent book by Robert Boyd [14], but

extended it to include ideal voltage sources,

current sources, transformers and gyrators.

Using Mathematica, I was able to create two

versions: one numeric and the other symbolic.

This proved invaluable for creating

complicated acoustical designs such as a

combined call and handsfree loudspeaker. I

have described this computation method in

the final chapter of the book. If one day

someone were to use it to create a proper

software tool with a nice GUI, it would

really make my day.

I should also mention Juha Backman who

has provided much support and encouragement

over the years as well as introducing

me to many people and important literature.

It has certainly been rewarding working

with someone as eminent as Leo Beranek.

What both Leo’s share is a generosity of

nature, energy and passion for acoustics as a

subject that makes it seem more like fun

than work. I hope that the new book inspires

future generations of students, engineers and

academics as the original inspired me. It

really is a most fascinating subject and there

is still plenty to explore.


References

[1] Beranek, L.L., Acoustics, McGraw-Hill

(1954).

[2] Rayleigh, J.W.S., The Theory of Sound,



Dover, New York, 1945, Vol. II, p. 163.

[3] Wiener, F.M., “On the relation between

the sound fields radiated and diffracted by

Plane Obstacles,” J Acoust. Soc. Am. vol.23,

no.6, pp. 697–700 (1951).

[4] Bouwkamp, C.J., “Theoretical and

numerical treatment of diffraction through a

circular aperture,” IEEE Transactions of



Antennas and Propagation, vol. AP18, no. 2,

pp. 152–176 (1970). (This is a translation of

his 1941 PhD thesis which was originally

published in Dutch.)

[5] Streng, J.H., “Sound radiation from a

circular stretched membrane in free space,” J



Audio Eng. Soc
. vol. 37, no. 3, pp. 107–118,

(1989).

[6] Streng, J.H., “Calculation of the surface

pressure on a vibrating circular stretched

membrane in free space,” J Acoust. Soc. Am.

vol. 82, no. 2, pp. 679–686 (1987).

[7] Mellow, T.J. and Kärkkäinen, L.M., “On

the sound field of an oscillating disk in an

open and closed circular baffle,” J Acoust.


Soc. Am., vol. 118, no. 3, pp. 1311–1325

(2005).


[8] Mellow, T.J., “On the sound field of a

resilient disk in an infinite baffle,” J Acoust.



Soc. Am., vol. 120, no. 1, pp. 90–101 (2006).

[9] Mellow, T.J. and Kärkkäinen, L.M., “On

the sound field of a circular membrane in

free space and an infinite baffle,” J Acoust.



Soc. Am., vol. 120, no. 5, pp. 2460–2477

(2006).


[10] Thiele, A. N., “Loudspeakers in vented

boxes,” Proc. IREE 22: 487 (1961); republished

in J. Audio Eng. Soc., vol. 19, no. 5, pp.

382–392 (1971) and vol. 19, no. 6, pp.

471–483 (1971).

[11] Small, R.H., “Vented-box loudspeaker

systems,” J. Audio Eng. Soc., vol. 21, no. 5,

pp. 363–372; vol. 21, no. 6, pp. 438–444; vol.

21, no. 7, pp. 549–554; vol. 21, no. 8, pp.

635–639 (1973).

[12] Small, R.H., “Direct Radiator Loudspeaker

System Analysis,” J. Audio Eng. Soc.

vol. 20, no. 5, pp. 383–395 (1972).

[13] Villchur, E.M., “Problems of bass

reproduction in loudspeakers,” J. Audio Eng.

Soc. vol. 5, no. 3, pp. 122-126 (1957).

[14] Boyd, R., Node List Tolerance Analysis



CRC Press, Boca Raton, 2006.

J. Audio

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