HC Verma Questions and Solutions: Chapter 16: Sound Waves- 1

Sound is a longitudinal wave produced by the oscillation of pressure at a point, thus, forming compressions and rarefactions. That portion of gas itself does not move but the pressure variation causes a disturbance.

When we clap, there is a change in pressure, which sets a disturbance and forms a wave. However, this variation is not uniform every time we clap (unlike in the case of a sound wave). Hence, we sum up all the disturbances.

If B is the bulk modulus and ρ is the density, then the velocity of sound is given by:

If both B and ρ are greater, then we cannot compare 
For proper comparison, we need numerical values.

The velocity of a sound wave varies with temperature as follows:

As the temperature increases, the speed also increases. However, since the frequency remains the same, its wavelength changes.

When a sound or light wave undergoes refraction, its frequency remains constant because there is no change in its phase.

Propagation of any wave through a medium depends on whether it is elastic and possesses inertia. A wave needs to oscillate (elastic property) for it to be propagated and if it does not have inertia, the oscillations won't keep on moving to and fro about the mean position.

Since the average power transmitted by a wave is independent of the wavelength, we have P₁ = P₂

The energy is redistributed due to the presence of interference. However, as the frequency and phase remain constant, the distribution also remains constant with time.

In the case of an open organ pipe vibrating in its fundamental mode, antinodes are formed at both ends. An open pipe supports a standing wave pattern that has an antinode (maximum displacement and minimum pressure variation) at each open end. The pressure variation is maximum at the nodes, which occur at the midpoint of the pipe in the fundamental mode. Hence option (b).

An open organ pipe has sound waves that are longitudinal. These waves undergo repeated reflections till resonance to form standing waves.

If v is the velocity of the wave and L is the length of the pipe,
then the fundamental frequency for an open organ pipe is v = v/2L
For a closed organ pipe of length L' = L/2,  the fundamental frequency is 
(When the pipe is dipped in water, it behaves like a closed organ pipe that is half the length)

When two or more waves of slightly different frequencies (v₁ – v₂ ≯ 10) travel with the same speed in the same direction, they superimpose to give beats. Thus, the waves may be longitudinal or transverse.

The frequency of beats ν = |ν₁ - ν₂| = 6 Hz where ν₁ = 512 Hz, ν₂ = frequency of sonometer. Thus ν₂ is either 506 or 518 Hz. 

When the tension in the string is slightly increased the frequency of the sonometer slightly increases. Since the beat frequency reduces it means ν₂ < ν₁ so ν₂ is 506 Hz. Option (a).

Since the source and the observer both move with the same speed there is no relative motion. Thus there is no Doppler's effect and no change in apparent frequency. 

It is clear from the equation that the change in frequency due to Doppler effect depends only on the relative motion and not on the distance between the source and the observer.

When the source is at C the source speed is perpendicular to the line joining the observer and the source. Thus at this instant, the separation between the two is not changing and the frequency heard ν₃ is same as the source. When the source is at A the separation between them is increasing, so due to the Doppler effect ν₁ < ν₃. But when the source is at B, the separation is decreasing and due to the Doppler effect ν₃ < ν₂. So ν₂ > ν₃ > ν₁. 

• The frequency, wavelength and amplitude do not have a unique value in the sound produced.
• The frequency (and wavelength) changes as the pitch of the sound varies, while the amplitude is different as the loudness varies. However, the speed of sound in the air at a particular temperature is constant, i.e., it has a unique value.

The velocity of sound V is directly proportional to the square root of the temperature. So if the temperature increases the velocity of the sound also increases. Option (c).     Since the frequency of the sound producing tuning fork is constant hence the frequency of the sound will also be constant. Also, the wavelength = (Sound velocity)/(Frequency). Thus the wavelength will also increase if the temperature increases. Option (a). 
Since frequency is constant, Time period will also be constant. 

The overtone frequencies may be a simple multiple of the fundamental frequency (ν = nν₀) or an odd multiple of the fundamental frequency {ν = (2n + 1)ν₀} depending upon whether it is an open organ pipe or closed organ pipe.
v₀ = Fundamental Frequency
Overtone frequencies for:
Open organ pipe = nv₀
{n = 2, 3, 4, .....]
Closed organ pipe = (2n + 1)ν₀
{n = 1, 2, 3, ....}

Since it is not given whether the pipe is open or closed, we cannot be sure that the first overtone is 400 Hz. So the option (a) is not true.       
The first overtone may be 400 Hz if the pipe is open because it is the first even multiple of the fundamental frequency 200 Hz. Option (b) is true.        
The first overtone may be 600 Hz if the pipe is closed because it is the first odd multiple of the fundamental frequency 200 Hz (3 * 200 Hz). Option (b) is true.      
Since 600 Hz is an even as well as odd multiple of 200 Hz, it is an overtone whether the pipe is open one or closed. Option (d) is true.

Due to Doppler effect, the frequency or wavelength of the sound changes towards the observer only. The actual frequency and wavelength of the source does not change.

The frequency does not change. Hence, the time period (inverse of frequency) also remains the same.
Due to wind, the relative velocity of sound changes. Thus, the wavelength also changes so as to keep the frequency the same. (As v = vλ)