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Experiment 1

Velocity of Sound

Apparatus:

Audio frequency generator, speaker, microphone, Cathode Ray Oscilloscope (CRO), meter scale,

large board (or wall), Thermometer (0-100C)

Purpose of experiment:

(i) To determine the velocity of sound (ii) To understand the operation of a CRO

Basic methodology:

Sound waves produced by an audio frequency generator connected to a speaker, are made to

reflect off a large reflecting board, forming standing waves. A pickup microphone connected

to a CRO serves to measure the amplitude of the sound. The wavelength of sound waves isobtained from the positions of the nodes.

I Theory

Standing waves are produced when two progressive sinusoidal waves of the same amplitude and

wavelengths, travel in opposite directions and superpose on each other. Consider two travelling

waves moving along the positive and negative x-directions respectively:

y1(x, t) = A sin(kx t) (1)

y2(x, t) = A sin(kx + t) (2)

where A is the amplitude of the waves, k = 2/is the wave number and = 2f is the

angular frequency. y(x, t) is the displacement of the medium at the point x and time t. By

the principle of superposition, the net displacement at any point is the sum of the individual

displacements. Thus,

y(x, t) = y1(x, t) + y2(x, t) = A sin(kx + t) + A sin(kx t) (3)

= y(x, t) = 2 sin kx cos t) (4)

The resulting displacement, eq. (3), represents a (standing) wave of frequency , and an

amplitude 2A sin kx, which varies with the position x. The amplitude is zero for values of kx

that give sin kx = 0. These values are

kx = n, n = 0, 1, 2,... (5)

Now k = 2/. Therefore,

x =n

2, n = 0, 1, 2 (6)

represents the positions of zero amplitude. These points are called nodes. Adjacent nodes are

separated by a distance /2, half a wavelength. The amplitude of the standing wave has a

maximum value of 2A, which occurs for values of kx that give sin kx = 1. These values are

kx = (n +1

2), n = 0, 1, 2,... (7)

1

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2 PHYSICS LABORATORY MANUAL

The positions of maximum amplitude are called antinodes of the standing wave. The antinodes

are separated by 2

and are located half way between pairs of nodes. Now, a sound wave is a

longitudinal wave caused by displacement of particles in the medium and the resulting pressure

variations. Travelling sound waves can also be represented by eqs. (1) and (2) with y(x, t)

denoting the longitudinal displacements of air particles. The velocity of the sound wave will begiven by

v = k = f (8)

In this experiment standing waves of sound are formed in air. The distance between successive

nodes or antinodes is /2. By measuring the distance between the nodes, i the wavelength can

be determined and hence the velocity v of sound (knowing the frequency f). The apparatus of

the experiment includes a cathode ray oscilloscope (CRO). One of the aims of the experiment

is to gain familiarity with the use of a CRO. The main features and controls of a CRO are

described in the appendix to this experiment.

II Set-up and Procedure:

C.R.O.

Sine Wave GeneratorMic

ReflectorA F

Scale

Speaker

Figure 1: Setup

The wavelength of sound waves will be measured by two different methods:

METHOD A

1. Connect the audio frequency generator to the loudspeaker (L) and adjust the controls of

the generator to produce a sine wave in the frequency range 1-10 kHz. Place the loud

speaker facing a large board B (or a wall) at a distance of about one meter.

2. Connect the microphone (M) to a channel of the CRO.

3. Select a proper scale for the horizontal time base to observe a stationary sinusoidal trace

on the screen of the CRO. Adjust the vertical and horizontal positions so that the trace

is symmetrically positioned on the screen.

4. Observe that the amplitude of the trace changes as the position of the microphone is

varied along the scale fixed to the table. Measure the period of the signal by reading the

number of horizontal divisions separating the minima of the signal on the CRO screen.

5. From the chosen scale for the time base, determine the time interval between successive

minima of the trace and hence calculate the frequency of the signal and compare with the

frequency generated by the audio generator.

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Experiment 1. Velocity of Sound 3

6. Next, start with the microphone close to the wall, and move it gradually away from the

wall. Note down the positions of the nodes, i.e., where the amplitude of signal on the

CRO screen becomes minimum. Note down the positions of at least five successive nodes.

7. Repeat this measurement for three different frequencies.

METHOD B

1. Connect the output of the speaker to the other available channel of the oscilloscope and

set the CRO to X-Y mode.

2. Observe the Lissajous pattern produced on the screen. Move the microphone along the

bench and observe different Lissajous patterns. Sketch the observed patterns for the cases

when the phase difference between X and Y signal are 0, 90, 180, 270. (Note: due to

small amplitude of the signals the observed pattern may be small in size. Also due to

attenuation of the reflected signal, there can be distortion of the pattern).

3. Select any one pattern for a fixed phase difference. Starting from the position nearest

to the speaker, move the microphone gradually towards the wall and note the positions

where the selected pattern repeats. The distance between two successive such positions

corresponds to the wavelength of the sound wave.

Precautions:

1. Learn the functions of the various control knobs of the CRO before operating the CRO.

2. The intensity of the CRO spot/pattern should be set LOW especially when the spot isstationary to avoid damage to the fluorescent screen.

3. The microphone should be moved along the perpendicular direction to the loudspeaker.

4. It is advisable to measure the position of nodes rather than antinodes.

III Exercises and Viva Questions:

1. What is a travelling wave? Write down equations representing waves travelling along +ve

and ve x directions.

2. What is a standing wave? Explain by superposing appropriate travelling waves.

3. What are nodes and antinodes? Draw a rough diagram depicting the standing wave

formed in the experiment. Is the point at the reflection board a node or an antinode?

4. On what factors does the velocity of sound depend? What is the effect of temperature,

pressure and humidity on the velocity of sound?

5. What are Lissajous figures? Explain by construction how Lissajous patterns are produced

when two perpendicular oscillations of phase differences 0

, 90

, 180

, 270

are superposed.

6. List the different sub-systems of a CRO and explain their operation and function.

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7. Explain how a CRO can be used for voltage, frequency and phase measurements? How

can a CRO be used for current measurement?

8. A periodic signal of 400 Hz is to be displayed so that 4 complete cycles appear on the

oscilloscope screen, which has 10 horizontal divisions. To what settings should the Trigger

source and sweep Time/div be set to allow this pattern to be displayed?

9. Explain what triggering, internal triggering and external triggering mean.

10. List the possible sources of error in this experiment and quantitatively estimate the error

caused in the velocity measurement.

References:

1. Fundamentals of Physics, 6th ed., D. Halliday, R. Resnick and J. Walker, John Wiley and

Sons Inc., New York, 2001.

2. Physics, M. Alonso and E.J. Finn, Addison Wesley, 1992.

Appendix: Cathode Ray Oscilloscope

A cathode ray oscilloscope (CRO) is a convenient and versatile instrument to display and

measure analog electrical signals. The basic unit of a CRO is a cathode ray tube (CRT) shown

in Figure ??. The cathode ray is a beam of electrons which are emitted by the heated cathode

Waveform

applied to horizontal

plates

Vs

screen

Voltage appliedto verticalplates

Sweep vs t trace on oscilloscope

Electron gun

Figure 2: Cathode Ray Tube

(negative electrode) and accelerated toward the fluorescent screen. The assembly of the cathode,

intensity grid, focus grid, and accelerating anode (positive electrode) is called an electron gun.

Its purpose is to generate the electron beam and control its intensity and focus. Between the

electron gun and the fluorescent screen are two pair of metal plates one oriented to provide

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horizontal deflection of the beam and the other pair oriented to give vertical deflection to the

beam. These plates are thus referred to as the horizontal and vertical deflection plates. The

combination of these two deflections allows the beam to reach any portion of the fluorescent

screen. Wherever the electron beam hits the screen, the phosphor is excited and light is emitted

from that point. This conversion of electron energy into light allows us to write with points orlines of light on an otherwise darkened screen.

In the most common use of the oscilloscope, the signal to be studied is first amplified and

then applied to the vertical (deflection) plates to deflect the beam vertically and at the same

time a voltage that increases linearly with time is applied to the horizontal (deflection) plates

thus causing the beam to be deflected horizontally at a uniform rate. The signal applied to the

verical plates is thus displayed on the screen as a function of time. The horizontal axis serves

as a uniform time scale.

The linear deflection or sweep of the beam horizontally is accomplished by use of a sweep

generator that is incorporated in the oscilloscope circuitry. The voltage output of such a gen-

erator is that of a sawtooth wave as shown in Fig. 2. Application of one cycle of this voltage

difference, which increases linearly with time, to the horizontal plates causes the beam to be

deflected linearly with time across the tube face. When the voltage suddenly falls to zero, as

at points (a) (b) (c), etc...., the end of each sweep the beam flies back to its initial position.

The horizontal deflection of the beam is repeated periodically, the frequency of this periodicity

is adjustable by external controls.

t

V

a b c d0

Figure 3: Sweep waveform

The CRO displays the signal as a voltage variation versus time graph on the CRT screen.

By properly interpreting the characteristics of the display the CRT can also be used to indicate

current, time, frequency and phase difference.

To obtain steady traces on the tube face, an internal number of cycles of the unknown signal

that is applied to the vertical plates must be associated with each cycle of the sweep generator.With such a synchronization of the two deflections, the pattern on the tube face repeats itself

and hence appears to remain stationary. The persistance of vision in the human eye and of the

glow of the fluorescent screen aids in producing a stationary pattern. In addition, the electron

beam is cut off (blanked) during flyback so that the retrace sweep is not observed.

CRO Operation

The basic subsystems of CRO are :

1. Display subsystem (CRT)

2. Vertical deflection subsystem

3. Horizontal deflection subsystem

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4. Power supplies

5. Calibration circuits

In general, the instrument is operated in the following manner. The signal to be displayed

Figure 4: Schematic diagram of CRO subsystems

is amplified by the vertical amplifier and applied to the vertical deflection plates of the CRT.

A portion of the signal in the vertical amplifier is applied to the sweep trigger as a triggering

signal. The sweep trigger then generates a pulse coincident with a selected point in the cycle

of the triggering signal. This pulse turns on the sweep generator, initiating the sawtooth wave

form. The sawtooth wave is amplified by the horizontal amplifier and applied to the horizontal

deflection plates. Usually, additional provisions are made for applying an external triggering

signal or utilizing the 60 Hz line for triggering. Also the sweep generator may be bypassed and

an external signal applied directly to the horizontal amplifier.

CRO Controls

(A) CRT

Focus: controls the focus of the spot on the screen

Intensity: Controls the brightness of the spot

X Shift/Y Shift: Changes the deflection voltages by a constant amount to shift the signal

vertically or horizontally.

(B) Vertical Deflection Subsystem

Vertical Sensitivity: Amplifier for the vertical deflection subsystem calibrated in terms of

sensitivity. The input voltage can be determined from the deflection of the signal. For eg., if

the vertical sensitivity is set at 50 mV/div and the vertical deflection is 4 div, then the input

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voltage is 50 4 = 200 mV.

Var (V/dir):Vernier control for continuous vertical sensitivity.

AC-DC-GND: Selects desired coupling (ac or dc) for incoming signal applied to vertical

amplifier, or grounds the amplifier input. Selecting dc couples the input directly to the amplifier;

selecting ac send the signal through a capacitor before going to the amplifier thus blocking anyconstant component.

(C) Horizontal Deflection Subsystem

Sweep time/cm: Selects desired sweep rate from calibrated steps or admits external signal to

horizontal amplifier.

Sweep time/cm Variable: Provides continuously variable sweep rates. Calibrated position

is fully clockwise.

Horizontal Variable: Controls the attenuation (reduction) of signal applied to horizontal

aplifier through Ext. Horiz. connector.

(D) TriggerSelects the timing of the beginning of the horizontal sweep.

Slope: Selects whether triggering occurs on an increasing (+) or decreasing (-) portion of

trigger signal.

Coupling: Selects whether triggering occurs at a specific dc or ac level.

Source: Selects the source of the triggering signal.

INT - (internal) - from signal on vertical amplifier

EXT - (external) - from an external signal inserted at the EXT. TRIG. INPUT.

LINE - 60 cycle trigger

Level: Selects the voltage point on the triggering signal at which sweep is triggered. It also

allows automatic (auto) triggering of allows sweep to run free (free run).

Operating Instructions:

Before plugging the oscilloscope into a wall receptacle, set the controls as follows:

1. Power switch at off

2. Intensity fully counter clockwise

3. Vertical centering in the center of range

4. Horizontal centering in the center of range5. Vertical at 0.2

6. Sweep times 1

Plug line cord into a standard ac wall receptacle. Turn power on. Do not advance the Intensity

Control.

Allow the scope to warm up for approximately two minutes, then turn the Intensity Control

until the beam is visible on the screen.

WARNING: Never advance the Intensity Control so far that an excessively

bright spot appears. Bright spots imply burning of the screen. A sharp focused

spot of high intensity (great brightness) should never be allowed to remain fixed

in one position on the screen for any length of time as damage to the screen may

occur.

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Lissajous Figures

When sine-wave signals of different frequencies are input to the horizontal and vertical amplifiers,

a stationary pattern is formed on the CRT, when the ratio of the two frequencies is an integral

fraction such as 1, 1/2, 2/3, 4/3, 1/5, etc. The nature of the pattern also depends on the relative

phases of the two signals. These stationary patterns are known as Lissajous figures and can

be used for comparative measurement of frequencies and phases. One can see how Lissajous

figures arise by superposing two SHMs in perpendicular directions (See Ex. (5)).

In this experiment, the two signals have the same frequency. The Lissajous patterns are

observed for phase differences of n/2.

References:

1. Student Reference Manual for Electronic Instrumentation, S. E. Wolf and R. F. M. Smith,

PHI, New Delhi, 1990.

2. Basic Electronic Instruments Handbook, C. F. Coombs, McGraw Hill Book Co., 1972.

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