INTERFERENCE OF SOUND WAVES

INTERFERENCE OF SOUND WAVES

PURPOSE: To measure the wavelength, frequency, and propagation speed of ultrasonic sound waves and to observe interference phenomena with ultrasonic sound waves.

APPARATUS: Oscilloscope, function generator, transducers, meter stick, angle board.

INTRODUCTION: In this experiment we deal with sound waves, produced by and detected with ultrasonic transducers. Sinusoidal waves can be characterized by the following parameters:

Wavelength: ...

Frequency: ...f Period: ...T = 1/f

Wave propagation speed: ...c = f

=

/T The speed of sound through air at 20 °C = 344 m/s

Ultrasonic transducers: A transducer is a device that transforms one form of energy into another, for example, a microphone (sound to electric) or loudspeaker (electric to sound). In this experiment the transducer is a "piezoelectric" crystal which converts electrical oscillations into mechanical vibrations that make sound. The piezoelectric material contracts (or expands) a small amount when a voltage is applied across the crystal. Conversely, if its dimension is changed, it produces voltage. The crystal has a natural resonance frequency, like a bell, at which it will vibrate when struck. If the frequency of the voltage applied to the piezoelectric crystal is the same as its natural frequency, the crystal will settle into steady large amplitude oscillations that produce high intensity sound waves. The oscillating frequency of the transducers you will use is near 40 kHz which is beyond what can be heard by the human ear (about 20 kHz).

Oscilloscope: The oscilloscope is an electronic device that acts as a voltmeter that can respond very rapidly to changes in the applied voltage. It is used here to display a graph of the instantaneous voltage applied to the crystal as a function of time.

Interference of Waves: Figure 1, on the next page, is a drawing of the basic concept of interference of coherent waves from two point sources. S1 and S2 are wave sources oscillating in phase (because the two transducers are driven by the same voltage signal generator) and separated by distance d. P is the place where we place a detector. At point P the path difference to S1 and to S2, is the distance S2P - S1P. When this path difference is an integral multiple of the wavelength

, waves arriving at P from S1 and S2 will be in phase and will interfere constructively.

S2P - S1P = n

(1)

where n = 0, 1, 2,··· is referred to as the order of the particular maximum. Note that constructive interference gives maximum intensity:

If we can assume that S1P and S2P >> d and

d sin



^{. (2)}

Figure 1 PROCEDURE:

1. Measuring frequency: The setup is shown in Fig. 2: a variable frequency signal generator drives one ultrasonic transducer; its output is also applied to channel B of the oscilloscope.

(Later in the experiment two transmitting transducers will be connected to channel B.) The output of a second receiving ultrasonic generator is applied to channel A of the scope. Channel B’s trace (pattern of the oscilloscope) shows the sinusoidal voltage applied to the transmitting generator; channel A's trace shows the sinusoidal voltage produced by the receiving ultrasonic crystal.

Set the signal generator to a frequency of 40 kHz. Adjust the scope controls (trigger, beam intensity, vertical amplification and horizontal sweep rate) so that trace A shows several, steady cycles of the sine waves. Place the transmitting transducer facing the receiver at a distance of a few centimeters. If you think the transducers are not functioning (nothing on trace A), it is most likely that the function generator is not set at the exact resonance frequency. Vary the frequency around 38 to 42 kHz. You should tune to resonance when the signal getting through to the receiving transducer (trace A) reaches a maximum.

Warning: Do not exchange your transducers with those from other tables; your

three transducers are a matched set.

Measure the period of oscillation directly from the oscilloscope's screen, using the sweep rate (milliseconds per centimeter) marked on oscilloscope's sweep control. To make the period measurement as accurate as possible, measure the time interval corresponding to several complete oscillations. Compare the frequency f

determined with the oscilloscope with the frequency fsg of the signal generator.

Figure 2

2. Measuring wavelength: The transmitting and receiving transducer stands fit over, and can slide along, a meter stick. With both transducers fixed in position, the two sinusoidal traces on the scope are steady. What happens to the scope trace from the receiving transducer when you move the receiving transducer away from the transmitting transducer?

Measure the wavelength by slowly shifting the receiving transducer a known distance away from the transmitter while noting on the oscilloscope screen by how many complete cycles of relative phase the wave pattern shifts. Don't choose just one cycle, but as many cycles as can conveniently be measured along the meter stick.

Use the measured period of ultrasonic oscillations from Part 1 and the wavelength from Part 2 to compute the speed of sound through air. The oscillation period measured with the scope sweep calibration is more accurate than the frequency readings on the signal generator. Compare your computed value with the standard value of 344 m/s for dry air at 20 °C.

3. Double Source Interference:

A. The setup will be similar to Fig. 2, but another transmitting transducer will be added. The pair of transmitters are placed side-by-side and driven in phase by a signal generator; a third receiving transducer is at an angle

which can be varied. See Fig. 3, which duplicates the arrangement shown in Fig. 1.

Dual Trace Oscilloscope

Function Generator Chnl B

Chnl A

Receiver Transmitter

We are interested in observing the amplitude of the resultant ultrasonic wave reaching the receiving transducer. When you move the detector you are receiving the combined intensity of the two interfering waves. Sometimes you will see a strong signal while other times little or none. Move the receiving transducer along a circular arc, maintaining a constant distance from the two transmitting transducers. Record the transmitter separation d which should be kept as small as possible, and the angular positions

max for interference maxima. Suggestion: rather than plotting

max readings directly from the protractor, try taking corresponding left-side and right-side values and average them eliminating any error in judging where the center line is.

Figure 3

Confirm the constructive interference relation, n

= d sin

max, by plotting sin

max as a function of the integer n. (Take values for n and

max on the right side as positive and those on the left as negative so your plot is a straight line (sin

= -sin

) rather than a "V"). The slope of your best straight line will enable you to calculate (

), and then the separation d in terms of the theoretical wavelength

= c/f.

B. Now we will use a setup with a different geometry where Eq. (2) does not apply. Set up the two transmitters, separated a distance d as shown in Fig. 4, and the receiver at P a distance z from S1. Keeping S1 and P fixed vary the separation d (by tilting the support for S

) and record the values of d for which maxima and minima in intensity are observed at the receiver. Use Eq.

(1) (appropriately modified) to calculate the expected values and compare with your data.

Figure 4

4. Other Interference Effects(Make qualitative observations. Report on this section is not required):

Now we use only one transmitting transducer, and one receiver.

Figure 5

See Fig. 5A which is a view of the setup from above. Both a single transmitting and a receiving transducer are fixed in position about 75 cm apart. A flat object, such as the side of book, that can act as a "mirror" for ultrasonic waves is oriented parallel to the line joining the transducers, and is moved towards and away from the meter stick. Observe on the scope the resulting wave signal at the receiving transducer. What do you see? How do you account for the effect? What happens when you block the direct path with paper? (Keep your body away to avoid reflection effects.) You may also keep the "mirror" fixed in position and vary transmitter- receiver separation, starting at about 50 cm. What do you observe?

See Figure 5B. Remove the "mirror". With the transmitting transducer 4 cm away facing the receiver, slowly change the separation. Note that amplitude variations are observed as the separation changes (two per wavelength shift of separation-why?). The maxima correspond to sound antinodes at the ends of the space when linear boundary conditions are satisfied, as with strings or sound tubes excited at a fixed frequency. Count several amplitude variations and record the corresponding separation shift. Calculate the wavelength

from your amplitude observations and compare with your previous determination from phase variations.

5. Optional

A. In the discussion in the introduction we noted that when there is a maximum in the intensity of the sound, constructive interference occurs. Intensity is the sound power per unit area. On the other hand, sound is a pressure wave. That is, as the sound wave passes the average air pressure oscillates about its average value. The two quantities are related in that the intensity is proportional to the square of the amplitude of the pressure oscillation. Which quantity does the ultrasonic transducer measure -- intensity or pressure amplitude? Explain specifically what it is about your data that motivates your answer.

B. In Fig. 1 we make an implicit assumption about the angular variation of the intensity of

the sound waves emitted by S1 or S2. What is this assumption? Set up the apparatus as shown

in Fig. 6 and record the signal as a function of angle. (Up to now you have only looked for

maxima and minima.) Plot the signal versus angle. On your plot show the predicted angular dependence implicitly assumed in Fig. 1. Discuss why the actual transmitter signal has a different angular dependence from the expected value.

Figure 6

INTERFERENCE OF SOUND WAVES

Name: ____________________________________________________ Section: _______

Partners: ____________________________________________________ Date: _______

Include units for numerical values where appropriate.

1) Frequency Measurement

Oscilloscope Time Base Per Div (TB): __________

Number of Waves Counted on Screen (NW):_________

Number (& fractional parts) of Divisions Covered by Waves (ND): ________

Period of One Wave = TB * ND / NW = _________ frequency(f

) = _________

frequency output of signal generator(f

) = _________

fractional difference (100x|f

– f

|/f

) = _________ % 2) A. Wavelength Measurement:

Number of Waves Moved on Oscilloscope Nw: _________

Initial Position of Movable Sensor Pi: ___________

Final Position of Movable Sensor Pf: ___________

Distance sensor moved, Pi - Pf = D: _____________

Length of One Wave, Wavelength = D/Nw : ___________

2) B. Speed of Sound: use your wavelength and frequency to calculate c.

c =

f

= __________ Ratio c/(344m/s) = ________

3) A. Double Source Interference of Sound: Separation of Sources, d: ________

 max sin( max) n

Use Graphical Analysis on the computer to graph sin(

max) vs. n. Do your data support the

assumption that Eq. (2) is applicable? Explain.

Determine the slope of line fitted to the data.

slope

/d _______ Use the measured d to get

_______

Compare the wavelength measured on the oscilloscope (Part 2) with the value from interference (Part 4):

R (ratio) = 

oscilloscope)

interference) = __________

3) B. Show a sample calculation for the theoretical values for d.

d(max) exp d(max) thy d(min) exp d(min) thy n