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Letters

Journal of the Physical Society of Japan 90, 083602 (2021)
https://doi.org/10.7566/JPSJ.90.083602

Interaction of Acoustic and Quasi-Elastic Modes in Liquid Wateron Nanometer Length Scales
Daisuke Ishikawa1,2 and Alfred Q.R. Baron1,2+1Materials Dynamics Laboratory, RIKEN SPring-8 Center, Sayo, Hyogo 679-5148, Japan
Precision Spectroscopy Division, CSRR, SPring-8=JASRI, Sayo, Hyogo 679-5198, Japan

2(Received May 31, 2021; accepted June 21, 2021; published online July 21, 2021)

We investigate the presence of an acoustic-quasi-elastic interaction contribution in the IXS spectra of liquid water at
301 K using inelastic x-ray scattering with sub-meV energy resolution at momentum transfers 0.77 ≦ Q ≦ 4.20 nm−1.
The contribution appears due to the overlap the acoustic mode with the tail the quasi-elastic mode and is fully consistent
with hydrodynamic theory. Incorporating this interaction allows us to describe the dynamic structure factor, SðQ; !Þ,without introducing an extra mode, and may help explain earlier contradictory interpretations. The sound velocity, andrelative intensity of the quasi-elastic and acoustic mode, plateau for Q > 2 nm−1 at values consistent with a viscoelastic
generalization of the Landau–Placzek relation.

The dynamics of liquids on nm length scales is afascinating subject for which consensus is still evolving. At
long wavelengths, equilibrium liquids are often described bya linearized Navier–Stokes equation (see, e.g., Ref. 1) giving
a Rayleigh–Brillouin triplet for the dynamic structure factor,
SðQ; !Þ, with a quasi-elastic (Rayleigh) mode at zero energy
transfer bracketed between the Stokes and anti-Stokes peaksof the acoustic (Brillouin) mode. As the probed length scale
shortens into the regime accessible by inelastic neutron or
x-ray scattering (INS or IXS), approximately Q > 0:5 nm−1,
one often still observes three peaks in SðQ; !Þ, but liquid
spectra can blur out, with the acoustic mode width increasing
so it sometimes appears as a rather broad feature on the tail ofa quasi-elastic peak (Refs. 1–3 and references therein).
Further, there is a rich panoply of eﬀects (diﬀerent types ofrelaxation, fast sound=positive dispersion, onset of observable transverse dynamics) that potentially impact the behavior at ∼nm length scales. Thus, the meso-scale region of
momentum space, where the crossover from continuum toatomistic dynamics occurs in liquids, is both interesting, andchallenging.
Interpretations of experimental work on liquid water
demonstrate some of these complexities. In particular,
measurements of SðQ; !Þ at ∼nm−1 momentum transfers
have been carried out by INS and IXS (including4–14)). All
the measurements show a large quasi-elastic central peakbracketed by the Stokes and anti-Stokes peaks of a dispersing
acoustic mode — a triplet as described above — and the data
appears broadly similar across diﬀerent work. Common
features of that work include (1) a quasi-elastic peak thatis much bigger than the acoustic mode peaks and (2) the
presence of what has been called “fast sound”, or “positive
dispersion”, or “high-frequency sound”, as the acoustic mode
disperses with a velocity higher than the hydrodynamic (longwavelength) sound velocity. The presence of fast sound is
expected from work on other materials (e.g., Ref. 15 is anearly example) and also viscoelastic theory or generalizedhydrodynamics (e.g., Refs. 1–3, and 16) but the magnitude inwater is notably large, with the fast sound velocity roughly
double the hydrodynamic value. However, beyond thesecommon features, interpretation of the spectra of water is
far from consistent. In particular, as data quality improved,
analysis of the data using a simple model (the sum of aLorentzian and a damped harmonic oscillator, the “L+DHO”
model) was augmented by a second DHO mode (an“L+2DHO” model) for SðQ; !Þ for Q  2{3 nm−1. This
extra mode has been suggested to be related to transverse
dynamics,7) or to an anti-crossing behavior10,12) — while both
pictures are discussed in Ref. 17. Meanwhile, analysis ofspectra using a memory function approach,9,13) found noevidence of an additional mode for 2  Q  7 nm−1. Thus,
interpretations of the work on liquid water have inconsistencies that have not been reconciled, even over the limited
range of momentum transfers (0:77  Q  4:20 nm−1) that
we focus on in the present work.
In fact, the choice of the model used to ﬁt measurements ofSðQ; !Þ is important for interpreting most liquid spectra, notonly water: acoustic lines for liquids are usually intrinsicallybroad, and often overlap the quasi-elastic contribution, so thatthe model choice inﬂuences the values determined forparameters, and thus the interpretation of the spectra. Whilethe case of water mentioned above (with at least 3 diﬀerent
interpretations, two with extra modes, 1 without) is extreme,
more generally models are used to get information fromspectra, to determine mode dispersion, and to suggest thepresence, or not, of additional modes (e.g., Refs. 18–20).
Here we demonstrate that interaction between the quasielastic mode and the acoustic mode is an importantcomponent of the IXS spectra of liquid water for 0:77 Q  4:20 nm−1, and, we suggest, provides a good basis for acommon interpretation of the data in this region momentum
transfer. Taking this interaction (which appears similar to an
extra mode at an energy below the acoustic mode, but with anintensity and position determined by the quasi-elastic andacoustic components) into account, we ﬁnd the behavior ofwater at 301 K in this range of momentum transfer can bewell described without introducing additional modes. We
ﬁnd the cross-over from hydrodynamic to fast sound occurs
at Q < 2 nm−1 and that the ratio of the intensity of theacoustic mode and the quasi-elastic relaxation is constant for
2 < Q  4:2 nm−1, and consistent with a viscoelastic modiﬁcation to the Landau–Placzek relation.
The L+DHO or the L+2DHO models used previously may
be interpreted in terms of independent modes: the quasielastic Lorentzian term is typical of relaxation while the DHO
contribution(s) describes lifetime broadened (decaying)083602-1

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Letters

J. Phys. Soc. Jpn. 90, 083602 (2021)

Q = 1.18 nm-1

D. Ishikawa and A. Q.R. Baron

Q = 2.54 nm-1

30-3

Q = 4.20 nm-1

3
0
-3

3
0
-3

Interacting

Interacting22

χ /ν=1.09

-10

-5

0

5

10

-10

-5

0

5

10

3
0
-3

-20

-10

0

10

20

Energy Transfer [meV]
3
0
-3

L+DHO

L+DHO

L+DHO

2

χ /ν=1.56

2χ /ν=2.21

χ /ν=1.40

0χ /ν=1.15

Energy Transfer [meV]

3
0
-3

-5

2

χ /ν=0.92

Energy Transfer [meV]-10Interacting

5

Energy Transfer [meV]

10

-10

-5

0

5

10

Energy Transfer [meV]

2

-20

-10

0

10

20

Energy Transfer [meV]Fig. 1. (Color online) Water spectra at Q ¼ 1:18, 2.54, and 4.20 nm−1 as ﬁt either with a model including interaction (top) or without it (bottom). Insets
show the full scale. The lines show the individual components after convolution with the resolution, with the quasi-elastic contribution a dashed line, the DHO
solid, and the interaction term (top only) the thick line. The upper panels show the residuals ðydata  yﬁt Þ=. The inclusion of the interaction term signiﬁcantly
improves the ﬁt quality without increasing the number of free parameters. The residuals for the L+DHO model also suggest systematic errors, showing
correlated oscillations in their deviation from 0, which are removed by using the interacting model.

mode(s).21) However, naively, there could be interaction
between the quasi-elastic and acoustic modes, since theacoustic mode overlaps the tail of the quasi-elastic contribution. One can also make an analogy to Raman scattering
where a phonon resonance overlapping a continuum (i.e.,
non-resonant, energy transfer independent) from electronic
scattering can lead to an asymmetric Fano line shape22) fromthe interference of the resonant scattering, which changes
phase through the transition, and the continuum. In nonresonant IXS, as we use here, the electronic scattering fromvalence electrons is too weak to observe such interference,
however, the tail of the quasi-elastic scattering provides asmooth background that is non-negligible compared to thephonon intensity: one then might expect a similar asymmetriccontribution may appear on passing through the acoustic
phonon resonance.
In fact, if one returns to hydrodynamic theory [e.g., startingfrom Eq. (5.3.15) of Ref. 1] the dynamic structure factor may
be written as

SðQÞ
z0
2z1 2
SðQ; !Þ ¼
þ
I
I0 2
1

! þ z20
ð!2  2 Þ2 þ 4z21 !2
2  !2
þ I0 z0 2;
ð1Þ
ð!  2 Þ2 þ 4z21 !2
where ω is the energy transfer and SðQÞ is the static structure
factor [the integral of SðQ; !Þ over energy transfer]. I0 ; z0
are the intensity and width (half width at half maximum,
HWHM) of the central Lorentzian mode and I1 ; z1 ;  are theintensity, width and energy of the sound mode, respectively[normalization requires I0 þ I1 ¼ 1, and relations between
the parameters of Eqs. (1) and (5.3.15) of Ref. 1 may befound in the Supplemental Materials23)]. The ﬁrst two terms
constitute the “L+DHO” model. The last term is theinteraction of the acoustic mode with the quasi-elastic mode:
its amplitude is proportional to the area and width of thequasi-elastic (Lorentzian) component, while its shape isdetermined by the position and width of the acoustic (DHO)
mode. In hydrodynamics this contribution is eﬀectively
included in the “asymmetry parameter”, sometimes denoted“b” (e.g., Ref. 24). It integrates to 0, changing sign at the
acoustic mode energy, but, to a ﬁrst approximation (seeFig. 1) may appear similar to a broad extra mode at an energy
below that of the acoustic mode. We note Eq. (1) can also be
converted to the same form used to describe the rotationtranslation coupling in crystals25) as has been applied toferroelectric materials,26) so this type of interaction term isgeneral.
Notably, Eq. (1) is derived in a limit where longitudinal
and transverse dynamics are entirely separated, so, a-priori,
the interaction is not related to transverse dynamics. Further,
the amplitude and shape are entirely determined by thequantities appearing in the Lorentzian and DHO terms, so,when considered as a line-shape for ﬁtting, the interacting
form, Eq. (1), has the same number of free parameters as the
L+DHO model. Thus, the fact that this expression reproduces water spectra well, as shown below, where the083602-2

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Letters

Vel. [km/s]

0

1

2

3

4

(a) Ω/Q

3
2
1

Width [meV]

L+DHO model ﬁts poorly, is excellent indication (Ockham’srazor) that the interaction is real.
We investigated the dynamic structure factor of water at
301 K at momentum transfers, 0:77  Q  4:20 nm−1, using
inelastic x-ray scattering at BL43LXU27) of the RIKEN
SPring-8 center. Runs were made with extremely goodenergy resolution (down to 0.84 meV, FWHM) using theSi(13 13 13) reﬂection at 25.7 keV28) for 0:77  Q < 3:2
nm−1 and using a higher ﬂux setup with the Si(11 11 11)
reﬂection at 21.7 keV for 2:6  Q  4:2 nm−1 (resolution 
1.33 meV). Specially designed analyzer masks29) allowed us
to improve count-rates while retaining good momentum
resolution. The sample was ultrapure water placed in a cellwith two 0.05 mm thick polished single crystal sapphire
windows. Additional experimental details can be found in the
Supplemental Materials.23)
The impact of the interaction term can be seen in Fig. 1,
where, without increasing the number of ﬁt parameters, there
is strong improvement in ﬁt quality, and substantial reductionof correlations in the residuals, relative to the L+DHO model
without interaction (see Supplemental Materials for additional information about the ﬁtting23)). Improvement is visible
across the momentum range investigated, with, always, theinteracting model providing a better ﬁt (see the Supplemental
Materials23)). The improvement in reduced chi-squared byincluding the interaction is signiﬁcant: e.g., at 2.54 nm−1, theprobability that the chi-squared per degree of freedom  2 = 
1:56 for the L+DHO model is p < 105 for the  ¼ 128
degrees in the spectrum (  2 = and p are given for several
models for all measured spectra in the Supplemental
Materials23)). Of course, one can suggest that the data should
be interpreted by adding one or more additional modes to themodel: one should pass to an L+2DHO model, or something
more complicated, e.g., L+nDHO. However, while this ispossible in principle, physical considerations also play a role:
not only are the ﬁts better in the interacting model, but thatmodel is directly suggested by theory. Also, as shown in the
Supplemental Materials23) even the L+2DHO, with its full
additional line, gives ﬁts that are usually worse than theinteracting model, and in some cases signiﬁcantly worse.
Better ﬁts, using a model with fewer parameters and atheoretical basis, is strong evidence of validity.
The Q-dependence of the ﬁt parameters are shown in
Fig. 2. While a detailed analysis of water is beyond the scopeof the present work, we discuss several features directly
visible in the plots. The sound speed, =Q, and the intensity
ratio, I0 =I1 , and, to some extent, the quasi-elastic linewidth,
z0 , saturate at constant values for Q  2 nm−1. The
magnitude of the fast or high-frequency sound, 2.91 km=sfrom Fig. 2(a), is about double that of the hydrodynamic
value, 1.50 km=s at this temperature. This is reasonably
consistent with, or slightly lower than, other work — thoughother work uses diﬀerent ﬁtting models. We note that theplateau in the sound speed, intensity ratio and quasi-elastic
width suggest that if one considers the water response in thecontext of a viscoelastic transition starting at small Q, thenthat transition has largely completed by Q  2 nm−1. Meanwhile, an approximately linear Q-dependence of the acoustic
mode width, z1 , is possible within viscoelastic models over alimited range of momentum transfers, though it represents aninterplay of diﬀerent quantities.16)

D. Ishikawa and A. Q.R. Baron

Width [meV]J. Phys. Soc. Jpn. 90, 083602 (2021)

0

0.4
0
3

0

(b) z0
0.5

1

1.5

2

2.5

3

3.5

4

0.5

1

1.5

2

2.5

3

3.5

4

1

1.5

2

2.5

3

3.5

4

(c) z1

2
1
0
4

0

0.5

(d) I0/I1

2
0

0

1

2

3

4

Momentum Transfer [nm -1]
Fig. 2. (Color online) Q-dependence of parameters from ﬁts to theinteracting model — see text for discussion. The lines are constant ﬁts tothe indicated regions, and are in good agreement with a viscoelastic
modiﬁcation of the Landau–Placzek relation.

We brieﬂy focus on the intensity ratio, I0 =I1 , as, viewed
from a long-wavelength limit, this is perhaps the mostsurprising characteristic of the water spectra. In particular,
light scattering measurements at Q  0:02 nm−1 show a verysmall quasi-elastic intensity, typically a few % of the acoustic
mode intensity.30,31) This is in good agreement with theLandau–Placzek relation, I0 =I1 ¼   1, where  ¼ cp =c isthe speciﬁc heat ratio:  ¼ 1:03 for water at room temperature
so one expects, and observes, I0 =I1  0:03.30,31) In contrast,at Q  2 nm we observe roughly a 100-fold increase to
I0 =I1  2:67 [the line in Fig. 2(d)]. However, this change isconsistent with a viscoelastic generalization of the Landau–
Placzek relation — something, that, to the best of ourknowledge has not previously been mentioned in the context
of spectra measured in the present range of momentum
transfer. In particular, when one is above the crossover to fastsound, a generalization of the Landau–Placzek relation is16)
 2
I0
c1

 1:
ð2Þ
I1
c0
Taking   1:03 for water at room temperature nearatmospheric pressure, c0 ¼ 1:50 km=s32) and c1 ¼ 2:91ð2Þ
km=s directly from Fig. 2(a), Eq. (2) predicts I0 =I1 
2:88ð4Þ which is in acceptable agreement with the 2.67(5)of the ﬁt in Fig. 2(d). This suggests that the relaxation
represented by the quasi-elastic mode in the spectra may be
predominantly that responsible for the transition to the higher
sound speed.
We now revisit some of the previous work measuringSðQ; !Þ for water in this range of momentum transfer. Very
early work using both neutrons and x-rays analyzed the datain terms of a L+DHO model (or, sometimes, 3 Lorentzians),
however, more recent work, with better quality data
introduced a second DHO mode, passing to an L+2DHO
model, where the second mode was attributed variously to the083602-3

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J. Phys. Soc. Jpn. 90, 083602 (2021)

Letters

onset of observable transverse dynamics or to the presence ofan anti-crossing.7,10,12,17) Meanwhile, more recently, someworkers have taken the interpretation in terms of a transverse
mode as being accepted (e.g., Ref. 14). We suggest that, atleast over the range of momentum transfers investigated here,
the extra mode may have been an artifact of using a noninteracting L+DHO model as a starting point for thedescription. In fact, our observed lack of any extra mode in
the investigated Q range is consistent with a memory function
approach to the analysis of water spectra where no extra
mode was observed for 2  Q  7 nm−1.9,13) This agreement
is not surprising given that the memory function eﬀectively
used in that work has been shown33) to be equivalent to thehydrodynamic line-shape. However, as has been our
experience, and noted by others, analysis using memory
functions is “usually aﬀected by strong correlations amongﬁtting parameters”34) so the direct frequency domain
approach used here, and its interpretation, may oﬀer
advantages in practical treatment of SðQ; !Þ. We also note,that above 3.9 nm−1 we do observe a slight increase in thequasi-elastic linewidth which may hint at a need to consider
an additional component at higher Q, and that component
might be related to transverse dynamics.
In sum, we have shown that including interaction between
the quasi-elastic mode and the acoustic mode in water
spectra, SðQ; !Þ, at momentum transfers of 0.77 to 4.20 nm−1
gives good ﬁts to the measured data without introducing
extra modes, simplifying the interpretation of the data. Theinteraction is fully consistent with hydrodynamic theory
from a linearized Navier–Stokes equation.1) We suggest thatEq. (1) is then a good, accurate and straightforward-tointerpret, form for spectra at low momentum transfers. This
will be useful for ﬁtting and understanding liquid spectra, andresults of molecular dynamics simulations, and as a starting
point for investigating more complex eﬀects. In the present
work we also show that the main cross-over from hydrodynamic long-wavelength behavior to the high frequency
behavior in ambient water occurs at small momentum
transfers, Q < 2 nm−1, and that a viscoelastic modiﬁcationof the Landau–Placzek relation accounting for fast soundreasonably gives the intensity ratio of the quasi-elastic andacoustic parts of the response, suggesting the quasi-elastic
mode observed in IXS in this range may be predominantly
due to the relaxation causing fast sound.
Acknowledgements IXS measurements were made at BL43LXU of the
RIKEN SPring-8 Center. A.B. is grateful to Hajime Tanaka for a comment at aworkshop reminding him of Mountain’s work and to Sergey Vakhrushev forrecalling the paper by Stock et al.
Author contributions Experimental work was done by D.I. and A.B. The
approach to data analysis via Eq. (1) was suggested by A.B. and the data analysis
and ﬁtting was carried out by A.B. with input from D.I. The paper was written by
A.B. with input from D.I.

D. Ishikawa and A. Q.R. Baron+

baron@spring8.or.jp
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083602-4

Societyof Japan©2021 The Physical©2021
The Author(s)Supplemental Materials forInteraction of Acoustic and Quasi-Elastic Modes in
Liquid Water on Nanometer Length Scales
Published as a Letter in the Journal of the Physical Society of Japan (2021)

Daisuke Ishikawa and Alfred Q.R. Baron*
Materials Dynamics Laboratory, RIKEN SPring-8 Center, 1-1-1 Kouto, Sayo, Hyogo 679-5148 JAPAN
Precision Spectroscopy Division, CSRR, SPring-8/JASRI, 1-1-1 Kouto, Sayo, Hyogo 679-5198 JAPAN

Section S1: Experimental Conditions and Data Treatment
For IXS measurements the water cell was mounted in a vacuum chamber with a cleanup
pinhole (0.3 mm diameter) at 25.7 keV (Si (13 13 13)) and a cleanup slit (0.3 x 2mm2) at 21.7 keV(Si (11 11 11)), placed just upstream of the cell (in vacuum) to reduce backgrounds. The samples
used were 5 mm thick at 25.7 keV, and 2 mm at 21.7 keV. With the pinhole in place, and a Soller
slit on the outgoing beam path 1), empty cell backgrounds were always much smaller than the signal,
and in fact were negligible for higher Q. The temperature at the sample was not actively controlled
but the hutch temperature, as monitored near the sample position, was measured to be an average of
27.4 C with a diurnal variation of < 0.5C from the average. Momentum transfers are expected to beaccurate to better than 0.02 nm-1 and the momentum resolution was flat topped with a width <20% of
the momentum transfer, and typically closer to 10%.
In order to compare our measured result to the various models of S ( Q,ω ) we took theclassical model function for S ( Q,ω ) given by eqn. 1, scaled it by a detailed balance factor()( DB (ω ) = x 1− e− x , x = !ω k BT ) and convolved the result with the energy resolution for the
relevant analyzer/mask combination. That resolution was determined in the conditions used in theexperiment by measuring scattering from plexiglass near its structure factor maximum and thenremoving the small phonon contribution to the plexiglass as described in 2). As the masks were
tailored to provide good resolution, momentum acceptances were near top-hat distributions (see 1)).
For completeness, fits were done both with and without averaging S ( Q,ω ) over the relevant rangeof momentum transfers, but this had small impact on the results – and the data presented includes theaveraging. The parameters of the model function were optimized by minimizing chi-squared with*baron@spring8.or.jp

1respect to the (background subtracted, when necessary) data using code based on the MINUIT
package 3) first using the MIGRAD routine and then MINOS to find the final minimum. Throughout
this paper, error bars represent 1-sigma deviations on counting statistics propagated, when needed,according to standard methods.
Section S2: Goodness of Fit
Figure S1 shows the goodness of fit parameter, χ 2 ν , where ν is the number of degrees of
freedom, for the best fits for several models. The interacting model uniformly gives a significantly
improved value of this goodness of fit parameter compared to an L+DHO model, and, in most cases
gives a comparable or better fit than even the L+2DHO model, with sometimes, the L+2DHO being
significantly worse. This can be emphasized by plotting the probability, p, of have χ 2 ν larger thanthe observed value (see, e.g., discussion in 4)) as may be calculated using an incomplete gammafunction 5): the interacting model usually gives fits with p>10%, while the L+DHO model often has
p<<1% and even occasionally p<10-4. The L+2DHO, which doubles the number of phonon lines
available, so is a major increase in fitting freedom compared to the interacting or L+DHO models,does better than the L+DHO model, but usually not as well as the single-phonon line interacting
model (eqn. 1), and, occasionally, the L+2HO is in fact significantly worse than the interacting model.

Figure S1. Goodness of fit for several models. The top plot shows the chi-squared, , per degree of freedom,, while the lower plot shows the probability of getting that value or larger for a data set assuming the measured
data points have normally distributed random errors. See discussion in text. Note that some p values for the
L+DHO fits are below 10-7 and do not appear in the lower plot.

2Section S3: Acoustic Mode Energy: Ω , ω s = Ω2 − z12 , ω pk
There have been different choices made as to what parameter is used to discuss the acoustic modeenergy in liquids, with some work choosing Ω (including much of the referenced work on water),other work focusing on ω s = Ω2 − z12 , and still other work choosing the peak, ω pk , of thelongitudinal current-current correlation function, J ( Q,ω ) = ω 2 S ( Q,ω ) Q 2 . The last is convenient
as it can be determined directly from the shape of S ( Q,ω ) , and, in principle, is independent of the
choice of the model for S ( Q,ω ) . We prefer Ω because (1) in the limit where the quasi-elastic mode
is weak or narrow ( I0 ≪ 1 or z0 ≪ Ω ) then ω pk ≈ Ω and (2) if the quasi-elastic mode is strong or
broad, or if one does make more complicated models with possible additional modes, it seems
reasonable to discuss the acoustic mode energy as a parameter that is not affected by tails of the othermodes. For completeness, we note that for the interacting model, one has
2⎡⎤⎛ I0 z0 z1 ⎞
I
z
z
1
5
0
0
1
⎢
ω pk,interacting = Ω 1 −+ ⎜+ H.O. ⎥⎢⎥2 I1 Ω28 ⎝ I1 Ω2 ⎟⎠
⎣
⎦

(SM1)

and we plot the various mode energies in figure S2. As suggested, ω pk ≈ Ω .

Figure S2. Different possible choices for acoustic mode energies as discussed in the text.

3Section S4: Conversion of Parameters
For convenience and completeness we give relations between the parameters of equation 5.3.15 of 6)
and equation 1 of the main text:

z0 = χ k 2(SM2a)()

z1 = Γk 2 = 12 ⎡⎣ν l + (γ − 1) χ ⎤⎦ k 2 = 12 ν l k 2 + 12 I0 I1 z0(SM2b)

ν l k 2 = 2z1 − (γ − 1) z0 = 2z1 − z0 I0 I1(SM2c)

γ = 1+ I0 I1(SM2d)

c 2 k 2 = ω s 2 = Ω2 − z12(SM2e)

Where k is used in 6) instead of Q , χ is the thermal diffusivity, ν l is the longitudinal viscosity andΓ is the acoustic mode width. It is perhaps worth noting that while equation 1 can be negative if z1
is small compared to z0 and I1 is small compared to I0 , the parametrization above prevents this, as,when I0 becomes large, z1 becomes large compared to z0 (eqn. SM2b):

the hydrodynamic

formulation does not allow for an acoustic mode that is simultaneously weak and narrow comparedto the central line, as is consistent with all spectra observed in our experience.

References for these Supplemental Materials
1) A.Q.R. Baron, D. Ishikawa, H. Fukui, and Y. Nakajima, AIP Conf. Proc. 2054, 20002 (2019).
2) D. Ishikawa and A.Q.R. Baron, J. Synch. Rad. 28, 804 (2021).
3) F. James and M. Roos, Comput. Phys. Commun. 10, 343 (1975).
4) P.R. Bevington and D.K. Robinson, Data Reduction and Error Analysis for the Physical Sciences,
Second Edi (WCB McGraw-Hill, Boston, 1992).
5) A. Gil, J. Segura, and N.M. Temme, SIAM J. Sci. Comput. 34, A2965 (2012).
6) J.P. Boon and S. Yip, Molecular Hydrodynamics (Dover Publications, Mineola, New York, 1980).

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