# Engineering Acoustics/Noise control with self-tuning Helmholtz resonators

## Introduction

Many engineering systems create unwanted acoustic noise. Noise may be
reduced using engineering noise control methods. One noise control
method popular in mufflers is the Helmholtz resonator, see
here. It is comprised
of a cavity connected to the system of interest through one or several
short narrow tubes. The classical examples are in automobile exhaust
systems. By adding a tuned Helmholtz resonator, sound is reflected back
to the source.

Helmholtz resonators have been exploited to enhance or attenuate sound
fields at least since ancient Greek times where they were used in
ancient amphitheaters to reduce reverberation. Since this time,
Helmholtz resonators have found widespread use in reverberant spaces
such as churches and as mufflers in ducts and pipes. The Helmholtz
resonator effect underlies the phenomena of sunroof buffeting seen
here.
One advantage of the Helmholtz resonator is its simplicity. However, the
frequency range over which Helmholtz resonators are effective is
relatively narrow. Consequently these devices need to be precisely tuned
to the noise source to achieve significant attenuation.

## Noise and vibration control

There are four general categories for noise and vibration control:[^1]

1.  **Active systems:** load or unload the unwanted noise by using
    actuators such as loudspeakers
    1
    and Acoustics/Active Control
2.  **Passive systems:** achieve sound attenuation by using
    2

    :   2.1. reactive devices such as Helmholtz resonators and expansion
        chambers.
    :   2.2. resistive materials such as acoustic linings and porous
        membranes
3.  **Hybrid systems:** use both active and passive elements to achieve
    sound reduction
    3
4.  **Adaptive-passive systems:** use passive devices whose parameters
    can be varied in order to achieve optimal noise attenuation over a
    band of operating frequencies.

## Lumped element model of the Helmholtz resonator

The Helmholtz resonator is an acoustic filter element. If dimensions of
the Helmholtz resonator are smaller than the acoustic wavelength, then
dynamic behavior of the Helmholtz resonator can be modelled as a lumped
system see 4.
It is effectively a mass on a spring and can be treated so
mathematically. The large volume of air is the spring and the air in the
neck is the oscillating mass. Damping appears in the form of radiation
losses at the neck ends, and viscous losses due to friction of the
oscillating air in the neck. Figure 1 shows this analogy between
Helmholtz resonator and a vibration absorber.

\[\[Image: HR and vibration absorber sys.JPG\|frame\|center\|

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Figure 1. Helmholtz resonator and vibration absorber

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## Parameters definition

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Parameter

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$M_a$

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Acoustic mass of the resonator

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$C_a$

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Acoustic compliance

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$\rho$

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Density of the fluid

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P

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pressure at the neck entrance

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ω

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Excitation frequency

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S

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Cross section area of the neck

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a

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Radius of the neck

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L

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Actual neck length

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y

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Displacement in the direction pointing inward the neck

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V

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Cavity volume of the Helmholtz resonator

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F

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Force applied in the y direction at the resonator neck entrance

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L~eff~

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Effective neck length (the mass inside the neck and the mass near the
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## Theoretical analysis of Helmholtz resonators

For a neck which is flanged at both ends, L~eff~ is approximately:

$L_\text{eff} = \ L + \ 1.7 \ a$

The acoustic mass of a Helmholtz resonator is given by

$M_a = L_\text{eff}\ \rho\ S$

The stiffness of the resonator is defined as the reciprocal of the
compliance, and it is defined as

$k_r = \frac{dF}{dy}$

where

$F = P\ S$

For adiabatic system with air as an ideal gas, the thermodynamic process
equation for the resonator is

$PV^{\gamma} = \text{constant} \,$

Differentiating this equation gives

$V^\gamma dP + P \gamma V^{\gamma -1} dV = 0 \,$

The change in the cavity volume is

$dV = -Sdy\$

substituting these into differential equation, it can be re-casted as

$V^\gamma \frac{dF}{S} - P \gamma V^{\gamma -1} S dy = 0 \Rightarrow \frac{dF}{dy} = \frac{P \gamma S^2}{V} = k_r$

or considering $P = \rho R\ T$ and $c=\sqrt{\gamma R T}$, resonator
stiffness is then:

$k_r =\frac{\rho c^2\ S^2}{V}$

where c is the speed of sound, and $\rho$ is the density of the medium.

Two source of damping in the Helmholtz resonator can be considered:
sound radiation from the neck and viscous losses in the neck, which in
many cases can be neglected compared to radiation losses.

1\. **Sound radiation from the neck:** Sound radiation resistance is a
function of the outside neck geometry. For a flanged pipe, the radiation
resistance is approximately[^2]

$R_r =\frac{\rho c k^2\ S^2}{2\pi}$

where k is the wave number, $k =\frac{w}{c}$

2\. **Viscous losses in the neck:** The mechanical resistance due to
viscous losses can be considered as[^3]

$R_v =2R_s S\frac{ \ (L+a)}{\rho c a}$

where R~s~ for a sufficiently large neck diameter is

$R_s =0.00083\sqrt{\frac{w}{2\pi}}$, where ω is the excitation
frequency.

The mechanical impedance of the mechanical system is defined as the
ratio of the driving force and the velocity of the system at the driving
point.The mechanical impedance of a driven mass-spring--damper system is

$\hat{Z}_m=\frac{\hat{F}}{\hat{u}}= R_m+j(\omega m - \frac{k_r}{w})$

according to the analogy between Helmholtz resonator and
mass-spring--damper system (vibration absorber), the mechanical
impedance of a Helmholtz resonator is obtained by replacing mass and
damping from Helmholtz resonator system in above equation:

$\hat{Z}_{mres}= (R_v+\frac{\rho c k^2\ S^2}{2\pi})+j(\omega \rho L_{eff} S - \frac{\rho c^2\ S^2}{wV})$

The natural frequency of a Helmholtz resonator,w~0~, is the frequency
for which the reactance is zero:

$w_0 =c\sqrt{\frac{S}{L_\text{eff}V}}$,

and the acoustic impedance of the Helmholtz resonator is

$\hat{Z}_{res}= \frac{\hat{Z}_{mres}}{\ S^2}$

Resonance occurs when the natural frequency of the resonator is equal to
the excitation frequency. Helmholtz resonators are typically used to
attenuate sound pressure when the system is originally at resonance.A
simple open-ended duct system with a side branch Helmholtz resonator and
the analogous electrical circuit of the system is shown below. For an
undamped resonator, the impedance at resonance is zero, and therefore
according to electrical analogy in Fig.2 the Helmholtz resonator becomes
a short circuit. There is no current flowing in the elements in the
right. On the other hand, the undamped Helmholtz resonator at resonance
causes all reflection of acoustic waves back to the source, while in
damped resonator some current will flow through the branch to the right
of the Helmholtz resonator and reduce the magnitude of attenuation.

\[\[Image: HR analogy.JPG\|frame\|center\|300px\|

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Figure 2. open duct system with a side branch Helmholtz resonator with
electrical circuit analogy near junction point A

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**1- Effect of Resonator Volume on sound attenuation**

Figure 3 shows the frequency response of the above duct system without
Helmholtz resonator, and with two different volume Helmholtz resonators
with the same natural frequency. The excitation frequency axis is
normalized with respect to the fundamental frequency of the straight
pipe system, which was also chosen as the natural frequency of the
resonator. The maximum attenuation of sound pressure for duct systems
with side branch Helmholtz resonators occurs when the natural frequency
of the resonator is equal to the excitation frequency. By comparing two
curves with different colors, blue and gray, it can be seen that to
increase the effective bandwidth of attenuation of a Helmholtz
resonator, the device should be made as large as possible. It should be
mention that in order to minimize the effects of standing waves within
the device, the dimensions do not exceed a quarter wavelength of the
resonator natural frequency .

\[\[Image: volume var HR.png\|frame\|center\|300px\|

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Figure 3. Effect of Resonator Volume on sound attenuation

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**2- Effect of Resonator Damping on sound attenuation**

The effect of Helmholtz resonator damping(Resulting from radiation
resistance and viscous losses in the neck) on the frequency response of
the duct system is shown in Figure 5. The lightly damped Helmholtz
resonator is not robust with respect to changes in the excitation
frequency, since the sound pressure in the duct system can be amplified
if the noise frequency shifts to the vicinity of either of the two
system resonances. To increase the robustness of Helmholtz resonator
with respect to changes in the excitation frequency, damping is useful
to add to the resonators to decrease the magnitude of the resonant
peaks. Such increase in robustness decreases performance, since the
maximum attenuation is significantly less for heavily damped Helmholtz
resonators. The motivation for creating a tunable Helmholtz resonator
stems from this trade off between robustness and performance. A tunable
Helmholtz resonator, capable of adjusting its natural frequency to match
the excitation frequency, would be able to guarantee the high
performance of a lightly damped Helmholtz resonator and track changes in
frequency.

\[\[<File:Damping> HR.png\|frame\|center\|300px\|

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Figure 4. Effect of Resonator Damping on sound attenuation

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## Adaptive Helmholtz resonator

The tunable Helmholtz resonator is a variable volume resonator, which
allows the natural frequency to be adjusted.As shown in Figure 5, a
variable volume Helmholtz resonator can be achieved by rotating an
internal radial wall inside the resonator cavity with respect to an
internal fixed wall. The movable wall is fixed to the bottom end plate
which is attached to a DC motor to provides the motion to change the
volume.

\[\[Image: movable HR4.JPG\|frame\|center\|100px\|

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Figure 5. Variable volume Helmholtz resonator

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To determine the sound pressure and volume velocity at any position
along the duct such as the microphone position in Figure 2, we should
first determine the pressure and velocity at the speaker.

\[\[Image: speaker analo.JPG\|frame\|center\|100px\|

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Figure 5. Equivalent circuit for duct system in figure 2

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\[\[Image: equivalent circuit.JPG\|frame\|center\|100px\|

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Figure 6. Simplified equivalent circuit

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The acoustic impedance for the system termination, which is an unflanged
open pipe is approximately

$Z_1 =\frac{\rho c}{S_p}(\frac{1}{4})(ka)^2+j 0.6 ka$

where S~p~ is the cross sectional area of the pipe, and a~p~ is the
radius of the pipe. The impedance at point 2 is

$\frac{Z_2}{\frac{\rho c} {S_p}}=\frac{\frac{Z_1}{\frac{\rho c} {S_p}}+j tan(kL_1)}{1+j\frac{Z_1}{\frac{\rho c} {S_p}}tan(kL_1)}$

where L~1~ is the length of the pipe separating termination form the
resonator. The resonator acoustic impedance is the same as what is shown
above. The acoustic impedance at point 3 is given by

$Z_3 =\frac{Z_2 Z_{res}}{Z_2+ Z_{res}}$

The impedance at point 4 can be determined by

$\frac{Z_4}{\frac{\rho c} {S_p}}=\frac{\frac{Z_3}{\frac{\rho c} {S_p}}+j tan(kL_2)}{1+j\frac{Z_3}{\frac{\rho c} {S_p}}tan(kL_2)}$

finally the impedance at the speaker is given by

$\frac{Z_{sys}}{\frac{\rho c} {S_{enc}}}=\frac{\frac{Z_4}{\frac{\rho c} {S_{enc}}}+j tan(kL_{enc})}{1+j\frac{Z_4}{\frac{\rho c} {S_{enc}}}tan(kL_{enc})}$

where S~enc~ is the cross section area of the speaker enclosure, and
L~enc~ is the length of the enclosure aperture from the speaker.

From figure 6 with the system impedance (Z\_{sys}), the pressure and
velocity at the speaker can be determined. considering transfer
matrices, the pressure and velocity at any location in the duct system
may be computed from the pressure and velocity at the speaker.The first
transfer matrix may be used to relate the pressure and velocity at the
point downstream in a straight pipe to the pressure and velocity at the
origin of the pipe.

$\begin{bmatrix} P_d(l) \\ U_d(l) \end{bmatrix}$=$\begin{bmatrix}  cos (k L) & -j sin (k L) \frac{\rho c}{S_p}  \\ -j \frac{sin (k L)}{\frac{\rho c}{S_p}} & cos (k L) \end{bmatrix}
\begin{bmatrix} P_d(0) \\ U_d(0) \end{bmatrix}$

The second matrix relates the pressure and velocity immediately
downstream of the sidebranch, to the pressure and velocity immediately
before the side branch.

$\begin{bmatrix} P_d(3) \\ U_d(3) \end{bmatrix}$=$\begin{bmatrix}  1 & 0  \\  -\frac {1}{Z_{res}} & 1 \end{bmatrix}
\begin{bmatrix} P_d(2) \\ U_d(2) \end{bmatrix}$

Correct combination of theses transfer matrices may be used to determine
the pressure occurring in the system at the location of the microphone
in figure 2.

## Appendix A: Matlab Code for straight pipe with Helmholtz resonator

``` {.matlab .numberLines startFrom="1"}
%This Matlab code is used for calculating pressure at the place of
%microphone for the pipe without Helmholtz resonator
clear all
clc
V0=1;
for f=10:1:200;
   omega=2*pi*f;
   freq(f-9)=f;
   c=343;                                     % speed of sound
   rho=1.21;                                  % density of the medium (air)
   ap=0.0254;                                 % radius of the pipe
   Sp=pi*(ap^2);                              % cross sectional area of the pipe
   k=omega/c;                                 % wave numbe
   Lenc=0.10;                                 %distance separating the enclosure aperture and the speaker face
   L1=0.34+(0.6*ap);                          % length of the pipe separating the termination from the resonator
   L2=0.62+(0.85*ap);                         % length of the pipe between the resonator and speaker enclosure
   Lx=0.254;                                  % length of the pipe from HR to micropjone
   Lm=Lx+L2;                                  %distance from speaker to microphon
   Ld=L1+L2;
   Rm=1;                                      % loudspeaker coil resistance
   Bl=7.5;
   Cm=1/2000;                                 %Complience of the speaker
   OmegaN=345;                                %Natural frequency of the speaker(55HZ)
   a=0.0425;                                  %effective radius of the diaphragm
   Senc=pi*(a^2);                             % cross sectional area of the speaker enclosure  (Transformer ratio)
   Mm=0.01;                                   %Air load mass on both side of the driver
   Gamma=1.4;                                 %Specific heat ratio for air
   P0=10^5;
   b=0.38;
   Pref=20e-6;                                %Reference pressure for sound pressure level
       
   %calculate the system impedance
   Z1=(rho*c*Sp)*(((1/4)*((k*ap)^2)+i*(0.6*k*ap))/(Sp^2));      %open ended unflanged pipe
   % impedance at point 4( after speaker)
   Z4=(rho*c/Sp)*(((Z1/(rho*c/Sp))+i*tan(k*Ld))/(1+i*(Z1/(rho*c/Sp))*tan(k*Ld))); 
   Zsys=(rho*c/Senc)*(((Z4/(rho*c/Senc))+i*tan(k*(Lenc)))/(1+i*(Z4/(rho*c/Senc))*tan(k*(Lenc)))); 
   %calculating the impedance of the loud speaker 
   Induct=(Mm/(Senc^2));                                                   
   Resist=((Bl^2)/((Senc^2)*(Rm)));
   Drivimp=Resist+(i*omega*Induct)+(1/(i*omega*Cm));           % impedance moadel for speaker
   Impedance=Zsys+ Drivimp;
   Voltage=V0*Bl/((Rm)*Senc);
   Velocity=Voltage/Impedance;
   Pressure=Velocity*Zsys;
   Pressure=Pressure*sqrt(2);
   Velocity=Velocity*sqrt(2);
   TR1=[Pressure;Velocity];
   TR15=[cos(k*(Lenc)) -i*(rho*c/Senc)*sin(k*(Lenc)); -i*(sin(k*(Lenc)))/(rho*c/Senc) cos(k*(Lenc))];
   TR3=[cos(k*Lm) -i*(rho*c/Sp)*sin(k*Lm); -i*(sin(k*Lm))/(rho*c/Sp) cos(k*Lm)];
   Tf=TR3*TR15*TR1;
   Pmike=Tf(1)/(sqrt(2));                                % pressure at microphone
   Vmike=Tf(2);                                          % velocity at microphone
   MagP(f-9)=20*log10(Pmike/Pref);
end;
nondim=freq/(OmegaN/2/pi);
plot(nondim,MagP,'k-.');
title('frequency response of system with Helmholtz Rsonator');
xlabel('Normalized Excitation Frequency (Hz)');
ylabel('Sound pressure level (dB)');
```

## References


[^1]: Robert J. Bernhard, Henry R. Hall, and James D. Jones.
    Adaptive-passive noise control. Proceeding of Inter-Noise 92, pages
    427-430, 1992.

[^2]: L. Kinsler, A. Frey, A. Coppens, and J. Sandesr. Fundamentals of
    Acoustics. John Wiley and Sons, New York, NY, Third edition, 1982.

[^3]: On the theory and design of acoustic resonators. The journal of
    the Acoustical society of America, 25(6):1037-1061, 1953.
