JEE Physics Sound Waves Notes Set 02

Nature of Sound and the Wave Equations

Sound is a mechanical, longitudinal wave that spreads in three dimensions. It needs a medium with inertia and elasticity, and it travels as a train of compressions (pressure above normal) and rarefactions (pressure below normal) created by a vibrating source such as a tuning fork prong or a loudspeaker cone.

For a source in simple harmonic motion the excess pressure and the displacement of the air particles are both sinusoidal waves, but they are a quarter of a cycle apart:

\[ s = s_0\sin(\omega t-kx),\qquad \Delta p = \Delta p_0\sin\!\left(\omega t-kx+\tfrac{\pi}{2}\right),\qquad \Delta p_0 = B k s_0 \]

1. A displacement antinode is a pressure node and a displacement node is a pressure antinode.
2. Here B is the bulk modulus of the medium, \( k = 2\pi/\lambda \) and \( \omega = 2\pi f \).
3. A pulse such as a clap is not a pure sine wave; it is the sum of many sine waves of different frequencies.
4. Audible range for humans: 20 Hz to 20 kHz. Below 20 Hz is infrasonic, above 20 kHz is ultrasonic. Pitch is the brain's subjective impression of frequency.

Speed of Sound

1. Temperature: \( v\propto\sqrt T \). Near room temperature the speed rises by about 0.6 m/s for each degree Celsius.
2. Pressure: at constant temperature the speed does not change, because pressure and density change together and their ratio stays fixed.
3. Humidity: damp air is lighter than dry air (water vapour has a smaller molar mass), so sound travels slightly faster in it.
4. For gases at the same temperature compare speeds with \( v\propto\sqrt{\gamma/M} \).

Intensity and the Decibel Scale

Intensity is the average power passing through unit area held perpendicular to the direction of travel. For a plane sound wave:

\[ I = \frac{\Delta p_0^2}{2\rho v} = \tfrac12\rho v\,\omega^2 s_0^2 \]

1. Intensity is proportional to the square of the pressure amplitude, the square of the displacement amplitude and the square of the frequency.
2. A point source spreads its power over a sphere, so the intensity falls as 1 over r squared and the amplitude falls as 1 over r. For a line source the intensity falls as 1 over r.
3. Sound level in decibel: \( \beta = 10\log_{10}(I/I_0) \) with threshold \( I_0 = 10^{-12}\ \text{W/m}^2 \). The threshold of hearing is therefore 0 dB.
4. Multiplying the intensity by 2 adds about 3 dB; multiplying by 10 adds 10 dB; multiplying by 100 adds 20 dB.
5. Independent (incoherent) sources add their intensities, not their amplitudes.

Interference of Sound

Two waves of pressure amplitudes \( p_1 \) and \( p_2 \) meeting with phase difference \( \phi \) give a resultant amplitude \( p_0=\sqrt{p_1^2+p_2^2+2p_1p_2\cos\phi} \), and for intensities \( I = I_1+I_2+2\sqrt{I_1I_2}\cos\phi \).

Phase difference from a path difference \( \Delta x \) is \( \phi = 2\pi\,\Delta x/\lambda \). Sources are coherent only if their phase difference does not change with time; two separate sources are usually incoherent.

Reflection and Standing Waves

1. At a rigid (closed) boundary the displacement wave is reflected with a phase change of pi, while the pressure wave is reflected without any change: a compression returns as a compression.
2. At an open end or a region of low pressure the displacement wave is reflected without a phase change and the pressure wave with a phase change of pi: a compression returns as a rarefaction.
3. Two equal waves travelling in opposite directions produce a standing wave. Its amplitude varies with position, with nodes and antinodes separated by lambda over 4 (adjacent nodes are lambda over 2 apart).

\[ p = 2p_0\cos\!\left(kx+\tfrac{\phi}{2}\right)\sin\!\left(\omega t+\tfrac{\phi}{2}\right) \]

Vibrations of Air Columns

Resonance occurs when the driving frequency equals a natural frequency of the air column. A closed end is a displacement node (pressure antinode) and an open end is a displacement antinode (pressure node). In practice the antinode lies a little beyond the open end, an effect called end correction, about 0.3 d (0.6 times the radius) at each open end of a tube of diameter d.

1. With an end correction of 0.3 d per open end, the effective length is \( L+0.3d \) for a closed pipe and \( L+0.6d \) for an open pipe.
2. In a resonance tube experiment, two successive resonance lengths differ by half a wavelength, so lambda = 2 (L2 - L1) and e = (L2 - 3 L1) / 2.
3. Pressure variation is greatest at displacement nodes and zero at displacement antinodes.
4. A closed pipe of length L has the same fundamental as an open pipe of length L/2. A small hole in the side wall forces a displacement antinode at that point.

Beats

Two waves of nearly equal frequencies \( f_1 \) and \( f_2 \) combine into a wave of frequency \( (f_1+f_2)/2 \) whose amplitude swells and fades at the rate \( |f_1-f_2| \). The intensity therefore rises and falls \( |f_1-f_2| \) times per second:

\[ f_{beat} = |f_1-f_2|,\qquad T_{beat} = \frac{1}{|f_1-f_2|} \]

1. The beats are audible only if the difference is below about 16 Hz.
2. Beats are used to find an unknown frequency by comparing it with a known one.
3. Loading a tuning fork (wax) lowers its frequency; filing its prongs raises it. Increasing the tension of a string raises its frequency.
4. The change of the beat rate after such a change tells you whether the unknown is above or below the reference.

Doppler Effect

Let v be the speed of sound relative to the ground, with the positive direction taken from the source towards the observer, and let \( v_S \) and \( v_O \) be the velocities of the source and the observer along this direction:

\[ f' = f\,\frac{v-v_O}{v-v_S},\qquad \lambda' = \frac{v-v_S}{f} \]

1. If the source and the observer approach each other the frequency rises; if they separate it falls.
2. A moving observer changes the frequency but not the wavelength; a moving source changes the wavelength as well.
3. If the medium moves (wind of speed \( w \) along the direction of sound), use \( v = c+w \) in the formula, where c is the speed of sound relative to the air.
4. For an echo from a stationary wall, treat the wall first as a stationary observer and then as a stationary source (two Doppler steps).
5. Only the component of velocity along the line joining source and observer matters.
6. For a source moving in a circle, extremes occur when its velocity is directed towards or away from the observer, so \( f_{max}=f\dfrac{v}{v-v_S} \) and \( f_{min}=f\dfrac{v}{v+v_S} \).

Worked Examples

Example 1. The speed of sound in oxygen is 450 m/s at a certain temperature. What is the speed in hydrogen at the same temperature?

1. Both are diatomic with the same gamma, so \( v\propto1/\sqrt M \).
2. \( v_{H_2}/v_{O_2} = \sqrt{32/2} = 4 \), hence \( v_{H_2} = 1800 \) m/s.

Example 2. A sound level increases by 20 dB. By what factor do the intensity and the pressure amplitude change?

1. \( 20 = 10\log_{10}(I_2/I_1) \) gives \( I_2/I_1 = 100 \).
2. Pressure amplitude is proportional to the square root of intensity, so it increases by a factor of 10.

Example 3. A closed pipe has fundamental frequency 100 Hz. List its next two resonant frequencies and the fundamental frequency of an open pipe of the same length.

1. The closed pipe allows odd harmonics: 300 Hz and 500 Hz.
2. An open pipe of the same length has fundamental \( v/2L \), twice the closed value: 200 Hz.

Example 4. A source of frequency 400 Hz moves at 20 m/s towards a stationary observer. The speed of sound is 340 m/s. What is the observed frequency and the apparent wavelength?

1. \( f' = 400\times\dfrac{340}{340-20} = 425 \) Hz.
2. \( \lambda' = \dfrac{340-20}{400} = 0.80 \) m, whereas the true wavelength is 0.85 m.

Example 5. A fork of 256 Hz gives 4 beats per second with a second fork. Waxing the second fork reduces the beat rate. What was the frequency of the second fork?

1. It is 252 Hz or 260 Hz.
2. Waxing lowers its frequency. From 252 Hz it would move further from 256 Hz and the beats would increase; from 260 Hz it moves closer and the beats decrease, so it was 260 Hz.

Quick Revision Points

1. Sound in air: longitudinal, speed about 332 m/s at 0 degrees C, independent of pressure, proportional to the square root of absolute temperature.
2. Intensity is proportional to amplitude squared and to frequency squared; level in dB is ten times the common logarithm of the intensity ratio.
3. Closed pipe: odd harmonics, frequencies in the ratio 1 : 3 : 5. Open pipe: all harmonics, ratio 1 : 2 : 3.
4. Beat frequency equals the difference of the two frequencies; use a small change in one source to decide the sign.
5. Doppler: approach raises the frequency, separation lowers it; wind modifies the speed of sound relative to the ground.