Answer

Mass of the string, $M=2.50 kg$

Tension in the string, $T=200 N$

Length of the string, $l=20.0 m$

Mass per unit length, $\mu=\frac{M}{l}=\frac{2.50}{20}=0.125 kg m^{-1}$

The velocity $(v)$ of the transverse wave in the string is given by the relation:

$ \begin{aligned} v & =\sqrt{\frac{T}{\mu}} \\ & =\sqrt{\frac{200}{0.125}}=\sqrt{1600}=40 m / s \end{aligned} $

$\therefore$ Time taken by the disturbance to reach the other end, $t=\frac{l}{v}=\frac{20}{40}=0.50 s$

Answer

Height of the tower, $s=300 m$

Initial velocity of the stone, $u=0$

Acceleration, $a=g=9.8 m / s^{2}$

Speed of sound in air $=340 m / s$

The time (${ }^{t_1}.$ ) taken by the stone to strike the water in the pond can be calculated using the second equation of motion, as:

$s=u t_1+\frac{1}{2} g t_1^{2}$

$300=0+\frac{1}{2} \times 9.8 \times t_1^{2}$

$\therefore t_1=\sqrt{\frac{300 \times 2}{9.8}}=7.82 s$

Time taken by the sound to reach the top of the tower, $t_2=\frac{300}{340}=0.88 s$

Therefore, the time after which the splash is heard, ${ }^{t=t_1+t_2}$

$=7.82+0.88=8.7 s$

Answer

Length of the steel wire, $l=12 m$

Mass of the steel wire, $m=2.10 kg$

Velocity of the transverse wave, $v=343 m / s$

Mass per unit length, $\mu=\frac{m}{l}=\frac{2.10}{12}=0.175 kg m^{-1}$

For tension $T$, velocity of the transverse wave can be obtained using the relation:

$v=\sqrt{\frac{T}{\mu}}$

$\therefore T=v^{2} \mu$

$=(343)^{2} \times 0.175=20588.575 \approx 2.06 \times 10^{4} N$

Answer

Take the relation:

$v=\sqrt{\frac{\gamma P}{\rho}}$

Where,

Density, $\rho=\frac{\text{ Mass }}{\text{ Volume }}=\frac{M}{V}$

$M=$ Molecular weight of the gas

$V=$ Volume of the gas

Hence, equation $(i)$ reduces to:

$v=\sqrt{\frac{\gamma P V}{M}}$

Now from the ideal gas equation for $n=1$ :

$P V=R T$

For constant $T, P V=$ Constant

Since both $M$ and $\gamma$ are constants, $v=$ Constant

Hence, at a constant temperature, the speed of sound in a gaseous medium is independent of the change in the pressure of the gas.

Take the relation:

$ \begin{equation*} v=\sqrt{\frac{\gamma P}{\rho}} \tag{i} \end{equation*} $

For one mole of an ideal gas, the gas equation can be written as:

$P V=R T$

$P=\frac{R T}{V} \ldots($ ii $)$

Substituting equation (ii) in equation (i), we get:

$ v=\sqrt{\frac{\gamma R T}{V \rho}}=\sqrt{\frac{\gamma R T}{M}} \ldots(iii) $

Where,

Mass, $M=\rho V$ is a constant $\gamma$ and $R$ are also constants

We conclude from equation $(i v)$ that $v \propto \sqrt{T}$.

Hence, the speed of sound in a gas is directly proportional to the square root of the temperature of the gaseous medium, i.e., the speed of the sound increases with an increase in the temperature of the gaseous medium and vice versa.

Let $v_m$ and $v_d$ be the speeds of sound in moist air and dry air respectively.

Let $\rho_m$ and $\rho_d$ be the densities of moist air and dry air respectively.

Take the relation:

$v=\sqrt{\frac{\gamma P}{\rho}}$

Hence, the speed of sound in moist air is:

$v_m=\sqrt{\frac{\gamma P}{\rho_m}}$

And the speed of sound in dry air is:

$v_d=\sqrt{\frac{\gamma P}{\rho_d}}$

On dividing equations $(i)$ and (ii), we get:

$ \frac{v_m}{v_d}=\sqrt{\frac{\gamma P}{\rho_m} \times \frac{\rho_d}{\gamma P}}=\sqrt{\frac{\rho_d}{\rho_m}} $

However, the presence of water vapour reduces the density of air, i.e.,

$ \rho_d<\rho_m $

$\therefore v_m>v_d$

Hence, the speed of sound in moist air is greater than it is in dry air. Thus, in a gaseous medium, the speed of sound increases with humidity.

Answer

No;

(a) Does not represent a wave

(b) Represents a wave

(c) Does not represent a wave

The converse of the given statement is not true. The essential requirement for a function to represent a travelling wave is that it should remain finite for all values of $x$ and $t$.

Explanation:

(a)

For $x=0$ and $t=0$, the function $(x-v t)^{2}$ becomes 0 .

Hence, for $x=0$ and $t=0$, the function represents a point and not a wave.

(b)

For $x=0$ and $t=0$, the function

$\log (\frac{x+v t}{x_0})=\log 0=\infty$

Since the function does not converge to a finite value for $x=0$ and $t=0$, it represents a travelling wave.

(c)

For $x=0$ and $t=0$, the function

$\frac{1}{x+v t}=\frac{1}{0}=\infty$

Since the function does not converge to a finite value for $x=0$ and $t=0$, it does not represent a travelling wave.

Answer

Frequency of the ultrasonic sound, $v=1000 kHz=10^{6} Hz$

Speed of sound in air, $v_a=340 m / s$

The wavelength $(\lambda_r)$ of the reflected sound is given by the relation:

$ \begin{aligned} & \lambda_r=\frac{v}{v} \\ & =\frac{340}{10^{6}}=3.4 \times 10^{-4} m \end{aligned} $

Frequency of the ultrasonic sound, $v=1000 kHz=10^{6} Hz$

Speed of sound in water, $v_w=1486 m / s$

The wavelength of the transmitted sound is given as:

$ \lambda_t=\frac{1486}{10^{6}}=1.49 \times 10^{-3} m $

Answer

Speed of sound in the tissue, $v=1.7 km / s=1.7 \times 10^{3} m / s$

Operating frequency of the scanner, $v=4.2 MHz=4.2 \times 10^{6} Hz$

The wavelength of sound in the tissue is given as:

$ \begin{aligned} & \lambda=\frac{v}{v} \\ & =\frac{1.7 \times 10^{3}}{4.2 \times 10^{6}}=4.1 \times 10^{-4} m \end{aligned} $

Answer

(a) Yes; Speed $=20 m / s$, Direction $=$ Right to left

(b) $3 cm ; 5.73 Hz$

(c) $\frac{\pi}{4}$

(d) $3.49 m$

Explanation:

The equation of a progressive wave travelling from right to left is given by the displacement function:

$y(x, t)=a \sin (\omega t+k x+\Phi) \ldots(i)$

The given equation is:

$y(x, t)=3.0 \sin (36 t+0.018 x+\frac{\pi}{4})\ldots(ii)$

On comparing both the equations, we find that equation (ii) represents a travelling wave, propagating from right to left.

Now, using equations $(i)$ and $(ii)$, we can write:

$\omega=36 rad / s$ and $k=0.018 m^{-1}$

We know that:

$v=\frac{\omega}{2 \pi}$ and $\quad \lambda=\frac{2 \pi}{k}$

Also,

$v=v \lambda$

$ \begin{aligned} \therefore v & =(\frac{\omega}{2 \pi}) \times(\frac{2 \pi}{k})=\frac{\omega}{k} \\ & =\frac{36}{0.018}=2000 cm / s=20 m / s \end{aligned} $

Hence, the speed of the given travelling wave is $20 m / s$.

Amplitude of the given wave, $a=3 cm$

Frequency of the given wave:

$v=\frac{\omega}{2 \pi}=\frac{36}{2 \times 3.14}=5.73 Hz$

On comparing equations $(i)$ and (ii), we find that the initial phase angle, $\phi=\frac{\pi}{4}$

The distance between two successive crests or troughs is equal to the wavelength of the wave.

Wavelength is given by the relation:

$k=\frac{2 \pi}{\lambda}$

$\therefore \lambda=\frac{2 \pi}{k}=\frac{2 \times 3.14}{0.018}=348.89 cm=3.49 m$

Answer

All the waves have different phases.

The given transverse harmonic wave is:

$y(x, t)=3.0 \sin (36 t+0.018 x+\frac{\pi}{4})$

For $x=0$, the equation reduces to:

$y(0, t)=3.0 \sin (36 t+\frac{\pi}{4})$

Also, $\omega=\frac{2 \pi}{T}=36 rad / s^{-1}$

$\therefore T=\frac{\pi}{8} s$

Now, plotting $y$ vs. $t$ graphs using the different values of $t$, as listed in the given table.

For $x=0, x=2$, and $x=4$, the phases of the three waves will get changed. This is because amplitude and frequency are invariant for any change in $x$. The $y$ - $t$ plots of the three waves are shown in the given figure.

Answer

Equation for a travelling harmonic wave is given as:

$y(x, t)=2.0 \cos 2 \pi(10 t-0.0080 x+0.35)$

$=2.0 \cos (20 \pi t-0.016 \pi x+0.70 \pi)$

Where,

Propagation constant, $k=0.0160 \pi$

Amplitude, $a=2 cm$

Angular frequency, $\omega=20 \pi rad / s$

Phase difference is given by the relation:

$\phi=k x=\frac{2 \pi}{\lambda}$

For $x=4 m=400 cm$

$\Phi=0.016 \pi \times 400=6.4 \pi rad$

For $0.5 m=50 cm$

$\Phi=0.016 \pi \times 50=0.8 \pi rad$

For $x=\frac{\lambda}{2}$

$\phi=\frac{2 \pi}{\lambda} \times \frac{\lambda}{2}=\pi rad$

For $x=\frac{3 \lambda}{4}$

$\phi=\frac{2 \pi}{\lambda} \times \frac{3 \lambda}{4}=1.5 \pi rad$

Answer

(a) The general equation representing a stationary wave is given by the displacement function: $y(x, t)=2 a \sin k x \cos \omega t$

This equation is similar to the given equation:

$y(x, t)=0.06 \sin (\frac{2}{3} x) \cos (120 \pi t)$

Hence, the given function represents a stationary wave.

(b) A wave travelling along the positive $x$-direction is given as:

$y_1=a \sin (\omega t-k x)$

The wave travelling along the negative $x$-direction is given as:

$y_2=a \sin (\omega t+k x)$

The superposition of these two waves yields:

$$ \begin{align*} & y=y_1+y_2=a \sin (\omega t-k x)-a \sin (\omega t+k x) \\ & =a \sin (\omega t) \cos (k x)-a \sin (k x) \cos (\omega t)-a \sin (\omega t) \cos (k x)-a \sin (k x) \cos (\omega t) \\ & =-2 a \sin (k x) \cos (\omega t) \\ & =-2 a \sin (\frac{2 \pi}{\lambda} x) \cos (2 \pi v t) \tag{i} \end{align*} $$

The transverse displacement of the string is given as:

$y(x, t)=0.06 \sin (\frac{2 \pi}{3} x) \cos (120 \pi t) \ldots {\text{(ii)}}$

Comparing equations ( $i$ ) and (ii), we have:

$\frac{2 \pi}{\lambda}=\frac{2 \pi}{3}$

$\therefore$ Wavelength, $\lambda=3 m$

It is given that:

$120 \pi=2 \pi \nu$

Frequency, $v=60 Hz$

Wave speed, $v=\nu \lambda$ $=60 \times 3=180 m / s$

The velocity of a transverse wave travelling in a string is given by the relation:

$v=\sqrt{\frac{T}{\mu}}$

Where,

Velocity of the transverse wave, $v=180 m / s$

Mass of the string, $m=3.0 \times 10^{-2} kg$

Length of the string, $l=1.5 m$

Mass per unit length of the string, $\mu=\frac{m}{l}$

$ \begin{aligned} & =\frac{3.0}{1.5} \times 10^{-2} \\ & =2 \times 10^{-2} kg m^{-1} \end{aligned} $

(c) Tension in the string $=T$

From equation $(i)$, tension can be obtained as:

$T=v^{2} \mu$

$=(180)^{2} \times 2 \times 10^{-2}$

$=648 N$

Answer

(i)

Yes, except at the nodes

Yes, except at the nodes

No

$0.042 m$

Explanation:

(i)

All the points on the string oscillate with the same frequency, except at the nodes which have zero frequency.

All the points in any vibrating loop have the same phase, except at the nodes.

All the points in any vibrating loop have different amplitudes of vibration.

The given equation is:

$ y(x, t)=0.06 \sin (\frac{2 \pi}{3} x) \cos (120 \pi t) $

For $x=0.375 m$ and $t=0$

$ \begin{aligned} \text{ Amplitude }=\text{ Displacement } & =0.06 \sin (\frac{2 \pi}{3} x) \cos 0 \\ & =0.06 \sin (\frac{2 \pi}{3} \times 0.375) \times 1 \\ & =0.06 \sin (0.25 \pi)=0.06 \sin (\frac{\pi}{4}) \\ & =0.06 \times \frac{1}{\sqrt{2}}=0.042 m \end{aligned} $

Answer

(a) The given equation represents a stationary wave because the harmonic terms $k x$ and $\omega t$ appear separately in the equation.

(b) The given equation does not contain any harmonic term. Therefore, it does not represent either a travelling wave or a stationary wave.

(c) The given equation represents a travelling wave as the harmonic terms $k x$ and $\omega t$ are in the combination of $k x-\omega t$.

(d) The given equation represents a stationary wave because the harmonic terms $k x$ and $\omega t$ appear separately in the equation. This equation actually represents the superposition of two stationary waves.

Answer

Mass of the wire, $m=3.5 \times 10^{-2} kg$

Linear mass density, $\mu=\frac{m}{l}=4.0 \times 10^{-2} kg m^{-1}$

Frequency of vibration, $v=45 Hz$

$\therefore$ Length of the wire, $\quad l=\frac{m}{\mu}=\frac{3.5 \times 10^{-2}}{4.0 \times 10^{-2}}=0.875 m$

The wavelength of the stationary wave $(\lambda)$ is related to the length of the wire by the relation:

$\lambda=\frac{2 l}{n}$

Where, $n=$ Number of nodes in the wire

For fundamental node, $n=1$ :

$\lambda=2 l$

$\lambda=2 \times 0.875=1.75 m$

The speed of the transverse wave in the string is given as:

$v=v \lambda=45 \times 1.75=78.75 m / s$

The tension produced in the string is given by the relation:

$T=v^{2} \mu$

$=(78.75)^{2} \times 4.0 \times 10^{-2}=248.06 N$

Answer

Frequency of the turning fork, $v=340 Hz$

Since the given pipe is attached with a piston at one end, it will behave as a pipe with one end closed and the other end open, as shown in the given figure.

Such a system produces odd harmonics. The fundamental note in a closed pipe is given by the relation:

$l_1=\frac{\lambda}{4}$

Where,

Length of the pipe, $l_1=25.5 cm=0.255 m$

$\therefore \lambda=4 l_1=4 \times 0.255=1.02 m$

The speed of sound is given by the relation:

$v=v \lambda=340 \times 1.02=346.8 m / s$

Answer

Length of the steel rod, $l=100 cm=1 m$

Fundamental frequency of vibration, $v=2.53 kHz=2.53 \times 10^{3} Hz$

When the rod is plucked at its middle, an antinode (A) is formed at its centre, and nodes $N)$ are formed at its two ends, as shown in the given figure.

The distance between two successive nodes is $\frac{\lambda}{2}$.

$\therefore l=\frac{\lambda}{2}$

$\lambda=2 l=2 \times 1=2 m$

The speed of sound in steel is given by the relation:

$v=v \lambda$

$=2.53 \times 10^{3} \times 2$

$=5.06 \times 10^{3} m / s$

$=5.06 km / s$

Answer

First (Fundamental); No

Length of the pipe, $l=20 cm=0.2 m$

Source frequency $=n^{\text{th }}$ normal mode of frequency, $v_n=430 Hz$

Speed of sound, $v=340 m / s$

In a closed pipe, the $n^{\text{th }}$ normal mode of frequency is given by the relation:

$v_n=(2 n-1) \frac{v}{4 l} \quad ; n$ is an integer $=0,1,2,3 \ldots$

$430=(2 n-1) \frac{340}{4 \times 0.2}$

$2 n-1=\frac{430 \times 4 \times 0.2}{340}=1.01$

$2 n=2.01$

$n \sim 1$

Hence, the first mode of vibration frequency is resonantly excited by the given source.

In a pipe open at both ends, the $n^{\text{th }}$ mode of vibration frequency is given by the relation:

$ \begin{aligned} v_n & =\frac{n v}{2 l} \\ n & =\frac{2 l v_n}{v} \\ & =\frac{2 \times 0.2 \times 430}{340}=0.5 \end{aligned} $

Since the number of the mode of vibration $(n)$ has to be an integer, the given source does not produce a resonant vibration in an open pipe.

Answer

Frequency of string A, $f_A=324 Hz$

Frequency of string $B=f_B$

Beat’s frequency, $n=6 Hz$

Beat’s frequency is given as:

$n=|f_A \pm f_B|$

$6=324 \pm f_B$

$f_B=330 Hz$ or $318 Hz$

Frequency decreases with a decrease in the tension in a string. This is because frequency is directly proportional to the square root of tension. It is given as:

$v \propto \sqrt{T}$

Hence, the beat frequency cannot be $330 Hz$

$\therefore f_B=318 Hz$

Answer

(a) A node is a point where the amplitude of vibration is the minimum and pressure is the maximum. On the other hand, an antinode is a point where the amplitude of vibration is the maximum and pressure is the minimum.

Therefore, a displacement node is nothing but a pressure antinode, and vice versa.

(b) Bats emit very high-frequency ultrasonic sound waves. These waves get reflected back toward them by obstacles. A bat receives a reflected wave (frequency) and estimates the distance, direction, nature, and size of an obstacle with the help of its brain senses.

(c) The overtones produced by a sitar and a violin, and the strengths of these overtones, are different. Hence, one can distinguish between the notes produced by a sitar and a violin even if they have the same frequency of vibration.

(d) Solids have shear modulus. They can sustain shearing stress. Since fluids do not have any definite shape, they yield to shearing stress. The propagation of a transverse wave is such that it produces shearing stress in a medium. The propagation of such a wave is possible only in solids, and not in gases.

Both solids and fluids have their respective bulk moduli. They can sustain compressive stress. Hence, longitudinal waves can propagate through solids and fluids.

(e) A pulse is actually is a combination of waves having different wavelengths. These waves travel in a dispersive medium with different velocities, depending on the nature of the medium. This results in the distortion of the shape of a wave pulse.