Ultrasonic diffraction at different single and double slit systems
Related topics
Longitudinal waves, Huygens’ principle,
Fraunhofer and Fresnel diffraction.
Interference,
Principle
A plane ultrasonic wave is subjected to diffraction at single
slits of various widths and at various double slits. The intensity of the diffracted and interfering partial waves are automatically recorded using a motor-driven, swivel ultrasound detector and a PC.
Equipment
Goniometer with reflecting mirror
Power supply for goniometer
Ultrasonic unit
Power supply f. ultrasonic unit, 5 VDC, 12 W
Ultrasonic transmitter on stem
Ultrasonic receiver on stem
Object holder for ultrasonic
Diffraction objects for ultrasonic
RS 232 data cable
Measuring tape, l = 2m
Screened cable, BNC, l = 75 cm
Adapter, BNC-socket/4 mm plug pair
Measure Software Goniometer
PC, Windows® 95 or higher
13903.00
13903.99
13900.00
13900.99
13901.00
13902.00
13904.00
13905.00
14602.00
09936.00
07542.11
07542.27
14523.61
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Tasks
1. Record the intensity of an ultrasonic wave diffracted by various slits and double slits as a function of diffraction angle.
2. Determine the angular positions of the maximum and minimum values and compare them with the theoretical results.
Set-up and Procedure
Set up the experiment as shown in Fig. 1. Exact adjustment of
the experimental set-up is important!
Adjustment of the goniometer:
– Use the adjusting screws at the back of the mirror and
under its stem to set the mirror by eye to a vertical position
and align it to the zero line of the goniometer table.
– Slide the transmitter tightly against the mirror and align it
to the height of the centre of the mirror.
– Slide the transmitter back to fit the 16 cm long adjusting
rod in the hole in the centre of the mirror. The rod must
point directly to the middle of the transmitter. Should this
not be the case, again use the adjusting screws to readjust the mirror. Remove the rod so that the transmitter
can be brought to the focal point of the mirror. The distance from the centre of the mirror must be exactly
15.5 cm (measuring tape).
Fig. 1: Experimental set-up.
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Ultrasonic diffraction at different single and double slit systems
To adjust the height of the receiver, turn it with its swing
arm as near as possible to the mirror. It might be necessary here to first unlock the transport stop (to do this, pull
the yellow screw of the swing arm beneath the goniometer
table). Bring the receiver to the same height as the transmitter.
Set the receiver swing arm to zero. The axis of the receiver must correspond with the goniometer zero line.
Bring the receiver to the end of the swing arm.
When adjustment is properly made, the axes of the mirror,
transmitter and receiver must be on a common line and
this must be exactly above the zero line of the goniometer
table.
Fit the object holder with its centering pin in the central
socket of the goniometer table, with the feet of the holder
pointing to the mirror. Position the holder on the 90° line on
the goniometer table.
To prepare the slit and double slit, carefully insert the corresponding metal sheets in the guide grooves of the object
holder, then align these exactly centrally symmetrical with
the zero line of the goniometer table.
To avoid interfering sound reflections, use the carrier foam
as wave absorbent. Place it between the object holder and
mirror, tightly against the holder and with its opening symmetrically towards the diffraction object.
Connection of instruments:
Connect the transmitter to the diode socket of the ultrasonic
unit that is marked TR1, and operate it in “Con“ continuous
mode. Connect the receiver to the left BNC socket (prior to the
amplifier). Further, use the BNC cable to connect the analog
output of the ultrasonic unit with the input of the control unit
(pay attention to the polarity of the adapter), and the latter unit
to the PC by means of the RS 232 data cable.
For control of the goniometer, connect the socket underneath
the goniometer plate with the control unit.
With the “Cal“ key of the control unit pressed (release of the
motor drive) position the swing arm at 0°. Following this,
deactivate the “Cal“ function.
Use the software to set the range of swing of the receiver to
±50°.
To ensure proportionality between the input signal of the
receiver and its analog output signal, avoid operating the
ultrasonic unit amplifier in the saturation range. Should such a
case occur and the “OVL“ diode light up, reduce either the
transmitter amplitude or the input amplification of the receiver. It is purposeful here to adjust the amplification at the zero
position of the receiver so that the “OVL“ diode just no longer
lights up.
5 Measurements are described in the following.
1st Measurement (see Fig. 3):
Interference pattern of a slit of width b = 6 cm.
2nd Measurement (see Fig. 4):
Interference pattern of a slit of width b = 4 cm.
3rd Measurement (see Fig. 6):
Interference pattern of a double slit of slit width b = 2.5 cm
and slit separation s = 5.5 cm.
4th and 5th Measurement (see Fig. 7):
Set the measurement mode to repeat measurement.
Interference pattern of a double slit of slit width b = 2.0 cm
and slit separation s = 5.0 cm. Subsequently repeat measurement with a single slit of width b = 2.0.
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Note:
Faulty intensity modulation may occur in spectra as a result of
interference in the measurement field. To keep such interference as small as possible, do not carry out experiments in too
narrow rooms or in the direct vicinity of reflecting surfaces
(walls, cupboards etc.). It is recommended that the measuring
and supply instruments be installed behind the mirror if possible. Further to this, the person carrying out the experiment
should not stand too close to the measurement field.
Should asymmetries occur in the intensities in spectra, these
can as a rule be avoided by slightly turning the object holder
around the 90° line on the goniometer table.
Theory and Evaluation
When a wave hits a slit then, acc. to Huygens’ principle,
spherical waves are emanated from each point of the slit
opening. The individual partial waves interfere with each other
behind the obstacle. According to their phase position, they
intensify each other in certain directions, or extinguish each
other. In the direction of the incident waves (w = 0) (see Fig. 2),
all partial waves have the same phase and intensify each
other.
Fig. 2: Diagram of diffraction at a slit.
With sound waves, the sound pressure p(w) is represented as
a function of the diffraction angle w by the so-called slit function:
p1w 2 b
pb
sin w
pb
l
sin u
using u sin w (1)
b
pb
u
l
sin w
l
sin
Where b is the slit width, l the wavelength of the sound and p
the alternating sound pressure that is recorded by the sound
receiver.
Equation (1) is also valid, in squared form, for transversal electromagnetic waves (optics), as in this case the intensity is
given by the square of the amplitude.
For w = 0, an indefinite expression is obtained, as both the
numerator and the denominator are null. Application of the
l’Hospital rule shows, however, that for w = 0, the quotient
assumes the value 1. Zero positions are at sin u = 0, i.e. at
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
Ultrasonic diffraction at different single and double slit systems
u = np (n = 1, 2, 3,...). From this, the positions of the minima
are given by:
w1n2 min arcsin
nl
b
where n = 1, 2, 3,...
(2)
The numerator in equation (1) becomes 1 when u = 1/2(2n+1)l,
i.e. when u is an uneven multiple of p/2. Intensity maxima
therefore lie at:
w1n2 max arcsin
2n 1 nl
·
2
b
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As the central maximum does not generally lie exactly at 0°, it
is purposeful to determine the angular distance 2 w between
two extremes lying symmetrically to the zero line. Table 1 lists
both 2 w angle of extremes determined from Fig. 3 and Fig. 4,
as well as the wavelength values l calculated using equations
(2) and (3). The In/I0 ratios are also listed.
Table 1: Evaluation of the interference curves shown in Figs. 3
and 4.
Split width b = 6 cm, see Fig. 3:
where n = 1, 2, 3,...
(3)
From equation (1) it also follows for the ratio of the intensities
of the maxima that: I1/I0 = 0.21, I2/I0 = 0.13, I3/I0 = 0.09,
I4/I0 = 0.07. Further to this, I0 r b = is valid.
Fig. 3 and Fig. 4 each show interference patterns of ultrasonic waves at single slits, but with different split widths b. The
two curves were recorded using the same transmitter performance.
Maxima
Minima
n
2w/°
l/mm
In/I0
2w/°
l/mm
1
24.1
8.35
0.15
18.1
9.44
2
40.7
8.35
0.11
32.7
8.45
3
59.2
8.47
0.09
50.7
8.56
4
78.7
8.45
0.07
69.4
8.54
90.9
8.55
5
Split width b = 4 cm, see Fig. 4:
Maxima
Fig. 3: Interference pattern of ultrasonic waves diffracted at a
slit of b = 6 cm.
Minima
n
2w/°
l/mm
In/I0
2w/°
l/mm
1
37.4
8.55
0.17
27.1
9.37
2
63.9
8.47
0.10
52.3
8.81
3
96.9
8.55
0.07
77.8
8.37
The mean value of the wavelength values listed in Table 1 is:
l = (0.862±0.034) cm.
The transmitter operates at a frequency of f = 40 kHz. From
c = f · l (c = 343.4 ms-1 at T = 20°C) it follows that, in complete agreement with the experiment, l = 0.858 cm.
The curves in Figs. 3 and 4 also show that, at a constant emitter performance, the intensity of the maximum of zero order
becomes less when the slit width is reduced. For the corresponding quotients the experiment gives:
I0(6 cm)/I0(4 cm) = 3.45 V/2.63 V = 1.3 b(6 cm)/ b(4 cm) = 1.5.
Fig. 5 shows a diagram of diffraction at a double slit. The slit
width is again given by d and the distance apart of homologous partial waves by the slit separation s.
Fig. 4: Interference pattern of ultrasonic waves diffracted at a
slit of b = 4 cm.
Fig. 5: Diagram of diffraction at a double slit.
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Ultrasonic diffraction at different single and double slit systems
With a double slit, the intensity distribution is given by:
p
p1w2 r 2 cos a s · sin w b ·
l
pb
sin w b
l
pb
a
sin w b
l
sin a
Table 2: Evaluation of the interference curve shown in Fig. 6.
Double split s = 5.5 cm and b = 2.5 cm.
(4)
The slit function is now additionally modulated by a cos function. The minima of the individual slits (1st class minima) are
still as previously. Additional 2nd class minima occur, however, namely there where the cos function is null. This is always
the case for:
2k 1
w 1k2 min arcsin
· l where k = 0, 1, 2, 3,... (5)
2s
Additional 2nd class maxima occur when the cos factor is 1,
i.e. when:
w 1k 2 max arcsin
k·l
s
where k = 0, 1, 2, 3,... (6)
Fig. 6 shows the interference pattern of a double slit having
s = 5.5 cm and b = 2.5 cm.
Table 2 shows the corresponding evaluation and the values for
l calculated from equations 5 and 6. A mean value of l =
(0.848±0.035) cm is obtained. In Fig. 6, at w 20°, a slight
indentation is to be seen, that is concordant to the position of
the first minimum of the single slit.
Fig. 6: Interference pattern of ultrasonic waves diffracted at a
double slit of s = 5.5 cm and b = 2.5 cm.
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Maxima
k
Minima
2w/°
l/mm
0
2w/°
l/mm
9.0
8.63
1
16.9
8.08
27.6
8.75
2
32.9
7.79
47.1
8.79
3
57.0
8.75
67.0
8.67
4
74.1
8.28
89.5
8.60
With a little skill, the dimensions of the double slit can now be
so chosen, that the 1st minimum of the single slit (1st class
minimum) coincides with a minimum of the double slit (2nd
class minimum). This is the case, for example, when s = 5/2 b,
as the 1st minimum of the single slit then coincides with the
3rd minimum of the double slit. Fig. 7 shows such a case, with
the recording of the double slit curve superimposed on the
single slit curve.
It can be seen that the single slit curve envelopes the double
split curve.
Fig. 7: Interference pattern of ultrasonic waves diffracted at a
double slit of s = 5.0 cm and b = 2.0 cm. (The single slit
curve always envelopes the double split curve)
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen