Sound Wave:Displacement Equation

Q. When a sound wave of frequency $300Hz$ passes through a medium, the maximum displacement of a particle of the medium is $0.1cm$. The maximum velocity of the particle is equal to

1. $60\pi \text{cm/sec}$
2. $30\pi \text{cm/sec}$
3. $30\text{cm/sec}$
4. $60\text{cm/sec}$

Q.

A travelling wave is produced on a long horizontal string by vibrating an end up and down sinusoidally. The amplitude of vibration is 1.0 cm and the displacement becomes zero 200 times per second. The linear mass density of the string is $0.10kg{m}^{-1}$ and it is kept under a tension of 90 N. (a) Find the speed and the wavelength of the wave. (b) Assume that the wave moves in the positive x-direction and at t = 0, the end x = 0 is at its positive extreme position. Write the wave equation. (c) Find the velocity and acceleration of the particle at x = 50 cm at time t = 10 ms.

Q. Standing waves are set up in a string of length $240cm$ clamped horizontally at both ends. The separation between any two consecutive points where displacement amplitude is $3\sqrt{2}cm$ is $20cm$. The standing waves were set by two travelling waves of equal amplitude of $3cm$. The overtone in which the string is vibrating will be

Q. A string is clamped at both the ends and it is vibrating in its <!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> $4^{\text{th}}$ harmonic. The equation of the stationary wave is $Y=0.3\sin \left(0.157x\right)\cos \left(200\pi t\right).$ The length of the string is
(All quantities are in SI units.)

1. $40\text{m}$
2. $20\text{m}$
3. $60\text{m}$
4. $80\text{m}$

Q.

A car moves with a speed of 54 km $h^{-1}$ towards a cliff. The horn of the car emits sound of frequency 400 Hz at a speed of 335 m $s^{-1}$. (a) Find the wavelength of the sound emitted by the horn in front of the car. (b) Find the wavelength of the wave reflected from the cliff. (c) What frequency does a person sitting in the car hear for the reflected sound wave ? (d) How many beats does he hear in 10 seconds between the sound coming directly from the horn and that coming after the reflection ?

Q. Two sound sources $S_1$ and $S_2$ emit pure sinusoidal waves in phase. If the speed of sound is $330\text{m/s}$, then for what frequencies does constructive interference occur at P?

1. $500\text{Hz}$
2. $1000\text{Hz}$
3. $1200\text{Hz}$
4. $1500\text{Hz}$

Q.

A train approaching a platform at a speed of 54 km/h sounds a whistle. An observer on the platform finds its frequency to be 1620 Hz. The train passes the platform keeping the whistle on and without slowing down. What frequency will the observer hear after the train has crossed the platfrom ? The speed of sound in air = 332 m/s.

Q. how can we represent displacement x of SHM in terms of sine function other than cosine.?

Q. The displacement equation of a travelling sound wave along $x$ axis is given by $s=5\times {10}^{-5}\sin (500t-2x)$, where $s$ and $x$ are in meters and $t$ in seconds. Choose the correct option(s) among the following.

1. Ratio of displacement amplitude of the particles to the wavelength of wave is $1.1\times {10}^{-5}$.
2. Ratio of displacement amplitude of the particles to the wavelength of wave is $1.59\times {10}^{-5}$.
3. Ratio of the velocity amplitude of the particles to the wave speed is $2\times {10}^{-4}$.
4. Ratio of the velocity amplitude of the particles to the wave speed is ${10}^{-4}$.

Q. The equation of a travelling sound wave along $x-axis$ is given as $S=9\sin (400t-2.7x)$, where $S$ is measured in ${10}^{-4}\text{m}$, $t$ in seconds and $x$ in metres. Then:

1. Ratio of displacement amplitude to wavelength of wave is $3.86\times {10}^{-4}$.
2. The velocity amplitude of the particle is $36\times {10}^{-2}\text{m/s}$.
3. Ratio of the velocity amplitude of the particle to the wave speed is $2.43\times {10}^{-3}$.
4. The velocity amplitude of the particle is $3.6\times {10}^{-2}\text{m/s}$.

Q.

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.

Q. Two sound waves, originating from the same source, travel along different paths in air and then meet at a point. If the source vibrates at a frequency of $1\text{kHz}$ and one path is $83\text{cm}$ shorter than the other, then the phase difference between the waves arriving at the point of observation is ____$\pi$. The speed of sound in air is $332{\text{ms}}^{-1}.$

​​​​​​​

Q. A whistle of frequency 540 Hz is moving in a circle of radius 2 ft. at a constant angular speed of 15 $rad/s$. If the listener is at rest, standing far away from the centre, then which of the following is/are true?
(Consider velocity of sound in air as 1100 ft/s)

1. Lowest frequency heard by the listener is 515 Hz
2. Lowest frequency heard by the listener is 525 Hz
3. Highest frequency heard by the listener is 545 Hz
4. Highest frequency heard by the listener is 555 Hz

Q. The angle between wave velocity and particle velocity in a travelling wave may be

1. Zero
2. $\frac{\pi }{2}$
3. $\pi$
4. All of these

Q. A string is producing transverse vibration whose equation is $y=0.021\sin (x+30t)$, where $x$ and $y$ are in metre and $t$ is in second. If the linear density of the string is $1.3\times {10}^{-4}kg/m$, then tension in string in newton will be

1. $10$
2. $0.5$
3. $1$
4. $0.11$

Q. A taut string for which linear mass density is $\mu =5\times {10}^{-2}\text{kg/m}$ is under a tension of $80\text{N}$. How much power must be supplied to the string to generate sinusoidal waves at a frequency of $60\text{Hz}$ and an amplitude of $6\text{cm}$ ?

1. $312\text{W}$
2. $512\text{W}$
3. $412\text{W}$
4. $312\text{W}$

Q. A source of sound and a detector are placed at the same place on ground. At $t=0$ , the source $S$ is projected towards reflector with velocity $v_0$ in vertical upward direction and reflector starts moving down with constant velocity $v_0$. At $t=0$ , the vertical separation between the reflector and source is $H(>\frac{v_0^2}{2g})$. The speed of sound in air is $v(>>v_0)$. Take $f_0$ as the frequency of sound waves emitted by source. Find the frequency of sound received by detector after being reflected by reflector at $t=\frac{v_0}{2g}$.

1. $f_2=f_0\left(\frac{v^2}{(v-v_0)(2v-v_0)}\right)$
2. $f_2=2f_0\left(\frac{v^2}{(v-v_0)(2v-v_0)}\right)$
3. $f_2=f_0\left(\frac{v(v+v_0)}{(v-v_0)(2v-v_0)}\right)$
4. $f_2=2f_0\left(\frac{v(v+v_0)}{(v-v_0)(2v-v_0)}\right)$

Q. A source and an observer move away from each other with speed of $10m/s$ with respect to ground. Apparent frequency of the source is $1950Hz$. The natural frequency of the source is (velocity of sound is $340m/s$)

1. $1950Hz$
2. $1832Hz$
3. $1650Hz$
4. $2068Hz$

Q. Beats are produced by two waves given by $y_1=a\sin 2000\pi t$ and $y_2=a\sin 2008\pi t$. The number of beats heard per second is

1. two
2. One
3. Four
4. Eight

Q. A bat flies perpendicular to a wall at $6m/s,$ emitting a sound of frequency $4.5\times {10}^{4}Hz$. The number of beats it hears per second is (velocity of sound in air $=340m/s$)

1. $1616$
2. $21.37$
3. $1200$
4. $92.7$

Q. If two waves, originating from sources represented by $y_1=4\sin \left(\omega t\right)$ and $y_2=3\sin (\omega t+\frac{\pi }{3})$ interfere at a point. The amplitude of that point will be about,

1. $7$
2. $6$
3. $5$
4. $3.5$

Q. When one mole of monoatomic gas is mixed with one mole of a diatomic gas, Then the equivalent value of $\gamma$ for the mixture will be (vibration mode neglected)

1. $1.33$
2. $1.40$
3. $1.50$
4. $1.6$

Q. A simple harmonic wave of amplitude 2 units is travelling along positive x – axis. At a given instant, displacement for two different particles on the wave are 1 unit and $\sqrt{2}$ unit. If the wavelength of wave is $\lambda$, find the distance between them.

1. $\frac{\lambda }{12}$
2. $\frac{\lambda }{8}$
3. $\frac{\lambda }{24}$
4. $\frac{\lambda }{16}$

Q. In a medium in which a transverse progressive wave is travelling, the phase difference between two points with a separation of $1.25\text{cm}$ is $\left(\pi /4\right)$. If the frequency of wave is $1000\text{Hz}$, its velocity in $\text{m/s}$ will be

Q. An AC source of angular frequency $\omega$ is applied across an inductor. The instantaneous power developed in the circuit, has an angular frequency of,

1. $\omega$
2. $2\omega$
3. $\frac{\omega }{2}$
4. $\frac{\omega }{4}$

Q. A child playing with a long rope ties one end and holds the other. The rope is stretched taut along the horizontal. The child shakes the end he is holding , up and down, in a sinusoidal manner with amplitude $10cm$ and frequency $3Hz.$ Speed of the wave is $15m/s$ and at $t=0,$ displacement at the child's end is maximum positive. Assuming that there is no wave reflected from the fixed end, so that the waves in the rope are plane progressive waves, answer the following questions.
(Also assume that the wave propagates along the positive x-direction).
Equation of displacement of a point $2.5m$ from the child's end can be expressed as

1. $y=-\left(0.1m\right)sin\left(10rad/s\right)t$
2. $y=\left(0.1m\right)sin\left(4rad/s\right)t$
3. $y=-\left(0.1m\right)cos\left(18.8rad/s\right)t$
4. $y=0.1mcos\left(12.5rad/s\right)t$

Q. A $200\text{Hz}$ wave with amplitude $1\text{mm}$ travels on a long string of linear mass density $6{\text{g m}}^{-1}$ kept under a tension of $60\text{N}$. The average power transmitted across a given point in the string is

1. $0.63\text{W}$
2. $0.83\text{W}$
3. $0.47\text{W}$
4. $0.89\text{W}$

Q.

Q. A sine wave is travelling in a medium. The minimum distance between the two particles always having same speed speed, is

1. $\frac{\lambda }{4}$
2. $\frac{\lambda }{3}$
3. $\frac{\lambda }{2}$
4. $\lambda$

Q. The equation of a transverse wave is given by $y=10sin\pi (0.01x-2t)$ where x and y are in cm and t is in second. Its frequency is

1. 10sec^-1
2. 2sec^-1
3. 1sec^-1
4. 0.01sec^-1