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15.12)  For the wave on a string described in Exercise 15.11, do all the points on the string oscillate with the same (a) frequency, (b) phase, (c) amplitude? Explain your answers. (ii) What is the amplitude of a point 0.375 m away from one end?

(i) (a) All the points vibrate with the same frequency of 60 Hz. (b) They all have the same phase as it depends upon time. (c) At different points, the amplitude is different and is equal to A(x) given by   (ii)

Q.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 ?

Apparent frequency striking the wall and getting reflected is The frequency emitted by the bats is  Speed of sound is v Speed of bat is 0.03v Frequency of sound as heard by the bat

Q.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?

Let us assume the earthquake occurs at a distance s. The origin of the earthquake is at a distance of 1960 km.

Q.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 $1450 \: m\: s^{-1}$.

Frequency of SONAR =40 kHz Speed of enemy submarine vo=360 km h-1 = 100 m s-1 This is the frequency which would be observed and reflected by the enemy submarine but won't appear the same to the SONAR(source) as again there is relative motion between the source(enemy submarine) and the observer(SONAR) The frequency which would be received by the SONAR is

Q.15.24     One end of a long string of linear mass density $8.0\times 10^{-3}kg\: m^{-1}$ 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.

A=0.05 m Tension in the string is T=mg The speed of the wave in the string is v Angular frequency of the wave is Since at t=0, the left end (fork end) of the string x=0 has zero transverse displacement (y=0) and is moving along the positive y-direction, the initial phase is zero.  Taking the left to the right direction as positive we have Here t is in seconds and x and y are in metres.

Q.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 $1/20$ or $0.05\: Hz$ ?

(a) The pulse does not have a definite frequency or wavelength however the wave has definite speed given the medium is non-dispersive. (b) The frequency of the note produced by the whistle is not 0.05 Hz. It only implies the frequency of repetition of the pip of the whistle is 0.05 Hz,

Q.15.22     A travelling harmonic wave on a string is described by

$y(x,t)=7.5\: \sin (0.0050\: x+12t+\pi /4)$
(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.

Wavelength of the given wave is The points with same displacements and velocity at same instant of time are seperated by distances . The points of the string which have the same transverse displacements and velocity as the  point at 5 s and 11 s would be at a distance of  from x = 1cm. Therefore all points at distances  from the point x=1cm would have the same transverse displacements and...

Q.15.22     A travelling harmonic wave on a string is described by

$y(x,t)=7.5\: \sin (0.0050\: x+12t+\pi /4)$
(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?

The displacement of oscillation of a point at x = 1 cm and t = 1 s is The general expression for velocity of oscillation is k=0.005 cm-1 The velocity of propagation of the wave is The velocity of oscillation of point at x = 1 cm and t = 1 cm is not equal to the propagation of the wave.

Q.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 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 $340\: m\: s^{-1}$

Speed of the wind vw = 10 m s-1 Speed of sound in still air va = 340 m s-1 Effective speed with which the wave reaches the observer = v = vw + va = 10 + 340= 350 m s-1 There is no relative motion between the observer and the source and therefore the frequency heard by the observer would not change. The wavelength of the sound as heard by the observer is The above situation is not identical to...

Q.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 $340\: m\; s^{-1}$.

where  is the frequency as observed by the observer,  is the frequency of the source, v is the speed of the wave, vo is the speed of the observer and vs is the speed of the source. (i) (a) When source is moving towards the observer and the observer is stationary.   (b) (ii) The speed of the sound does not change as it is independent of the speed of observer and source and remains equal to 340 m s-1.

Q.15.19     Explain why (or how) :

(e) the shape of a pulse gets distorted during propagation in a dispersive medium.

As we know a pulse contains waves of different wavelengths, these waves travel at different speeds in a dispersive medium and thus the shape of the pulse gets distorted.

Q.15.19     Explain why (or how) :

(d) solids can support both longitudinal and transverse waves, but only longitudinal waves can propagate in gases,

Transverse waves produce shear, gases don't have shear modulus and cannot sustain shear and therefore can only propagate longitudinal waves. Solids have both shear and bulk modulus of elasticity and can propagate both transverse and longitudinal waves.

Q.15.19     Explain why (or how) :

(c) a violin note and sitar note may have the same frequency, yet we can distinguish between the two notes,

We can distinguish between the two notes with the same frequency as the harmonics they emit are different.

Q.15.19     Explain why (or how) :

(b) bats can ascertain distances, directions, nature, and sizes of the obstacles without any “eyes”,

Bats emit ultrasonic waves and when these waves strike the obstacles and get reflected back to the bats they ascertain distances, directions, nature, and sizes of the obstacles without any “eyes”.

Q.15.19     Explain why (or how) :

(a) in a sound wave, a displacement node is a pressure antinode and vice versa,

In the propagation of a sound wave the pressure increases at points where displacement decreases, Therefore maximum pressure at points of minimum displace and vice-versa i.e. a displacement node is a pressure antinode and vice versa.

Q.15.18     Two sitar strings A and B playing the note ‘Ga’ are slightly out of tune and produce beats of frequency                   $6\: Hz$. The tension in the string A is slightly reduced and the beat frequency is found to reduce to $3\: Hz$.  If the original frequency of A is $324\: Hz$, what is the frequency of B?

Since frequency increases with an increase in Tension, the frequency of string A must have decreased. Therefore .(If it were 330 Hz the beat frequency would have increased with decrease in Tension in string A)

Q.15.17     A pipe $20\: cm$ long is closed at one end. Which harmonic mode of the pipe is resonantly excited by a source? Will the same source be in resonance with the pipe if both ends are open? (speed of  sound in air is $340\:m\: s^{-1}$).

Let the nth harmonic mode of the pipe get resonantly excited by a 430 Hz source. The pipe resonates with a 430 Hz source in the fundamental mode when one end is open. Let the mth harmonic mode of the pipe get resonantly excited by a 430 Hz source when both ends are open. Since m is coming out to be less than 1 the same source will not be in resonance with the pipe if both ends are open.

Q.15.16     A steel rod $100\: cm$ long is clamped at its middle. The fundamental frequency of longitudinal vibrations of  the rod are given to be $2.53\: kHz$. What is the speed of sound in steel?

When the rod is clamped at the middle at is vibrating in the fundamental mode, a node is formed at the middle of the rod and antinodes at the end. i.e. where L is the length of the rod. Speed of sound in steel is

Q.15.15     A metre-long tube open at one end, with a movable piston at the other end, shows resonance with a fixed frequency source (a tuning fork of frequency $340\: Hz$) when the tube length is $25.5\: cm$ or $79.3\: cm$.  Estimate the speed of sound in air at the temperature of the experiment. The edge effects may be neglected.

The pipe behaves as a pipe open at one end and closed at one end. Such a pipe would produce odd harmonics i.e. Two consecutive modes of vibration are given in the question For l1 = 25.5 cm For l2 = 79.3 cm Since at both these modes the system resonates with the same frequency we have (our approximation is correct since the edge effects may be neglected)

Q.15.14     A wire stretched between two rigid supports vibrates in its fundamental mode with a frequency of $45\: Hz$. The mass of the wire is $3.5\times 10^{-2}\: kg$ and its linear mass density is $4.0\times 10^{-2}\: kg\: m^{-1}$. What is  (b) the tension in the string?

Tension in the string is given by
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