From non-plane wave modes in a cylindrical hard-walled cavity to whispering
and rotating waves, and consideration of a change in cross-section
Tristan Nerson1, a)
LAUM, CNRS, Le Mans Université, Avenue O. Messiaen, 72085 Le Mans, France
(Dated: 16 November 2022)
The theoretical pressure field generated by point sources inside a cylindrical cavity is derived using the modal
theory, and compared with experimental data. A proper arrangement of sources permits to create rotating
waves (acoustic vortices) inside the cavity. We focus on their study in a normal plane in the cylinder. When
the pressure oscillations are located near the circular edge in this plane, we show the appearance of the socalled whispering-gallery waves. Finally, a short introduction to mode matching is made in a two-dimensional
cavity with a change of cross-section. We also compare a Finite Element Method model using COMSOL and
experimental data of a 3D cylindrical cavity with a changing cross-section.
PACS numbers: 43.20.+g, 02.10.Ud, 02.30.Px, 02.40.Dr, 43.55.n
I.
II. FREE OSCILLATIONS INSIDE A CYLINDRICAL
CAVITY
INTRODUCTION
In the following, we will study the propagation of waves
inside a cylindrical cavity. The first step will be to derive
an expression for the inner pressure field in free oscillations, and then with sources, by making use of the modal
theory. Once that it is made, we compare it to some experimental data, and shortly discuss the phenomena of
rotating waves and whispering-gallery waves inside the
cavity.
The geometry of the considered system is the one described in Fig. 1, with Cartesian coordinates (x, y, z). It
is a cylindrical cavity of radius a and length l in the zdirection, submitted to external forcing in the form of
linear oscillations at an angular frequency ω created by
one or several point sources placed on the upper wall.
For convenience, we define the radial distance r and the
azimuth θ as x = r cos θ and y = r sin θ, so r = (r, θ, z).
First, we can derive an expression for the acoustic
field inside a hard-walled cavity without considering the
source term. The general solution of the Helmholtz equation in cylindrical coordinates for the complex pressure
field p over the time t is1
X
p(r, θ, z, t) =
[Ar Jm (kω r) + Br Ym (kω r)]
m,n
(1)
× [Aθm cos(mθ) + Bθm sin(mθ)]
h
i
× Az e−ikz (z−l) + Bz eikz (z−l) eiωt ,
where Jm and Ym respectively denote the first and second kinds Bessel functions of order m. The parameters kω and kz are components of the wavenumber that
are to be defined. Because the cavity is axisymmetric,
p(r, θ, z) ≡ p(r, θ + 2π, z) so m ∈ N.
Moreover, the position r = 0 is included within the
cavity domain and
lim Ym (kω r) = ∞.
(2)
kω r→0
Hence, Br = 0. The other constants Ar , Aθm , Bθm , Az
and Bz need to be determined in the following.
A Neumann boundary condition is defined on the upper and lower walls such as
∂p
∂z
= 0,
(3)
z=0,z=l
leading to
FIG. 1. Schematic drawing of the cavity. A source can be
placed at the position r0 = (r0 , θ0 , 0).
Bz = Az = Az e2ikz l
⇒ e2ikz l = 1
⇒ kz = kzq ∋ kzq l = qπ,
q ∈ N.
(4)
Finally, the radial velocity at r = a vanishes so
a) tristan.nerson.etu@univ-lemans.fr;
https://github.com/elyurn
∂Jm (kω r)
∂r
= 0.
r=a
(5)
2
Therefore, kωmn ≜ kω and kωmn a is the (n + 1)st root of
∂Jm /∂r. One can rewrite Eq. (1) omitting the temporal
dependence as
X
p(r, θ, z) =
tion to the inhomogeneous wave equation
∂ 2 p 1 ∂p
1 ∂2p ∂2p
+
+ 2 + k2 p
+
∂r2
r ∂r
r2 ∂θ2
∂z
q0
= − iω δ(r − r0 )δ(θ − θ0 )δ(z),
r
Amnq Jm (kωmn r) cos kzq (z − l)
(6)
m,n,q=0
× [Aθm cos(mθ) + Bθm sin(mθ)] ,
2
where2 Amnq = 2Ar Az and kmnq
≜ kω2 mn + kz2q . It is
expressed in terms of the spatial modal functions of the
cavity so they form a complete basis for the space of
solutions satisfying the above boundary conditions.
(7)
where k = ω/c0 , c0 is the phase velocity and δ denotes
the Dirac delta distribution.
Reporting the form of solution of Eq. (6) in Eq. (7)
leads to
X
2
k 2 − kmnq
[amnq ψmnq (r) + bmnq φmnq (r)]
m,n,q=0
= − iω
q0
δ(r − r0 )δ(θ − θ0 )δ(z),
r
(8)
III.
A.
with
(
EXTERNAL FORCING AND MODAL THEORY
ψmnq = Jm (kωmn r) cos(mθ) cos kzq (z − l)
.
φmnq = Jm (kωmn r) sin(mθ) cos kzq (z − l)
(9a)
(9b)
Inhomogeneous wave equation
B.
In order to determine Amnq , Aθm and Bθm , we need to
make use of the modal theory while introducing an external forcing provided in our case by loudspeaker modeled
by a point source on the upper position r0 = (r0 , θ0 , 0)
(see Fig. 1). The coefficients can be obtained if the modal
spatial functions are orthogonal for a suitable definition
of inner product3 . Considering that a given mass flow
rate q0 is imposed at the angular frequency ω, the steadystate pressure response as a function of position is solu-
ZaZ2πZL X
Application of the inner product
Let’s define the inner product ⟨ζ, ζ ′ ⟩ such that
⟨ζ, ζ ′ ⟩ =
ZaZ2πZL
ζζ ′ dr dθ dz.
(10)
0 0 0
We can apply the inner products ⟨·, ψm′ n′ q′ ⟩ and
⟨·, φm′ n′ q′ ⟩ to Eq. (8) so
ψm′ n′ q′
ψ
2
k 2 − kmnq
[amnq ψmnq (r) + bmnq φmnq (r)] φ ′ ′ ′ dr dθ dz = −iωq0 φ(r0 ).
mnq
(11)
0 0 0 m,n,q=0
The left-hand side (LHS) of Eq. (11) can be computed using respectively Eq. (9a) and Eq. (9b). Some
helpful formulas can be found in Appendix A. Moreover,
a famous formula in Storm-Liouville theory states4 that
when Eq. (5) holds,
′
Za
Za
if n ̸= n ,
where
rJm (kωmn r)Jm (kωmn′ r) dr = 0.
(12)
0
0
2
rJm
(kωmn r) dr =
kω2 mn a2 − m2 2
Jm (kωmn a).
2kωmn
(14)
!
One can deduce that5 (m, n, p) = (m′ , n′ , p′ ) so LHS ̸=
0. In these cases,
Za
LHS ≡
0
ZL
×
0
2
rJm
(kωmn r) dr
Z2π
cos2 (mθ) dθ
sin2
0
cos2 kzq (z − L) dz,
(13)
Finally, one can write the expressions of amnq and bmnq
by rearranging Eqs. (11), (13) and (14) as
3


1,



ωµmnq ψ(r0 ) 


× 2,

amnq = k 2 − k 2

mnq
4,



2,

ωµmnq φ(r0 )


×
b
=
 mnq
2
2
k − kmnq
4,
µmnq
if m = 0 and q = 0
if (m = 0 and q ̸= 0) or (m ̸= 0 and q = 0)
if m ̸= 0 and q ̸= 0
,
(15a)
if m ̸= 0 and q = 0
if m =
̸ 0 and q ̸= 0
(15b)

1

 ,
if m = 0 and n = 0
iq0  a2
=−
×
.
kω2 mn

πl

 2
,
elsewhere
Jm (kωmn a) [(kωmn a)2 − m2 ]
One can notice that for m = 0, φmnq = 0 and bmnq is
indefinite. The pressure field is now expressed as
X
p(r, t) =
amnq ψmnq (r) + bmnq φmnq (r)
(17)
LS 1
LS 2
Mic. 4
5 cm
m,n,q=0
and only the first term of Eq. (17) is kept when m = 0.
Mic. 3
18.5 cm
33 cm
IV.
(16)
COMPARISON WITH EXPERIMENTAL DATA
17.6 cm
Description of the setup
17 cm
A.
Mic.1
In order to confirm our model, we build an experimental setup. A sketch of the experimental setup is proposed
in Fig. 2. For the sake of availability of construction materials, two ducts were glued together. We assume in a
first approximation that their respective cross-section is
the same (2a = 18.5 cm). Four microphones measure the
pressure field at four cardinal points over a cross-section
of the whole cavity. Their sensibility is believed to be
constant (50 mV Pa−1 ). The microphone 1 is chosen to
be located at r1 = (a, 0, 41.5) cm.
Moreover, the cut-off frequency of the mode (m, n, q)
corresponds to
fmn =
kωmn c0
,
2π
(18)
since the condition for propagation of non-evanescent
acoustic waves inside the cavity is that kmnq must be
real and
q
kmnq = k 2 − kω2 mn .
(19)
One can deduce that the three modes with the lower cutoff frequencies are at
f10 ≈ 1.84
c0
c0
c0
, f20 ≈ 3.05
, f01 ≈ 3.83
. (20)
2πa
2πa
2πa
For the mode (m, n, q), m represents the number of
nodal diameters and n the number of nodal circles. We
Mic. 3
Mic. 2
Mic.1
FIG. 2. Scheme of the experimental setup. We use up to two
loudspeakers (LS) and four microphones (Mic.).
therefore expect to have a plane wave at the mode (0, 0,
q), one nodal line between the microphones 2 and 4 at
the mode (1, 0, q), and two nodal lines in the diagonals
of the microphones at the mode (2, 0, q). If p̃i denotes
the pressure measured by the ith microphone, one can
approximate the pressure field at the microphones – if
the studied frequency range is [50, fmax ] Hz with f20 <
fmax < f01 – as

p̃1



 p̃
2

p̃3



p̃4
≈ P00 + P10 + P20
≈ P00 − P20
,
≈ P00 − P10 + P20
≈ P00 − P20
(21a)
(21b)
(21c)
(21d)
where Pnm denotes the contribution of the nmth mode
to the total pressure field. Such expressions for the estimated measured pressures are tied down to Eqs. (9a)
and (17) when taken in the above frequency range (for
m up to 2 and n = 0) and at a given axial position z0 ,
4
p(a, θ, z0 ) ≈ P00 (ω) + P10 (ω) cos(θ) + P20 (ω) cos(2θ).
(22)
The contributions of the modes can be expressed as

p

|p̃21 |
p̃2
p̃3
p̃4



(23a)
1
+
+
+
P
≈
00


4
p̃1
p̃1
p̃1


p


|p̃21 |
p̃2
p̃3
p̃4
.
(23b)
P10 ≈
1−
−
+

2
p̃1
p̃1
p̃1


p



|p̃21 |
p̃2
p̃3
p̃4



(23c)
1−
+
−
 P20 ≈ 4
p̃1
p̃1
p̃1
The frequencyp
response functions p̃i /p̃1 as well as the
power spectrum |p̃21 | are measured between 50 Hz and
2.5 kHz, frequency by frequency (for more accuracy compared to computing the Fast Fourier Transform (FFT) of
the response to a sweep signal or a white noise).
B.
Note on the Sound Pressure Level in a short cavity
Relative Magnitude [dB]
In order to ensure that the coding of expressions for the
pressure using Python is correct, we can choose a very
small length l to get rid of some axial dependence. If we
only compute the modes for m up to 2 and n = q = 0,
resonances should occur at the frequencies f10 and f20
defined in Eq. (20) as well as in the limit f00 = 0 Hz.
One can introduce the Helmholtz number He = ka, so
we can plot the Sound Pressure Level (SPL) as a function
of it in Fig.3. This permits to highlight the values of
kωmn a independently of the speed of sound. The crosssection a is seen here as a characteristic length of the
short cavity.
Max
Min
0
1
2
Helmholtz number He = ka
3
FIG. 3. Relative magnitude of the Sound Pressure Level at
r = a and θ = 0 when l ≪ 1, as a function of the Helmholtz
number. The first roots of ∂Jm /∂r are show with the dotted
line.
One can retrieve the first values of the roots of ∂Jm /∂r,
kωmn a ≈ {0, 1.84, 3.05, ...}. Since the model behaves as
expected, we can look into the propagation of waves inside the real cavity in the following.
C.
The real cavity for three configurations
1.
One loudspeaker
The first configurations studied is when we use one
single loudspeaker at position (5, 0, 0) cm. The SPL in
both the measure and the model are plotted in Fig. 4.
Concerning the model, we predict the modes that are
contributing to the sound field by only taking the ones
that respect
q
2πfmax
(24)
≥ kω2 mn + kz2q .
c0
Sound Pressure Level L [dB]
leading to
100
80
60
40
20
Model
Measure
0
0
500
1000
1500
Frequency f [Hz]
2000
2500
FIG. 4. Comparison between the model (dotted line) and
the measure (solid line) of the Sound Pressure Level at the
microphone 1, with one loudspeaker.
We find 27 modes with c0 = 342 m s−1 . As detailed in
Appendix B, the speed of sound in the laboratory has an
impact on the number of modes, and changing it could
somehow explain some discrepancies between the model
and the measure in the very end of the frequency range
(above 2.4 kHz).
Moreover, the mass flow rate q0 = 2 × 10−7 kg s−1 has
been determined to properly fit the measure. As it is just
a constant coefficient to add to the pressure, it amounts
to control the offset of the curve. However, since we don’t
take losses into account, the SPL can take some infinite
values at the resonance frequencies, so the fitting tends
to be a bit hit-and-miss. Because of losses, the measure
data is contained in a smaller range of levels. Anyhow,
it would only matter if some absolute levels wanted to
be described, but then the sensibility of the microphones
would should be correctly defined at each frequency.
Apart from the previous remarks, the analytical results
correctly predict the real behavior of the field, in particular at low frequencies where only a few modes contribute.
Still, it seems that the model underestimate some low frequencies measured in the cavity. An explanation can be
found in the response of the microphones and the loudspeaker that could be very bad for these low frequen-
2.
Two loudspeakers oscillating in phase
When the second loudspeaker at position (5, π, 0) cm
oscillates in phase with the first one, we expect to excite
predominantly the modes (0, 0, q) and (2, 0, q) but not
the modes (1, 0, q). The higher modes (0, 1, q), (2, 1,
q) and (0, 2, q) should also be excited for example. This
can be deduced from looking at the diametral (or radial)
modes m, the circular (or azimutal) modes n and the axial (or longitudinal) modes q.6 In order to find analytical
results, we make use of the superposition principle so as
to sum two pressure fields.
The analytical and measured results are given in Fig. 5.
One can notice that the measured signal is nosier than the
one with one loudspeaker. It is caused by a change in the
experimental method: a white noise was used here and
the response was computed using a FFT. It also seems
that our measure has a problem in the low frequencies
(a huge gap near 150 Hz), probably due to a localized
bad response of the microphones or loudspeakers in low
frequencies.
The same general comments as in the previous configuration hold to explain some of the differences between the
model and the measure. Apart from this, discrepancies
appear at some discrete values, for instance near 1 kHz
and 1.5 kHz. At exactly 1026 Hz, the model predicted
(see Tab. I) a resonance corresponding to the mode (0,
0, 3), but it doesn’t appear. We suppose that a length
correction in the cavity could be taken into account because the whole model spectrum seems to be shifted to
100
80
60
40
20
Model
Measure
0
0
500
1000
1500
Frequency f [Hz]
2000
2500
FIG. 5. Comparison between the model (dotted line) and
the measure (solid line) of the Sound Pressure Level at the
microphone 1, with two loudspeakers in phase.
the right. This deviation in the behavior of the experimental setup is believed to be caused by the impedance
of the walls that are not taken into account in the model.
3.
Two loudspeakers oscillating out of phase
In this last configuration, the second loudspeaker is out
of phase compared to the first one. The expected results
are the opposites of the ones in the last configuration.
In fact, we expect no contribution of the modes (0, 0, q)
and (2, 0, q) but a high contribution of the modes (1, 0,
q). The results are shown in Fig. 6.
Sound Pressure Level L [dB]
cies (under 250 Hz). In addition, it was seen during the
measurements that the microphones 2,3 and 4 were not
perfectly calibrated.
In the higher part of the spectrum, we find some big
differences between the measure and the model. It appears that we sometimes see a resonance instead of an
anti-resonance, and vice versa. We can maybe explain it
because of the lack of information about the exact axial
position of the microphones. Indeed, looking at Fig. 2,
we suppose that the microphones are placed at 8.5 cm
from the bottom. However, we are not sure that the microphones are exactly positioned in the middle of the 17
cm indicated length. With that in mind, one can easily understand that the field can be completely different,
because some modes can be excited or not depending on
their z position.
In this configuration, we expect to excite many modes
since the second loudspeaker in Fig. 2 doesn’t impose any
pressure field. We can easily deduce from explanations
in Section IV A that the modes (0, 0, 0), (1, 0, 0), and
(2, 0, 0) for example are well excited. In practice, one
can compare the peaks in Fig. 4 with the values of resonance frequencies of Tab. I and see that it works for
many modes. At high frequencies, it becomes difficult to
discriminate one mode to another, so the comparison is
tougher.
Sound Pressure Level L [dB]
5
100
75
50
25
0
Model
Measure
−25
0
500
1000
1500
Frequency f [Hz]
2000
2500
FIG. 6. Comparison between the model (dotted line) and
the measure (solid line) of the Sound Pressure Level at the
microphone 1, with two loudspeakers out of phase.
In this case, the major discrepancies concern the nonexpected contribution of modes (0, 0, q) in low frequencies. It is most likely that the two loudspeakers are not
perfectly identical, i.e. different in term of modulus and
6
not a phase of exactly ϕ = π. This could lead to the
observed generation of some in-phase modes. Still, the
model predicts with a high accuracy the behavior of the
system in the medium frequencies (between 800 and 1400
Hz). However, it seems once again that a length correction should be added so the model fits better the experience, because it seems to be shifted to the right.
V.
ROTATING WAVES
Now that the model has been tested using experimental data, one can try to ”play” with it and generate some
acoustic vortices inside the resonator. These have the
particularity of exhibiting a helical wavefront, hence calling them rotating (or spinning) waves. In 1992, Ceperley
proposed to generate a rotating wave using two independent standing waves7 . The equation for a pure rotating
wave in a given plane normal to the z-axis is shown to
be
p⟲ (r, θ, t) = AJm (kr) cos(mϕ − ωt).
(25)
In this case, the wave propagates in the ϕ direction.
Using two spherical point sources on top of cavity of
opposite phase and position (like in the last section), one
can create a standing wave, by making use of the superposition principle. Let the imposed wavenumber k take a
value very near kω10 so one nodal line appears. The corresponding pressure field is plotted along one cross-section
in Fig. 7 for different times.
Then, one can construct a rotating wave using four
point sources that create two standing waves that are out
of phase8 . This is shown in Fig. 8. The four loudspeakers
have a relative phase shift of π/2.
The direction of rotation of the wave can be changed by
inverting the phase of LS2 and LS4. One can also show
that such a rotating wave can be created in the same way
by another number of loudspeakers uniformly positioned
around the center and uniformly shifted between 0 and
2π. For example, three sources placed at respectively
(r0 , θ0 = 0), (r0 , θ0 = 2π/3) and (r0 , θ0 = 4π/3) produce
the same type of rotating waves when the two last sources
have a phase shift of 2π/3 and 4π/3 relative to the first
loudspeaker.
One can plot the phase along the cross-section, so the
projection of the vortex in this plane is easily understandable in one visualization. This is done in Fig. 9. In order
to describe the critical point in the center of cross-section,
one can define the topological charge defined as
I
I
1
1
S=
∇ arg (p(r)) dr =
d arg(p(r)), (26)
2π
2π
C
C
as provided by9 . The counterclockwise contour C is taken
to be a simple contour (it does not intersect itself) around
a critical point. For a first-order left-handed vortex, S =
1, while for a first-order right-handed vortex, S = −1.
For any critical point that is not a vortex, S = 0. Higher
order vortices can be created using the same arrangement
as above, but with more sources (for example, with eight
sources for finding S = ±2).8
VI.
WHISPERING-GALLERY MODES
When rotating modes only carry energy near the circular edge of the cavity, this permits to make some information travel from one point close the edge to another. A simple review of this phenomenon has be made
by Wright10 . It has been popularized in the St Paul’s
Cathedral in London, where it is possible for two people
to discuss in the gallery without making it possible for
someone in the middle to hear them. We call these waves
”whispering-gallery waves”.
As proved by Lord Rayleigh in11 , it is possible to build
such waves from a good choice of index m in Eq. (25).
In fact, he show that increasing m permits to keep the
pressure oscillations away from the center of the circular cross-section. We claim that a good way to build
whispering-gallery modes is to have a high m and low
n. Three configurations are shown in Fig. 10, for three
choices of m and n: (15, 0), (15, 3) and (5, 0). The optimal case is the first one, since the pressure extrema are
kept near the edge.
VII.
A.
CHANGE OF SECTION
Mode matching in two dimensions
Now that the pressure in the cavity has been properly
studied, one can have an interest in figuring out the impact of the change in cross-section described in Fig. 2. In
the following, we will give an introduction to the topic of
mode matching between two cavities. Since this formulation is somehow difficult, we will only see in this section the two-dimensional case in Cartesian coordinates
an not the three-dimensional case in cylindrical coordinates. This work undertaken below is unfinished so no
pressure field is plotted at this level.
For simpler computation, we redefine in the following
the coordinate z = 0 at the position of change in the
cross-section. The three conditions at z = 0 are

(27a)

 p|z=0− ,x = p|z=0+ ,x , x ∈ a1



∂p
 ∂p

=
, x ∈ a1
(27b)
∂z z=0− ,x
∂z z=0+ ,x
,



∂p



= 0, x ∈
/ a1
(27c)
 ∂z
z=0− ,x
where a1 denotes the radius of the small cavity (1) and
a2 denotes the radius of the big cavity (2). In this new
set of coordinates, the position of the small wall (1) is at
z = L and the big wall (2) is at z = l (so l < 0). One can
7
FIG. 7. Pressure field in a cross-section of the cavity for the mode (1, 0, q) at different times. a) ωt = π/2 b) ωt = 0 c)
ωt = 3π/2. The position of the loudspeaker 1 (LS1) is symbolized by a grey circle. The second loudspeaker is located at the
symmetric point of LS1 with respect to the nodal line.
FIG. 8. Pressure field in a cross-section of the cavity for a rotating wave carrying the mode (1, 0, q) at different times. a)
ωt = 0 b) ωt = π/2 c) ωt = π d) ωt = 3π/2. The four loudspeakers are symbolized by black circles.
notice than we cannot directly consider the continuity of
the volume velocity at z = 0. In fact, this only holds for
the planar mode.
be written as
p|z≤0,x =
∞
X
(2)
gn(2) (x)pn (z) = An eikn
(l−z)
(2)
+ Bn eikn z ,
n=0
Moreover, the two pressures fields in the cavities can
(28)
8
nuity of the velocity at z = 0:
∂p
∂z
⇒
∂p
∂z
Za1
⇒
∂p
∂z
=
z=0− ,x
z=0+ ,x
(2)
gm
(x) =
z=0− ,x
∂p
∂z
0
∂p
∂z
(2)
gm
(x)
z=0+ ,x
Za1
∂p
∂z
(2)
gm
(x) dx =
z=0− ,x
0
(2)
gm
(x) dx.
z=0+ ,x
One can add a term Λ(1,2) on the left-hand side of this
last equation so
(1,2)
Λ
FIG. 9. Phase of a the generated vortex of topological charge
S = 1.
p|z≥0,x =
∞
X
(1)
gn(1) (x)pn (z) = Cn e−ikn
z
(2)
+ Dn eikn
(z−L)
,
Za2
=
∂p
∂z
a1
(2)
gm
(x) dx,
(33)
z=0− ,x
and from Eq. (27c), Λ(1,2) = 0. One can finally link the
z-derivative of the pressure in the two domains using the
internal addition rule of integrals as
n=0
where the functions
(i)
gn
(29)
are chosen to be orthonormal,
k2 −
nπ
ai
2
,
(30)
(1)
(1)
⇒ p|z=0− ,x gm
(x) = p|z=0+ ,x gm
(x)
Za1
Za1
(1)
(1)
⇒ p|z=0− ,x gm
(x) dx =
p|z=0+ ,x gm
(x) dx
0
pn |z=0− Fmn =
∞
X
pn |z=0+ δmn ,
where the right-hand side is obtained using the orthogonality relation and the matrix
Za1
Fmn =
Za1
dx =
z=0− ,x
p′n |z=0− δmn =
∂p
∂z
0
∞
X
(2)
gm
(x) dx
z=0+ ,x
p′n |z=0+ Fnm ,
n=0
where p′ (z) is a vector containing all the p′n (z) values.
One can finally write
(1)
gm
(x)gn(2) (x) dx.
where F T is the transposed matrix of F.
We define the elements of diagonal matrices

(1)
(1)
−ikn
L

δmn

 Emn = e
(2)
(2)
Emn
= eikn l δmn


 (i)
Ymn = ikn(i) δmn
n=0
n=0
(2)
gm
(x)
p′ |z=0− = F T p|z=0+ ,
p|z=0− ,x = p|z=0+ ,x
⇒
⇒
∞
X
n=0
and the vectors A, B, C and D are to be determined.
Applying the condition Eq. (27a) leads to write
0
∞
X
∂p
∂z
0
s
kn(i) =
Za2
(31)
.
(34)
(35a)
(35b)
(35c)
The two Neumann boundary conditions at z = L and
z = l lead to

(2)
(2)

 Ymn
Emn
B−A =0
(36a)
,

(1)
(1)
 Ymn
D − Emn
C =0
(36b)
0
This leads to the following expression for describing the
continuity of the pressure:
F p|z=0− = p|z=0+ ,
(32)
where p(z) is a vector containing all the pn (z) values.
Now we can derive an analog expression for the conti-
and the conditions at z = 0 are expressed from Eqs. (32)
and (34) as
 (2)
(1)

 F Emn
A + B = C + Emn
D
(37a)
.

(2)
(2)
(1)
(1)
 Ymn
(37b)
B − Emn
A = F T Ymn
Emn
D−C
We can finally write the system to be solved:
9
FIG. 10. Whispering-gallery modes in the cavity. a) m = 15, n = 0 b) m = 15, n = 3 c) m = 5, n = 0.
cavity with a changing cross-section, one can simulate
the model using Finite Element Method (FEM) with the
F
−1
software COMSOL Multiphysics. This is done in the fol(1)
(2)
F T Ymn
Ymn
   = 0, lowing.
(2) (2)
 C
0
0
Ymn Emn
In COMSOL, we use both an eigenfrequency solver and
(1)
(1) (1)
D
Ymn
0
0
−Ymn Emn
an analysis in the frequency domain between 5 Hz and 2.5
(38)
kHz to plot the pressure inside the cavity of the Fig. 2
which can be rewritten for convenience
with and without a point source located at LS1. We
probe the SPL at the location of the microphone 1 (one
MX = 0.
(39)
must keep in mind that the absolute level is dependent
of the mass flow rate that we choose the point source
This equation has a non-trivial solution if det(M) = 0,
to deliver). We use a tetrahedral mesh, refined near the
i.e. we find modes when det(M) = 0. We believe that
change of cross-section. Several meshes with different
it is possible to numerically solve such an equation in
number of elements are tested in order to ensure that
order to find the elements of X using the Singular Value
the solution is convergent.
Decomposition (SVD) of M:
X
The results of the simulation are plotted in Fig. 11,
M = USV † =
Un σn Vn† ,
(40)
and compared to the measure (same configuration as in
n
Fig. 4). The vertical red lines correspond to the eigenfrequencies.
where S is a diagonal matrix containing singular values,
σi = Sii and V † is the Hermitian conjugate of V. The
vectors in X are in the null space of M, which is spanned
by the right singular vectors Vn corresponding to the zero
singular values. Let σN be the smallest singular value. If
125
we choose
(2)
FEmn

(2) (2)
−Ymn Emn

(2)
 −Ymn
(1)
 
−Emn
A

(1)
(1)
−F T Ymn Emn   B 
X = VN ,
(41)
then
MX = σN UN .
(42)
If σN = 0, then VN is a solution of Eq. (38). In our
case, it is quite difficult to compute since M is a multidimensional matrix, and σN depends on the wavenumber
k. The idea is typically to find the wavenumbers leading
to σN = 0.
B.
Finite Element Method using Comsol
Since the previous method has not been properly studied in details in order to find the pressure field in the
Sound Pressure Level L [dB]


100
75
50
25
Comsol
Measure
0
0
500
1000
1500
Frequency f [Hz]
2000
2500
FIG. 11. Comparison between the COMSOL simulation (dotted line) and the measure (solid line) of the Sound Pressure
Level at the microphone 1, with one loudspeaker. The red
lines correspond to the eigenfrequencies found by COMSOL.
10
It is interesting to see that we now have more eigenfrequencies since some modes resulting from both cavities
appear. However, the results are not as good in the lowfrequency range as with the model detailed in the first
section. It was also chosen here to show as red lines all the
eigenfrequencies in order to highlight the fact that their
density increases when the frequency increases. This is
why a modal approach is very useful in low frequencies
but sometimes not preferred in high frequencies in comparison with a ray acoustics approach.
Once again, since we used some hard-wall boundary
conditions, the resonance can take an infinite energy. If
we add some impedant walls in the model, then the peaks
are lowered. It would also be interesting to take into account the oscillations of the walls, as it is done sometimes
in literature12 (acoustic-solid coupling).
VIII.
CONCLUSION
To sum up, we studied acoustic waves inside a cylindrical cavity with and without an abrupt change in cross
section. We show the appearance of rotating modes and
whispering-gallery modes when we make use of some
proper arrangement of sources. An introduction to mode
matching is made in 2D, without any concluding results.
This would need to be examined more in detailed, since
no it would be interesting to extend the computations to
the 3D case and to compare the two modal approaches
(with and without the change of cross-section).
ACKNOWLEDGMENTS
I thank Dr. Guillaume Penelet, MCF, for his useful
guidance throughout this project, and for the correction
of the present paper. I also thank Dr. Simon Félix,
CNRS Scientist, for very useful discussions concerning
mode matching.
Z2π
0

0, if q ̸= q ′



l
cos(kzq (z−l)) cos(kzq′ (z−l)) dz =
, if q = q ′ =
̸ 0 .

2


′
l, if q = q = 0
Appendix B: Finding the contributing modes
In order to find all the contributing modes in the
studied frequency range, we use a loop in Python that
tests Eq. (24). We arbitrary choose to test it up to
m, n, q = 20, which seems to be sufficient since the maximum value of m, n or q that we find is q = 7. The 27
modes contributions are given in Tab. I.
Mode Frequency [Hz] kωmn a
(0,
(0,
(0,
(0,
(0,
(0,
(0,
(0,
(0,
(0,
(0,
(0,
(1,
(1,
(1,
(1,
(1,
(1,
(1,
(2,
(2,
(2,
(2,
(2,
(2,
(3,
(3,
0,
0,
0,
0,
0,
0,
0,
0,
1,
1,
1,
1,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0)
1)
2)
3)
4)
5)
6)
7)
0)
1)
2)
3)
0)
1)
2)
3)
4)
5)
6)
0)
1)
2)
3)
4)
5)
0)
1)
0
342
684
1026
1368
1710
2052
2394
2255
2281
2356
2477
1083
1136
1281
1492
1745
2024
2320
1797
1829
1923
2069
2259
2481
2472
2496
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
3.83
3.83
3.83
3.83
1.84
1.84
1.84
1.84
1.84
1.84
1.84
3.05
3.05
3.05
3.05
3.05
3.05
4.20
4.20
Appendix A: Useful formulas for modal theory
In order to derive the left-hand side of Eq. (11), one
can make use of the following formulas:

if m ̸= m′

Z2π
0,
̸ 0 ,
cos(mθ) cos(m′ θ) dθ = π, if m = m′ =

2π, if m = m′ = 0
0
Z2π
0

′

0, if m ̸= m
′
̸ 0 ,
sin(mθ) sin(m θ) dθ = π, if m = m′ =

0, if m = m′ = 0
TABLE I. Modes contributions and their resonance frequency
for c0 = 346 m s−1 .
One can notice that we find two contributions with
m = 3, and it it means that we don’t respect Eq. (22).
In fact, the modes are found with an estimated speed
of sound c0 = 342 m s−1 . If we reestimate the speed of
sound as c0 = 346 m s−1 , we now don’t find the modes
(0, 1, 3), (2, 0, 5), (3, 0, 0) and (3, 0, 1) anymore.
With the resulting modes, we compute again the analytical pressure field and plot it together with the measured pressure in Fig. 12. It seems somehow to reduce some discrepancies at the very end of the frequency
range. It doesn’t have a big impact on the pressure field
11
2 The
below 2.4 kHz, when compared to Fig. 4.
sign “≜” can be read as “is equal by definition”.
Nolan and J. L. Davy, “Two definitions of the inner product of
modes and their use in calculating non-diffuse reverberant sound
fields,” The Journal of the Acoustical Society of America 145,
3330–3340 (2019).
4 N. H. Asmar, enPartial Differential Equations with Fourier Series and Boundary Value Problems: Third Edition (Courier
Dover Publications, 2017).
Sound Pressure Level L [dB]
3 M.
100
80
60
5 The
40
20
0
Model
Measure
−20
0
500
1000
1500
Frequency f [Hz]
2000
2500
FIG. 12. Same plot as Fig 4 (see caption) but with a modeled
speed of sound of 346 m s−1 (against 342 m s−1 ), resulting to
less contributions of modes.
We choose to keep in the main text c0 = 342 m s−1 ,
because it seems to be more consistent with what is expected in a laboratory at a temperature of roughly 19 ◦ C.
A measure of the speed of sound could permit to rule on
the issue.
1 E.
W. Weisstein, “Helmholtz Differential Equation–Circular
Cylindrical Coordinates,” MathWorld – A Wolfram Web Resource.
!
sign “=” can be read as “must be equal to”.
Schilt, L. Thévenaz, M. Niklès, L. Emmenegger,
and
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Spectroscopy 60, 3259–3268 (2004).
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60, 938–942 (1992).
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momentum to matter,” American Journal of Physics 77, 209–215
(2009).
9 M. R. Dennis, K. O’Holleran, and M. J. Padgett, en“Chapter 5
Singular Optics: Optical Vortices and Polarization Singularities,”
in enProgress in Optics, Vol. 53, edited by E. Wolf (Elsevier,
2009) pp. 293–363.
10 O. Wright, en“Gallery of whispers,” Physics World 25, 31 (2012).
11 L. Rayleigh, “CXII. The problem of the whispering gallery,”
The London, Edinburgh, and Dublin Philosophical Magazine and
Journal of Science 20, 1001–1004 (1910).
12 D. d. O. França Júnior, P. M. V. Ribeiro, and L. J. Pedroso,
en“Simplified expressions for dynamic behavior of cylindrical
shells uncoupled and coupled with liquids,” Latin American Journal of Solids and Structures 16 (2019), 10.1590/1679-78255546.
6 S.