Updated By SaraNextGen

On March 11, 2024, 11:35 AM

Question 15.20:

A train, standing at the outer signal of a railway station blows a whistle of frequency 400 Hz in still air. (i) What is the frequency of the whistle for a platform observer when the train (a) approaches the platform with a speed of 10 m s^–1, (b) recedes from the platform with a speed of 10 m s^–1? (ii) What is the speed of sound in each case? The speed of sound in still air can be taken as 340 m s^–1.

Answer:

(i) (a)Frequency of the whistle, ν = 400 Hz

Speed of the train, v_T= 10 m/s

Speed of sound, v = 340 m/s

The apparent frequency  of the whistle as the train approaches the platform is given by the relation:

(b) The apparent frequency  of the whistle as the train recedes from the platform is given by the relation:

(ii) The apparent change in the frequency of sound is caused by the relative motions of the source and the observer. These relative motions produce no effect on the speed of sound. Therefore, the speed of sound in air in both the cases remains the same, i.e., 340 m/s.

Question 15.21:

A train, standing in a station-yard, blows a whistle of frequency 400 Hz in still air. The wind starts blowing in the direction from the yard to the station with at a speed of 10 m s^–1. What are the frequency, wavelength, and speed of sound for an observer standing on the station’s platform? Is the situation exactly identical to the case when the air is still and the observer runs towards the yard at a speed of 10 m s^–1? The speed of sound in still air can be taken as 340 m s^–1.

Answer:

For the stationary observer: 400 Hz; 0.875 m; 350 m/s

For the running observer: Not exactly identical

For the stationary observer:

Frequency of the sound produced by the whistle, ν = 400 Hz

Speed of sound = 340 m/s

Velocity of the wind, v = 10 m/s

As there is no relative motion between the source and the observer, the frequency of the sound heard by the observer will be the same as that produced by the source, i.e., 400 Hz.

The wind is blowing toward the observer. Hence, the effective speed of the sound increases by 10 units, i.e.,

Effective speed of the sound, ve = 340 + 10 = 350 m/s

The wavelength (λ) of the sound heard by the observer is given by the relation:

For the running observer:

Velocity of the observer, v_o = 10 m/s

The observer is moving toward the source. As a result of the relative motions of the source and the observer, there is a change in frequency ( ).

This is given by the relation:

Since the air is still, the effective speed of sound = 340 + 0 = 340 m/s

The source is at rest. Hence, the wavelength of the sound will not change, i.e., λ remains 0.875 m.

Hence, the given two situations are not exactly identical.

Question 15.22:

A travelling harmonic wave on a string is described by

(a) What are the displacement and velocity of oscillation of a point at x = 1 cm, and t = 1 s? Is this velocity equal to the velocity of wave propagation?

(b) Locate the points of the string which have the same transverse displacements and velocity as the x = 1 cm point at t = 2 s, 5 s and 11 s.

Answer:

(a) The given harmonic wave is:

For x = 1 cm and t = 1s,

The velocity of the oscillation at a given point and time is given as:

Now, the equation of a propagating wave is given by:

∴

Hence, the velocity of the wave oscillation at x = 1 cm and t = 1 s is not equal to the velocity of the wave propagation.

(b) Propagation constant is related to wavelength as:

Therefore, all the points at distances nλ

, i.e. ± 12.56 m, ± 25.12 m, … and so on for x = 1 cm, will have the same displacement as the x = 1 cm points at t = 2 s, 5 s, and 11 s.

Question 15.23:

A narrow sound pulse (for example, a short pip by a whistle) is sent across a medium. (a) Does the pulse have a definite (i) frequency, (ii) wavelength, (iii) speed of propagation? (b) If the pulse rate is 1 after every 20 s, (that is the whistle is blown for a split of second after every 20 s), is the frequency of the note produced by the whistle equal to  or 0.05 Hz?

Answer:

Answer:

(a) (i)No

(ii)No

(iii)Yes

(b) No

Explanation:

(a) The narrow sound pulse does not have a fixed wavelength or frequency. However, the speed of the sound pulse remains the same, which is equal to the speed of sound in that medium.

(b) The short pip produced after every 20 s does not mean that the frequency of the whistle is  or 0.05 Hz. It means that 0.05 Hz is the frequency of the repetition of the pip of the whistle.

Question 15.24:

One end of a long string of linear mass density 8.0 × 10^–3 kg m^–1 is connected to an electrically driven tuning fork of frequency 256 Hz. The other end passes over a pulley and is tied to a pan containing a mass of 90 kg. The pulley end absorbs all the incoming energy so that reflected waves at this end have negligible amplitude. At t = 0, the left end (fork end) of the string x = 0 has zero transverse displacement (y = 0) and is moving along positive y-direction. The amplitude of the wave is 5.0 cm. Write down the transverse displacement y as function of x and t that describes the wave on the string.

Answer:

The equation of a travelling wave propagating along the positive y-direction is given by the displacement equation:

y (x, t) = a sin (wt – kx) … (i)

Linear mass density, 

Frequency of the tuning fork, ν = 256 Hz

Amplitude of the wave, a = 5.0 cm = 0.05 m … (ii)

Mass of the pan, m = 90 kg

Tension in the string, T = mg = 90 × 9.8 = 882 N

The velocity of the transverse wave v, is given by the relation:

Substituting the values from equations (ii), (iii), and (iv) in equation (i), we get the displacement equation:

y (x, t) = 0.05 sin (1.6 × 10³t – 4.84 x) m

Question 15.25:

A SONAR system fixed in a submarine operates at a frequency 40.0 kHz. An enemy submarine moves towards the SONAR with a speed of 360 km h^–1. What is the frequency of sound reflected by the submarine? Take the speed of sound in water to be 1450 m s^–1.

Answer:

Operating frequency of the SONAR system, ν = 40 kHz

Speed of the enemy submarine, v_e = 360 km/h = 100 m/s

Speed of sound in water, v = 1450 m/s

The source is at rest and the observer (enemy submarine) is moving toward it. Hence, the apparent frequency ( ) received and reflected by the submarine is given by the relation:

The frequency ( ) received by the enemy submarine is given by the relation:

Question 15.26:

Earthquakes generate sound waves inside the earth. Unlike a gas, the earth can experience both transverse (S) and longitudinal (P) sound waves. Typically the speed of S wave is about 4.0 km s^–1, and that of P wave is 8.0 km s^–1. A seismograph records P and S waves from an earthquake. The first P wave arrives 4 min before the first S wave. Assuming the waves travel in straight line, at what distance does the earthquake occur?

Answer:

Let vSand vP be the velocities of S and P waves respectively.

Let L be the distance between the epicentre and the seismograph.

We have:

L = vStS (i)

L = vPtP (ii)

Where,

tS and tP are the respective times taken by the S and P waves to reach the seismograph from the epicentre

It is given that:

vP = 8 km/s

vS = 4 km/s

From equations (i) and (ii), we have:

vS tS = vP tP

4tS = 8 tP

tS = 2 tP (iii)

It is also given that:

tS – tP = 4 min = 240 s

2tP – tP = 240

tP = 240

And tS = 2 × 240 = 480 s

From equation (ii), we get:

L = 8 × 240

= 1920 km

Hence, the earthquake occurs at a distance of 1920 km from the seismograph.

Question 15.27:

A bat is flitting about in a cave, navigating via ultrasonic beeps. Assume that the sound emission frequency of the bat is 40 kHz. During one fast swoop directly toward a flat wall surface, the bat is moving at 0.03 times the speed of sound in air. What frequency does the bat hear reflected off the wall?

Answer:

Ultrasonic beep frequency emitted by the bat, ν = 40 kHz

Velocity of the bat, v_b = 0.03 v

Where, v = velocity of sound in air

The apparent frequency of the sound striking the wall is given as:

This frequency is reflected by the stationary wall ( ) toward the bat.

The frequency ( ) of the received sound is given by the relation: