torsional waves in a bowed string
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Torsional waves in a bowed string.
This page is an appendix for the scientific paper:Bavu, E., Smith, J. and Wolfe, J. "Torsional waves in a bowed string" (2005, Acustica, 91,
241-246).
It has sound files of the translational and rotational waves, an animationof idealisedHelmholtz motion in translation and a briefsummary of the paper.This figure, reproducedfrom the paper, shows the velocity, in time and frequency domain representation, of the
transverse and torsional waves in a bass E string, bowed by an experienced string player.
For comparison, the torsional wave is represented by r, the product of the string radiusand the angular velocity.
Sound of the transverse wave (2.7 M) (74k)Sound of the torsional wave (2.7 M) (33k)
Notice that both sounds have the clear pitch associated with a highly periodic
signal and that, in both cases, the pitch is that of the transverse fundamental. This pitch
corresponds to the period (26 ms) that is clearly visible in the time domain graphs. Thepitch frequency (40 Hz) is the strongest harmonic in the transverse wave, and is also of
course the spacing between adjacent harmonics. In the torsional wave, it is also the
spacing of harmonics, but the fundamental at this frequency is very weak. The transverse
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velocity wave sounds rather like a bowed bass string. The similarity is not surprising:
both are bowed strings. (The spectra are very different, of course. Although the force
exerted on the bridge increases with frequency when compared with velocity, this is inpart offset by the lack of filtering by the radiativity of the instrument.) The angular
velocity wave also sounds somewhat like a bowed bass that has been filtered in an odd
way: the rich harmonic content and the initial transients suggest a bowed string. Theformants around 225 and 450 Hz are near the frequencies of the natural resonances of the
torsional wave, in the absence of a bow. These are so strong, however, that one or both of
the harmonics may be heard individually in the torsional sound file.The animation (made by Heidi Hereth) shows idealised Helmholtz motion of a transverse
wave.
A briefintroduction and
summary
The main
function of aviolin or bass
bow is to induce
a sideways or
transversemotion of the
string. Rosin
placed on a bowensures that
static frictionwith the stringmay be much
greater than
kinetic friction.
Consequently, ina cycle of
normal playing, the string at the position of the bow travels with the bow at a nearly
constant, low velocity in one direction (the stick phase), then slides rapidly past the bowin the opposite direction (the slip phase), as shown in the animation.However the bow
acts on the surface of the string, rather than at its centre, and so also must exert a twisting
or torsional force. This torque excites additional torsional or twisting waves that travel upand down the string. These torsional waves exert only a small torque on the bridge and so
produce little sound by themselves. Nevertheless, they can have an important effect on
the overall sound produced.
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The motion of the point of contact between bow and string
depends on both the transverse speed v of the string, and on
the torsional velocity (its speed is v+r, where r is the
radius of the string). During the stick phase, v+r must
equal the bow speed. The component waves of the familiar
transverse modes of the string are in harmonic ratios and so
produce a periodic wave: one that repeats exactly after oneperiod. However, there is no a priori harmonic relationship
between the torsional and transverse waves. Consequently,
the torsional waves may produce non-periodic motion or
jitter at the bow-string contact. Because the ear is very
sensitive to jitter, this can have a considerable effect on the
perceived sound.
The bowed string has been studied for centuries by scientists, including Helmholtz and Raman. It is
thus a little surprising to discover that the relative magnitudes and phases of the torsional and
transverse motion had not been measured. We did this electromechanically by attaching tiny sensing
coils, using a low bass string to minimise perturbation.
The magnitude of the torsional waves was surprising: they may contribute as much as
tens of percent of the speed at the contact point with the bow, as shown in the figure
above. In the first experiments, the strings were bowed by experienced players. Inmusically acceptable bowing regimes, the torsional motion was always phase-locked to
the transverse waves, producing highly periodic motion. The spectrum of the torsional
motion includes the fundamental and harmonics of the transverse wave, with strongformants at the natural frequencies of the torsional standing waves in the whole string.
Volunteers with no experience on bowed string instruments, on the other hand, often
produced non-periodic motion. This suggests that finding (quickly) the subtlecombination of force and speed that controls the non-harmonic torsional waves is a skill
that string players must learn.
More detail is given in the paper: Torsional waves in a bowed string.
Links with background information
Waves in strings, reflections, standing waves and harmonics. Bows and strings, which includes an animation of Helmholtz motion. An introduction to violin acoustics.
Eric Bavuworked on this project as an undergraduate research project. Other
undergraduates who worked on earlier projects on the bowed string, and who thereforecontributed to this project, are Pierre-Yves PlacaisandManfred Yew.
Harmonic singing (or overtone
singing) vs normal singing
Harmonic singing shares
techniques with diphonicsinging, overtone singing,
xoomi singing, sygyt singing,
throat singing, Tuva singing etc.
We explain some of the
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acoustics of this style of singing in terms of the measured acoustical response of the vocal
tract. In this technique, the singer emphasises one high harmonic of the voice to such an
extent that it is heard separately from the low pitched note being sung. Different notes inthe harmonic series may be chosen by changing the frequency of the resonance in the
vocal tract that gives rise to it.
For background information on speech and ordinary singing, see ourIntroduction to theacoustics of the vocal tract. For background about our research and techniques, see this
link. On this page, we begin by looking at how the vocal tract behaves for a whisper,
where the resonances of the tract are most clear, then fornormal singing, then forharmonic singing.
Whisper. In the first figure, a subject whispers the vowel in 'hoard'. We show the
frequency response of the vocal tract (For an explanation of the measurements, follow
this link.) The sound of the whisper itself is masked by the injected signal used tomeasure the vocal tract resonances. The figure shows several peaks, indicated by the
arrows. At these frequencies, the sound produced at the vocal folds is most effectively
transmitted as sound produced in the external air. (Technically, these are peaks in the
acoustic impedanceof the vocal tract. At these resonant frequencies, the tract operatesmost effectively as an impedance transformer between the relatively high acoustic
impedance of the tract and the low impedance of the radiation field at the mouth.)
Normal singing. In the figure below, the subject sings the same vowel at the pitch Bb3(117 Hz). In this graph, you can see the harmonics of the voice, and you can see that the
fourth and sixth harmonics appear stronger in the sound spectrum because they are near
resonances of the tract.
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Over the range shown and for this vowel, this subject's vocal tract has six resonances,which are indicated by the arrows. Note that the subject changes the first two resonances
a little between whispering and singing. The frequencies of these two resonances
determine the vowel in a particular accent. It is not unusual for people to have differentaccents when whispering, speaking and singing. The higher resonances are also
substantially changed, probably because rather different vocal mechanisms are used in
whispering and singing.Harmonic singing. The next graphs show two examples of harmonic singing. In this
technique, one of the vocal tract resonances is made much stronger, while all the others
are weakened. The strong resonance can be made so strong that it selects one of the
harmonics and makes it so much stronger than its neighbours that we can hear it as aseparate note. Hear it is the eighth harmonic that is amplified. Although the fundamental
is only 8 dB lower than the selected harmonic, the fundamental lies in a range in which
our ears are much less sensitive, so it sounds much less loud.
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For some harmonic singers, more complicated effects than those described here may beinvolved. It has been suggested that, for some sygyt singers, the strong resonance in the
vocal tract may drive an oscillation in the false vocal folds. This could produce a stronger
signal at the high pitch. Further, because the false vocal folds would be nonlinearoscillators, they would produce strong components at integral multiples of the high pitch
frequency, ie at n*f0, 2n*f0, 3n*f0 etc. An example of such a spectrum and an
explanation of the false vocal fold mechanism is given by Chen-Gia Tsai at this link.This research is part of a project investigation the acoustics of singing in general. It is
undertaken byNathalie Henrich,John Smith and Joe Wolfe.
Some related pages and explanatory notes
This style of singing was first popularised in the West by David Hykes, whose
page is at this link. He points out that "harmonic singing" refers to a broader range oftechniques than just the emphasis of an overtone. Chen-Gia Tsai's page on"acoustics of overtone singing" Some interesting results about thetuning of the vocal tract by
Sopranos: resonance tuning and vowel changes
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In the higher part of their range, sopranos face some
challenges that don't occur in normal speech, and which
don't affect other singers so severely. The problem is that
the harmonics of their voice become so widely spaced
that, unless they make appropriate adjustments, they are
likely to lose the benefit of the resonances of the vocal
tract.The vocal tract usually has two strong resonances in the
range 0-2 kHz. A bass singing a low G has harmonics at
100 Hz, 200 Hz, 300 Hz etc, all the way to several
thousand Hz. He has so many harmonics in this range
that some of them are bound to fall close to the tract
resonances and so he will benefit from enhanced
transmission as sound outside the mouth. A soprano
singing a high C has harmonics at 1000 Hz, 2000 Hz,
3000 Hz up to a similar upper limit. So for a soprano, the
harmonics could 'miss' the resonances for some note-
vowel combinations, and 'hit' for others.
Missing the resonances would be a problem: they are
needed when singing with loud accompaniment, andhaving a resonance near the frequency of the voice can
also make it easier to sing. The distinguished voice
researcher Johan Sundberg hypothised that sopranos
might learn to tune one of their vocal tract resonances to
the pitch of the note they are singing.
This creates a phonetic problem, because the tract
resonances produce bands of increased power
(calledformants) in the voice, and these carryinformation about vowels, information which is mainly
carried in the range 300 to 2000 Hz. For a more detailed
explanation of this point, see Introduction to vocal
tract acoustics. This page provides sound files to demonstrate the effect, background information and a non-technical introduction to the effect.
There is also a soprano challenge. The researchreported here is a spin-off from technologywe
developed for use in language training and speech
therapy.
Abrief scientific reportof the application to sopranosinging is published in the journal Nature and a moredetailed reportis in JASA.
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New results: Wagner makes it easier for sopranos
Wagner is well-known, even notorious, for writing operas that can challenge bothperformers and listeners. Both groups might be surprised to learn that Wagner was
helping both the performers and the listeners by taking the acoustics of the soprano voice
at high pitch into account when he set his text to music. A recent paper from this lab
suggests that this is indeed the case:Smith,. J. and Wolfe, J. (2009) "Vowel-pitch matching in Wagner's operas:
implications for intelligibility and ease of singing", JASA Express Letters, J. Acoust.
Soc. America, 125, EL196. A non-technical account follows.Each vowel in European languages is associated with a set of resonance frequencies of
the vocal tract. For the soprano voice at high pitch, both the intelligibility to listeners and
the ease of production by singers could be improved if the pitch of the note written for avowel corresponded with its usual range of resonance frequencies.
We tested this hypothesis by investigating whether Wagner used certain vowels more
often for the high notes than the low notes, and vice versa. A study of the two great
Wagnerian soprano rles, Brnnhilde and Isolde, indeed found the vowels that required
an open mouth were used more often for the very high notes. Similar studies on someoperas by Mozart, Rossini and Richard Strauss showed no such effect.
We are unaware of any written evidence about Wagner's intentions nor of whether he wasadvised on this issue by sopranos, with whom he sometimes had close relations.
Of course we are not suggesting that Wagner was a better opera composer than others. He
was writing a different type of opera with a much larger orchestra, and making his verydemanding vocal parts somewhat easier.
Summary: It appears that Wagner, either consciously or unconsciously, took the acoustics
of the soprano voice at high pitch into account when setting text he had written to music.This is consistent with the increased importance of textual information in his operas, the
increasing size of his orchestras, and the more complex vocal parts.
More news: resonance tuning in the coloratura/ whistle voice rangeThe resonance tuningdescribed above tunes the first vocal tract resonance to the frequency of the note sung(R1:f0 tuning). It's fine up to about 1 kHz or C6 high C for sopranos. After that, it
becomes difficult to open your mouth any wider. Some sopranos manage for a further
couple of notes, but for many sopranos, the limit of R1:f0 tuning is the limit of their vocalrange.
Coloratura sopranos or the jazz and pop singers who practise the whistle registers,
however, have another technique: starting at around C6, they begin to tune the secondvocal tract resonance to the frequency of the note sung: R2:f0 tuning. Further, they appear
to use a different mechanism and thus to show another transition. This and some related
effects are described in these papers:
Garnier, M., Henrich, N., Smith, J. and Wolfe, J. (2010) "Vocal tract adjustments in thehigh soprano range" J. Acoust. Soc. America. 127, 3771-3780.
Garnier, M., Henrich, N., Crevier-Buchman, L., Vincent, C., Smith, J. and Wolfe, J.
(2012) "Glottal behavior in the high soprano range and the transition to the whistleregisters" J. Acoust. Soc. America. 131, 951-962.
Sound files
These sound files do not form part of the study; they are simply illustrative of
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the phonetic effect of the widely spaced harmonics and the resonance tuning. For these
recordings, an experienced soprano, who had no knowledge of the purpose of the study,
was asked to sing an ascending scale over two octaves, from Bb3 to Bb5. She was askedto sing the scale five times, each time using a different vowel sound. The vowels
(phonetic symbols in parentheses) are those in the English words* "hard" (//), "hoard"
(//), "who'd" (//), "heard" (//) and "heed" (//), but sung with an initial consonant "L". (Themusic and the phonemes to be sung were presented in writing.)Scale sung on "La":
Scale sung on "Lore":
Scale sung on "Loo":Scale sung on "Ler":
Scale sung on "Lee":
Let's now listen to the five vowel sounds at low pitch, then at high pitch. (These
files are assembled from the first and last notes on each of the scalesabove.)"La, Lore, Loo, Ler, Lee" at low pitch:
"La, Lore, Loo, Ler, Lee" at high pitch:
Notice that the vowel sounds are much more similar at high pitch+. Do
you think that you can tell them apart? If so, listen to this file, in whichthe order has been changed and see how sure you are.Five vowels at high pitch, order
changed:The difficulty the listener has in differentiating the vowels at high pitch is due to two
effects. One is that high pitch puts the harmonics further apart. (SeeWhat is a sound
spectrum?for details.) The information about which vowel is spoken or sung is (looselyspeaking) conveyed by the relative amplitudes of the harmonics of the voice that fall in
the range from 200 to 2000 Hz. If a soprano sings a high C, there is only one harmonic in
this range, so little vowel information is carried. Another effect is the tuning of the first
vocal tract resonance by sopranos, including this singer. We have studied these effects,which are described very briefly in apaperin Nature, and which are described in much
more detail in a longer paper.We thank soprano Kristen Butchatsky of the School of
Music and Music Education at the University of New South Wales for singing thesamples recorded here and for her help in other aspects of our research.
* English vowels are often presented and studied in the context h[vowel]d, because a set
of such words minimises the number of nonsense syllables. In fact, as soon as we canconvince enough people to call a Head Up Display a 'hud', the list will be complete.
+ Even the consonant is difficult to discern. Much of the information about consonants is
conveyed by the way they modify the vowel that follows (or precedes) them.
Sopranos tune resonances of their vocal tract when they sing in the high range
A non-technical version of reseach published as Joliveau, E., Smith, J. and Wolfe, J. "The
tuning of vocal tract resonances by sopranos", Nature,427, 116.In brief. In the top half oftheir range, but not in the lower half, classically trained sopranos adjust one of theresonant frequencies of the vocal tract to match the pitch of the note they are singing.
This gives greater loudness for given effort, and may have musical advantages, but it
contributes to the difficulty of understanding the words they are singing.
Background.
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In singing or speech, we can identify two separate effects. First, the vocal folds vibrate
and that produces a sound. The frequency of that sound--the number of vibrations per
second--determines the pitch, ie whether a sound is high or low.Second, the sound fromthe vocal folds passes through the vocal tract. It is modified by the shape of the tract so
that we can make different speech sounds: open your mouth wide and you get 'aah', close
it almost completely and get 'ooo' etc. This process is very familiar to us, but it is also abit subtle. Here's how it works.
The vibration from the vocal folds is a complex sound: we say it has lots of harmonics or
that it is made up of a range of different frequencies. The vocal tract resonates at severaldifferent frequencies and these resonances amplify some of the frequencies present in the
voice. For instance, when this author's mouth and tongue are in a neutral position (when I
say 'er'), it 'amplifies' frequencies of about 450 and 1400 vibrations per second
(approximately the notes A above middle C and the F above the treble clef--both notesbeyond my singing range). Different tract positions amplify different frequency
components of the voice and that allows us to identify different speech sounds. This is
explained in more detail, with diagrams and sound files, in our page Physics in Speech.
These processes are independent. One can keep the vocal tract constant and change thepitch (eg humming, vocalise or singing 'la la la la la'), or one can keep the pitch the same
and vary the vocal tract, which is what the Daleks on Dr Who do ('Ex-ter-min-ate').Normally we do both independently, so we can sing different words on different notes
(normal singing), and we can use different inflexions in the same words. (eg. 'You're not
going' vs 'You're not going?')However, there is a problem for sopranos. In the high range of women's voices*, the pitch
frequency of the notes enters the range of the lowest vocal tract resonance. If the singers
did nothing about this problem, then whenever the pitch of the note coincided with a
resonance in the tract, that note would be much louder than the others. So you would getuneven loudness, and also uneven voice quality. Some time ago, the Swedish acoustician
Johan Sundberg suggested that sopranos actually tune the resonance of their vocal tract to
the note that they are singing: the original evidence for this was that they tend to open themouth more as they sing successively higher notes. However, this could not be confirmed
directly because there was a technical difficulty in measuring the acoustics of the tract
while it was being used for singing.* What about the voices of young children, which are higher in pitch than women's
voices? Children have smaller heads and shorter vocal tracts, so one would expect that
the resonances of their vocal tracts to occur at higher frequencies, so the overlap of pitch
and resonance would occur at higher pitch.
The project on sopranos and resonance tuning
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In the Acoustics Group, we have developed
acoustic techniques for studying the tract during
speech (see Voice Acoustics). Our mainmotivation was to provide technologies for
speech pathology and for language training, but
the techniques are also applicable to singing. Sowe invited in a group of classically trained
sopranos. For these measurements, we produce a
carefully synthesised sound just outside thesinger's or speaker's mouth. A microphone records
not only their voice, but also the way in which
their vocal tract interacts with the synthesised
sound. From the latter, we can tell a lot about theacoustics of the tract. This project became a
research project for Elodie Joliveau, who is both a
physics student and a soprano.What we find with
sopranos is this: In the low range of the voice (below
about A4, but it depends on the vowel), they do just what
we all do in speech and singing: the pitch and the vocal
tract resonances are nearly independent. In the high
range, however, they tune the lowest resonance of the
vocal tract rather precisely to equal the pitch they are
singing. They perform the resonance tuning by gradually
lowering the jaw as they ascend, and/or by 'smiling' more
as they ascend in pitch. What surprised us was the
consistency and precision of the tuning for all vowels.
This resonance tuning gives them uniform loudness and
vocal quality, but it also means that vowel sounds
become very similar--we include above some recordings
that demonstrate this clearly. So sopranos sacrifice someintelligibility in the interests of musical quality.
However, the amount of intelligibility sacrificed is not
great. In the high range, it is very difficult to understand
vowel sounds anyway: because of the high pitch, there is
simply not much frequency information available to the
ear.
This has been remarked on by composers such as
Berlioz, whose book about orchestration warns opera
composers about the effect. Many composers seem to
heed the warning: the high parts in soprano solos
sometimes do not have words, or have a single word
slurred over several notes, or sometimes repeat words
that are heard in other ranges. or sometimes the wordsare simply not important. This is by no means a cricitism
of sopranos: it is just an inherent physical constraint on
the instrument. One doesn't ask a trombonist to play
pizzicato, one shouldn't ask a soprano to make vowel
distinctions in the altissimo. Nevertheless, the effect is
possibly one* of the contributing reasons why opera
houses use surtitles even when the words are in the
language of the audience.
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So, if you are a soprano, download the paper and keep it for defence against conductors who thinkyou should be able to distinguish 'bead' and 'bed' on high C. Or else quote Berlioz (Berlioz, H. Grand
Trait d'Instrumentation et d'Orchestration Modernes (1844; transl. Clarke, M. C., Novello, London,
1882).)
* Other reasons? There are several, starting with the design of the hall. Acoustic
engineers must compromise between a long reverberation time, which makes the soundlouder, and a short one, which makes it clearer. If the performance space was designed for
orchestral music (relatively long reverberation), it will be hard to understand singing.
Then there is the audience, who are not always as silent as may be hoped. (This is often aproblem in the Sydney Opera House: a substantial fraction of each audience is there just
because it is a famous building, not because they like opera.) The orchestra may be too
loud, perhaps because the opera was written centuries ago when string and wind
instruments were less loud than the modern equivalents. Or perhaps because theorchestration is excessive. It must be admitted that not all singers take sufficient care in
pronunciation. And finally, it is especially difficult for the chorus: when plosives (p, d, t,
k etc) are not pronounced in synchrony, it is difficult to discern them.
Soprano challenge
If you are a soprano and you think would like to test whether our observations reflectphysical limitations on all sopranos, or just on some of them, perhaps you would like to
try repeating the exercise recorded in the sound files above. All you need is a microphoneand a computer or tape recorder. (It would help if you had some editing facility such as
the Cool Edit software, but this is not necessary.) First, sing the scale below, senzavibrato, in your professional singing voice, with projection. Depending on your
comfortable range, you might want to make it C major, B major or Bb major.
Then do the same for "Lore", "Loo", "Ler" and "Lee". Then listen to the first notes ineach in each scale. (If you have editing tools, take the first note (the minim or half-note)
of each sample and put them together to make the low pitch file.) Then do the same for
the last note of each scale. Then get a friend to mix up the order of the notes in the finalsample and listen to it. If you can clearly discern them, then we should really like to hear
from you: that would be the basis of a very interesting study!
Reports of the application to soprano singing are published in: Joliveau, E., Smith, J. and Wolfe, J. (2004) "Tuning of vocal tract resonances by
sopranos", Nature, 427, 116. Joliveau, E., Smith, J. and Wolfe, J. (2004) Vocal tract resonances in singing: the
soprano voice, J. Acoust. Soc. America, 116, 2434-39. Garnier, M., Henrich, N., Smith, J. and Wolfe, J. (2010) "Vocal tract adjustments
in the high soprano range" J. Acoust. Soc. America. 127, 3771-3780. Garnier, M., Henrich, N., Crevier-Buchman, L., Vincent, C., Smith, J. and Wolfe,
http://www.phys.unsw.edu.au/jw/reprints/SopranoNat.pdfhttp://www.phys.unsw.edu.au/jw/reprints/SopranoNat.pdfhttp://www.phys.unsw.edu.au/jw/reprints/SopranoNat.pdfhttp://www.phys.unsw.edu.au/jw/reprints/Joliveauetal.pdfhttp://www.phys.unsw.edu.au/jw/reprints/Joliveauetal.pdfhttp://www.phys.unsw.edu.au/jw/reprints/highsoprano.pdfhttp://www.phys.unsw.edu.au/jw/reprints/highsoprano.pdfhttp://link.aip.org/link/?JAS/127/3771http://www.phys.unsw.edu.au/jw/reprints/SopranoNat.pdfhttp://www.phys.unsw.edu.au/jw/reprints/SopranoNat.pdfhttp://www.phys.unsw.edu.au/jw/reprints/SopranoNat.pdfhttp://www.phys.unsw.edu.au/jw/reprints/Joliveauetal.pdfhttp://www.phys.unsw.edu.au/jw/reprints/Joliveauetal.pdfhttp://www.phys.unsw.edu.au/jw/reprints/highsoprano.pdfhttp://www.phys.unsw.edu.au/jw/reprints/highsoprano.pdfhttp://link.aip.org/link/?JAS/127/3771 -
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J. (2012) "Glottal behavior in the high soprano range and the transition to the whistle
registers" J. Acoust. Soc. America. 131, 951-962.
http://www.phys.unsw.edu.au/jw/reprints/Garnier-JASA-2012.pdfhttp://www.phys.unsw.edu.au/jw/reprints/Garnier-JASA-2012.pdfhttp://link.aip.org/link/?JAS/129/1024http://www.phys.unsw.edu.au/jw/reprints/Garnier-JASA-2012.pdfhttp://www.phys.unsw.edu.au/jw/reprints/Garnier-JASA-2012.pdfhttp://link.aip.org/link/?JAS/129/1024
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