the velocity of sound

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THE VELOCITY OF SOUND

1B03

OBJECT: To measure the velocity of sound (for the conditions obtaining in the laboratory) by determining the wavelength of the sound waves emitted by a source of known frequency.

THEORY
A sound wave is a travelling longitudinal wave. Alternate rarefactions and compressions travel in the direction of the wave. At each point through which the wave passes, the air oscillates in the direction of travel of the wave. An undamped progressive sound wave of frequency f and wavelength  travelling in the positive x-direction can be represented by the equation x  y  a cos 2  ft     where y is the longitudinal displacement of a particle from its mean position produced by the disturbance and a is the maximum displacement. At the origin (x=0) the displacement is: y0  2a cos(2 ft ) The vibrations at the origin and at a point distant x from the origin differ in phase by an amount 2 x /  ; all points one wavelength apart along the x-axis will effectively vibrate in phase with each other. The object of this experiment is to locate such points, and hence measure the wavelength, by a method in which a parallel beam of sound is produced by placing the source at the focus of a parabolic mirror and allowing the disturbance to travel in free air, unconfined by any tube, to a distant parabolic mirror whence it is reflected to a microphone situated at its focus. The phase of the disturbance arriving at the microphone is compared with that of the source using a microscope. The velocity of sound, v, at a known frequency will then be given by v = frequency x wavelength

APPARATUS
In the laboratory it is convenient to work above audio frequencies, so a transducer that resonates at a frequency of about 40 kHz is used as the source; a microphone tuned to the same frequency is used as the receiver. An oscillator can be adjusted to supply voltage to the transducer at the correct frequency and also to the X-plates of the oscilloscope to produce a horizontal trace. The signal received by the microphone is amplified and fed to the Y-plates so that the cathode rays are subject to two deflections at right angles to each other: due to the source x  a cos ft due to the receiver y  b cos( ft   ) If the phases are equal, then   2 , 4 etc., and so x  a cos ft , y  b cos ft
and therefore

y  bx / a This is a straight line having a gradient b/a. The oscilloscope trace will thus be identical for two positions of the microphone that are a whole number of wavelengths apart.

The apparatus is arranged as shown in the diagram. The transducer is excited by an oscillator of good stability and the signal received at the microphone is amplified by an

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amplifier tuned to the oscillator frequency. The microphone and transducer are positioned at the foci of spherical mirrors M1 and M2 (focal length = 125 mm) whose axes should be parallel. This arrangement should be checked and adjusted if necessary. The movable mirror M2 is mounted on a motor-driven trolley that runs along a pair of rails, so that the amplitude of the received signal should remain fairly constant over the entire range of motion and the in-phase positions (the straight lines) should be easily countable.

Movable mirror

Transducer

Microphone

Fixed mirror

Digital frequency meter

Oscillator

Amplifier

EXPERIMENTS
Set the oscillator frequency to give maximum response at the detector. With the trolley near to the source, adjust its position so that a linear trace is obtained. Drive the trolley towards the other end of the scale, counting the number of times an identical trace is observed. Stop the trolley near the end of its travel and adjust by hand to a position that reproduces the original trace. The distance moved is n where n can be about 75. N.B. Most faulty measurements arise because of miscounting the number of wavelengths being observed. It is therefore advisable to make several initial measurements where n = 5. From the value of  thus obtained, an approximate value for 75 may be calculated. This can then be used to check that the full distance used is really 75 and not (say) 74 or 76 . Although the scale can be read to 0.1 mm it is probable that disturbing influences such as convection currents will limit the accuracy of the measurement. Several values of  should therefore be determined, and their mean value and probable error calculated. To approach an overall precision of 0.1%, a careful check must be kept on temperature fluctuations. The velocity of sound changes by about 0.6 m/s for each degree change in temperature. Thus 0.1%, which corresponds to about 0.33 m/s, equates to a change in temperature of about half a degree. The air temperature should be measured by a thermometer placed horizontally and close to the beam path; it should be shielded from direct radiation by

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wrapping the bulb in aluminium foil. An oscilloscope dissipates about 80 W (and so does a student!). These heat sources contribute to the rise in temperature of the laboratory of about one or two degrees which can be expected during a typical afternoon session. For each of your measurements, calculate the velocity of sound, v, at the ambient temperature and correct it to a value at 273 K using the relationship v T where T is the absolute temperature. Calculate the mean value and the standard deviation from your set of values for v. Compare your value with that given in any standard reference book available in the laboratory. Consider carefully the sources of errors in your measurements, and make an estimate of their relative importance.

The frequency of the oscillator driving the transducer can be varied over a small range, although the signal from the detector will fall off at the high and low ends of the range. Repeat the above measurements at frequencies at both ends of the available range at which the detector shows a measurable response.

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