Absorbing Materials Is Common Practice Biology Essay

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The use of absorbing materials is common practice in architectural acoustics. From designing Opera houses to car interiors, the choice of absorbing materials that cover the surfaces of the room under analysis plays an important role in the rise and decay of the reverberant sound filed. [1]

A common technique to calculate the absorption coefficient of a sample is the Impedance tube method. Other parameters such as the reflection coefficient and the surface impedance can also be deduced from results obtained.

METHOD

The apparatus employs a standing wave within a tube by use of a small loudspeaker. The standing wave is produced due to the constructive and destructive phase interference between the incident and reflected waves.

The sample is placed at one end of the tube opposite the loudspeaker. The amplitude of the incident and reflected waves will be different and the nodes will not be at zero pressure as opposed to total reflection of the incident wave resulting in nodes at zero pressure and antinodes at twice the pressure.

The pressure nodes and antinodes are formed by a plane wave. Measurements are made in third octave bands starting from 200 Hz. The upper frequency limit is determined by the diameter of the Impedance tube, above which the results will be corrupted due to extra modes being formed in the tube due to the diameter being the same size as half the wavelength (Equation 1) .The relationship between the upper frequency limit and diameter of the tube is inversely proportional.

fh = C/2D (1 ) [1]

fh is the upper frequency limit in Hertz, d is the diameter of the tube in meters and c is the speed of sound in meters/second.

For the apparatus used, the measured diameter is 96mm � 0.5mm. The upper frequency limit using equation 1 is 1770Hz, it is the frequency at which the first mode will form. Hence readings only up to 1600Hz are taken.

The Distance between Maxima and Minima of pressures is measured by a probe microphone starting at the sample position and gradually moving further away.

In order to evaluate error in data, all maxima and minima should be recorded at each 1/3 octave band frequency. In this report the data is used from just the first Pressure maxima and minima point, therefore the results obtained may contain inaccuracies.

THEORY AND RESULTS

Two samples are used. One with a reflective surface and one without and both are of soft porous absorbing material.

Appendix A contains data for both samples on pressure maxima and minima at a distance away from the test sample and the corresponding recorded voltages from the probe microphone.

STANDING WAVE RATIO

The standing wave ratio is calculated at each 1/3 octave band frequency. It is the ratio of Pmax / Pmin, i.e. maxima divided by minima.

SWR=Pmax/Pmin (2)

REFLECTION COEFFICIENT

It is the ratio of the incident and reflected wave pressures at the surface of the test sample.

Pressure at antinode (Pmax):

(Prms)i(1+|R|) (3)

Pressure at Node (Pmin):

(Prms)i(1-|R|) (4)

SWR= (Prms)i(1+|R|)/(Prms)i(1-|R|) (5)

Rearrange (5) to provide |R|:

SWR-(SWR|R|) = 1+|R|

&

|R|+(SWR|R|)=SWR-1

|R|(SWR+1)=SWR-1

Hence:

|R|=(SWR-1)/(SWR+1) (6)

ABSORPTION COEFFICIENT

The absorption coefficient is related to the reflection coefficient as shown below in equation 7 [2]

a=1-|R|^2 (7)

SURFACE IMPEDENCE

Z=(1+R)/(1-R) (?c/e^jT ) (8)

?=density of air 415 rayl

C= 343 m/s (speed of sound)

T= 2 kx � p (Phase angle )

K= ?/c (wave number )

? = angular frequency (Rad/s)

REFLECTION AND ABSORPTION COEFFICIENT PLOTS as a function of frequency (HZ)

Fig1 Absorption coefficient of Sample 1 (Hard surface)

Fig2 Reflection coefficient of sample 1

Fig3 Absorption Coefficient Sample 2 (Soft surface)

Fig4 Reflection Coefficient of sample 2

The above plots are produced using Matlab using the reflection coefficient and absorption coefficient equations mentioned in section 3.2 and 3.3.

Using equation (8) the surface impedance can be found . The surface impedance is a complex variable having both real and imaginary parts.

REAL AND IMAGINARY PARTS OF SURFACE IMPEDANCE as a function of frequency (hz)

Fig5 Real part of surface Impedance sample 1 (Hard surface)

Fig6 Imaginary part of surface impedance sample1

Fig7 Real part of surface impedance sample2 (Soft surface)

Fig8 Imaginary part of surface impedance sample2

PHASE ANGLE (Rad/s) AS A FUNCTION OF FREQUENCY (Hz)

Fig9 Phase angle vs. frequency Sample 1 (Hard surface)

Fig10 Phase angle vs. Frequency sample 2 (Soft surface)

OBSERVATIONS

Figures 1-4 show that both samples are inefficient at low frequencies and become more absorbent at higher frequencies. It is interesting to see that the first sample with the reflective hard surface is more efficient at lower frequencies than the first soft surface sample.

Sample 2 is reflective at 1 kHz and has a steep notch at the same frequency with absorption. The reactive component of surface impedance for sample 2 has a steep rise at 1 KHz (Fig8).

CONCLUSIOIN

The impedance tube method was used to measure the absorption characteristics of two samples. The effect of different surface treatments on porous absorbers changes the acoustic properties so that various kinds of absorbers specific at attenuating certain frequencies can be used to fine tune a reverberant space.

It was demonstrated that Impedance is a complex variable having both real (resistive) and imaginary(reactive) parts and that it is dictated by the wave number K and the distance away from the test sample boundary.

The results are based on insufficient data and cannot be verified due to the lack of quantifiable total error in readings taken.

REFRENCES

[1]

Daniel A Russell. (). Absorption Coefficients and Impedance. Available: http://www.google.co.uk/url?sa=t&rct=j&q=daniel%20a.%20russell%20absorption%20coefficients%20and%20impedance%20citation&source=web&cd=1&cad=rja&ved=0CC8QFjAA&url=http%3A%2F%2Fwww.acs.psu.edu%2Fdrussel. Last accessed 07/03/2013.

[2]

Trevor J.Cox and Peter D'Antonio. (2009). Measurement of Absorber Properties. In: Acoustic Absorbers and Diffusers. Oxon: Taylor and Francis. 70-82.

APPENDIX A

Table1 Sample 1 data

Table2 Test sample 2 data

APPENDIX B

MATLAB code used

SAMPLE 1:

maxV=[122e-3 40e-3 52e-3 11.1e-3 11e-3 29e-3 15.5e-3 21.5e-3 14e-3 2.7e-3 ];

minV=[5.5e-3 6e-3 7.7e-3 3.7e-3 2.6e-3 7.9e-3 6.1e-3 14e-3 9e-3 1e-3 ];

freq=[200 250 315 400 500 630 800 1000 1250 1600];

pMin=[39e-2 32e-2 23e-2 18.9e-2 15.5e-2 11e-2 7.7e-2 9.4e-2 8e-2 7.5e-2];

pMax=[85e-2 61e-2 47.9e-2 40e-2 31.7e-2 26.5e-2 19.2e-2 18e-2 17.5e-2 12.4e-2];

rho=1.21;

c=343;

zAir=rho*c;

SWR=(maxV./minV);

R=((SWR-1)./(SWR+1));

k=((2.*pi.*freq)./c);

theta=((2.*k.*pMin)-pi);

alpha=1-(abs(R).^2);

zs=((1+R)./(1-R)).*((zAir)./(exp(1i.*theta)));

figure

plot(freq,real(zs),'b','LineWidth',2);

grid on

xlabel('Frequency (Hz)')

xlim([200 1600]);

ylabel('Re(Zs)')

title('Sample 1 - Real part of Surface Impedence','FontSize',14)

figure

plot(freq,imag(zs),'r','LineWidth',2);

grid on

xlabel('Frequency (Hz)')

xlim([200 1600]);

ylabel('Im(Zs)')

title('Sample 1 - Imaginary part of Surface Impedence','FontSize',14)

figure

plot(freq,alpha,'k','LineWidth',2);

grid on

xlabel('Frequency (Hz)')

xlim([200 1600]);

ylabel 'Absorption coefficient '( \alpha )

title('Sample 1 - Absorption Coefficient','FontSize',14)

figure

plot (freq,R,'m','LineWidth',2);

grid on

xlabel('Frequency (Hz)')

xlim([200 1600]);

ylabel ('Reflection coefficient (R)')

title('Sample 1 - Reflection Coefficient ','FontSize',14)

figure

plot (freq,theta,'m','LineWidth',2);

grid on

xlabel('frequency (Hz)')

xlim([200 1600]);

ylabel ('theta (rad/s)')

title('Sample 1 - Theta Vs Frequency ','FontSize',14)

SAMPLE 2

maxV=[290e-3 85e-3 128e-3 44e-3 111e-3 109e-3 101e-3 435e-3 84e-3 8e-3 ];

minV=[8e-3 4e-3 6.2e-3 4.6e-3 10.4e-3 12.1e-3 14e-3 7.3e-3 26e-3 3.3e-3 ];

freq=[200 250 315 400 500 630 800 1000 1250 1600];

pMin=[39e-2 30.1e-2 23.7e-2 17.1e-2 12.7e-2 9.5e-2 8.2e-2 3.7e-2 2.3e-2 10.8e-2];

pMax=[77.1e-2 59.2e-2 44.5e-2 41.5e-2 28.9e-2 20.6e-2 15.9e-2 14.4e-2 9.2e-2 15.8e-2];

rho=1.21;

c=343;

zAir=rho*c;

SWR=(maxV./minV);

R=((SWR-1)./(SWR+1));

k=((2.*pi.*freq)./c);

theta=((2.*k.*pMin)-pi);

alpha=1-(abs(R).^2);

zs=((1+R)./(1-R)).*((zAir)./(exp(1i.*theta)));

figure

plot(freq,real(zs),'b','LineWidth',2);

grid on

xlabel('Frequency (Hz)')

xlim([200 1600]);

ylabel('Re(Zs)')

title('Sample 2 - Real part of Surface Impedence','FontSize',14)

figure

plot(freq,imag(zs),'r','LineWidth',2);

grid on

xlabel('Frequency (Hz)')

xlim([200 1600]);

ylabel('Im')

title('Sample 2 - Imaginary part of Surface Impedence','FontSize',14)

figure

plot(freq,alpha,'k','LineWidth',2);

grid on

xlabel('Frequency (Hz)')

xlim([200 1600]);

ylabel 'Absorption coefficient '( \alpha )

title('Sample 2 - Absorption Coefficient','FontSize',14)

figure

plot(freq,R,'m','LineWidth',2);

grid on

xlabel('Frequency (Hz)')

xlim([200 1600]);

ylabel('Reflection coefficient (R)')

title('Sample 2 - Reflection Coefficient','FontSize',14)

figure

plot (freq,theta,'m','LineWidth',2);

grid on

xlabel('frequency (Hz)')

xlim([200 1600]);

ylabel ('theta (rad/s)')

title('Sample 2 - Theta Vs Frequency ','FontSize',14)