Hasan G Pasha
ME06M045
DETERMINATION OF CO-EFFICIENT
WAVE APPARATUS
OF
ABSORPTION
USING
STANDING
OBJECTIVE
Determine the coefficient of absorption of the given acoustic sample
Plot the absorption coefficient as a function of frequency
APPARATUS
1. Standing wave apparatus
2. Frequency analyzer
3. Function/Waveform generator
4. Audio amplifier
THEORY ON STANDING WAVES
Standing Wave
A standing wave is a wave that remains in a constant position. It is also termed as a stationary
wave. This phenomenon can occur due to any one of the following conditions:
The medium is moving in the opposite direction to the wave
Interference between two waves traveling in opposite directions in a stationary medium
Waves traveling in opposite directions
Standing waves are observed in physical media such as strings and columns of air. Any waves
traveling along the medium will reflect back when they reach the end.
A standing wave is formed by the superposition of two waves propagating in opposite
directions. The effect is a series of nodes (zero displacement) and anti-nodes (maximum
displacement) at fixed points along the transverse axis.
Displacement of Standing wave
A traveling wave moves from one place to another. In stark contrast a standing wave appears
to stand still, vibrating in place. If it is assumed that two waves (with the same amplitude,
frequency, and wavelength) are traveling in opposite directions on a string, the using the
principle of superposition, the resulting string displacement may be written as:
y (x, t)
= ym sin (kx - ωt) + ym sin (kx + ωt)
⇒ 2 ym sin (kx) cos (ωt)
This wave is no longer a traveling wave because the position and time dependence have been
separated. The displacement of the string as a function of position has an amplitude of 2 ym sin
(kx). This amplitude does not travel along the string, but stands still and oscillates up and down
according to cos (ωt) with maximum displacement (antinodes) and locations with zero
displacement (nodes).
Standing Wave Ratio
In practice, due to losses a perfect reflection and a pure standing wave are never achieved. The
result is a partial standing wave, which is a superposition of a standing wave and a traveling
wave. The degree to which the wave resembles either a pure standing wave or a pure traveling
wave is measured by the Standing Wave Ratio (SWR).
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Hasan G Pasha
ME06M045
Standing Wave Ratio (SWR) is the ratio of the amplitude of a partial standing wave at an antinode (maximum) to the amplitude at an adjacent node (minimum). In other words the ratio of
the pressure maximum (anti-node) to the pressure minimum (node) is termed the standing
wave ratio SWR.
SWR always has a value greater than or equal to unity. It is used to determine the reflection
co-efficient amplitude (R), absorption co-efficient (α) and impedance (Z) of any acoustic
sample.
Theory
Assuming that a pipe of cross-sectional area S and length L is driven by a piston at x=0. If the
piston vibrates harmonically at a frequency sufficiently low that only plane waves propagate.
For a circular waveguide (pipe) filled with air, the highest frequency at which only plane waves
will propagate is given by fmax =100/ a where ‘a’ is the radius of the waveguide.
The pressure wave in the pipe will be of the form given below:
A and B are determined by the boundary conditions at x=0 and x=L.
By substituting
The pressure amplitude is given by
When the pipe has a rigid termination, the entire sound energy incident upon the termination is
reflected with the same amplitude.
Pipe with rigid termination
When the pipe is terminated with acoustic absorbing material, some of the incident sound
energy is absorbed by the material and the reflected waves do not have the same amplitude as
incident waves. In addition the absorbing material introduces a phase shift upon reflection.
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Hasan G Pasha
ME06M045
Pipe terminated acoustic absorbing material
The amplitude at a pressure anti-node (maximum pressure) is A+B, and the amplitude at a
pressure node (minimum pressure) is A-B. It is not possible to measure A or B directly.
However, the amplitude at a pressure node and anti-node can be measured using a microphone
probe which is set in a standing wave tube.
SWR = (A + B) / (A - B)
R=B/A
= (SWR + 1) / (SWR - 1)
α = 1 – R2
= 1 – (SWR - 1)2
(SWR + 1)2
A pressure minimum occurs when
or
⇒
(L-x)
→
Distance from the test sample to the first pressure minimum (n=1)
A+B
→
Amplitude of pressure anti-node
A-B
→
Amplitude of pressure node
The sound power absorption coefficient for the test sample at a given frequency is given by:
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Hasan G Pasha
ME06M045
SCHEMATIC DIAGRAM
PICTURE
Standing wave apparatus and analyzers
DESCRIPTION
Standing Wave Apparatus is as shown in the figure.
The apparatus consists of a pipe (P) with one end containing the acoustic sample (S) and the
other end connected with a loudspeaker (L).
The loudspeaker produces an acoustic wave which travels down the pipe and reflects from the
acoustic sample.
The phase interference between the waves in the pipe which are incident on and reflected by
the acoustic sample will result in the formation of a standing wave pattern in the pipe.
If 100% of the incident wave is reflected, then the incident and reflected waves have the same
amplitude; the nodes in the pipe have zero pressure and the antinodes have double the
pressure.
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Hasan G Pasha
ME06M045
However this is not the ideal case, as some of the incident sound energy is absorbed by the
sample. The incident and reflected waves have different amplitudes and the nodes in the pipe
no longer have zero pressure.
The pressure amplitudes at nodes and anti-nodes are measured with a microphone probe (M)
attached to a car (C) which slides along a graduated ruler.
PROCEDURE
1. Set the frequency of the wave generator to 250 Hz
2. Move the microphone probe using the microphone car such that the maximum displacement
of the pressure wave is (amplitude of pressure anti-node) is observed in the wave analyzer,
tabulate the corresponding readings
3. Move the microphone probe using the microphone car such that the minimum displacement
of the pressure wave is (amplitude of pressure node) is observed in the wave analyzer,
tabulate the corresponding readings
4. Set the frequency subsequently to 350, 500, 700 and 1000 Hz and repeat steps 1 through 3
5. Using the formulae calculate the standing wave ratio and the absorption co-efficient of the
given acoustic sample
6. Plot the absorption co-efficient of the given acoustic sample as a function of the frequency
PLOT
A graph is plotted with the absorption values (ordinate) as a function of frequency (abscissa).
FORMULAE
SWR
=
(A + B) / (A - B)
R
=
B/A
=
(SWR + 1) / (SWR - 1)
=
1 – R2
=
1 – (SWR - 1)2
(SWR + 1)2
α
SWR
→
Standing Wave Ratio
R
→
Sound reflection co-efficient
α
→
Absorption co-efficient
A+B
→
Amplitude of pressure anti-node
A-B
→
Amplitude of pressure node
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Hasan G Pasha
ME06M045
SAMPLE CALCULATION
A+B
=
0.05
V
A-B
=
0.018
V
SWR
=
=
=
(A + B) / (A - B)
0.05/0.018
2.7778
R
=
=
=
(SWR + 1) / (SWR - 1)
(2.7778 + 1) / (2.7778 – 1)
0.4706
α
=
=
=
1 – R2
1 – (0.4706)2
0.7785
TABULATION
Sl
No
Frequency
Hz
Amplitude
of
pressure
node
Location
of
minimum
pressure
Amplitude
of
pressure
anti-node
Location
of
maximum
pressure
(A-B) V
cm
(A+B) V
cm
Standing
Wave ratio
Reflection
co-efficient
Absorption
co-efficient
SWR
R
α
1
250
0.018
29.5
0.05
65.6
2.7778
0.4706
0.7785
2
350
0.024
19.5
0.12
44.5
5
0.6667
0.5556
3
500
0.036
48.1
0.135
66
3.75
0.5789
0.6648
4
700
0.036
8.5
0.09
21.3
2.5
0.4286
0.8163
5
1000
0.025
5.9
0.043
14.1
1.72
0.2647
0.9299
Mean value of absorption co-efficient: 0.7490
RESULT
The absorption coefficient of the given acoustic sample was determined. The mean value of
the absorption co-efficient is 0.7490.
A plot with the absorption coefficient as a function of frequency was generated
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