Resultant Wave Equation

Q. A wave is represented by the equation $y=Asin(10\pi x+15\pi t+\frac{\pi }{3})$ where x is in metre and t is in second. The expression represents

1. a wave travelling in positive x-direction with a velocity 1.5 m/s
2. a wave travelling in the negative x-direction having a wavelength 0.4 m
3. a wave travelling in negative x-direction with a velocity1.5 m/s
4. a wave travelling in positive x-direction of wavelength 0.1 m

Q. Two waves are simultaneously passing through a string and their equations are : $y_1=A_1\sin k(x-vt),y_2=A_2\sin k(x-vt+x_0)$ Given amplitudes $A_1=12\text{mm}$ and $A_2=5\text{mm}$, $x_0=3.5\text{cm}$ and wave number $k=6.28{\text{cm}}^{-1}$. The amplitude of resulting wave will be _____ mm.

Q. A wave equation which gives the displacement along the y-direction is given by $y={10}^{-4}sin(60t+2x)$, where $x$ and $y$ are in metre and $t$ is time in second. This represents a wave

1. travelling with a velocity of $30m/s$ in the negative x-direction
2. of wavelength $\pi m$
3. of frequency $\left(30/\pi \right)Hz$
4. of amplitude ${10}^{-4}m$ travelling along the negative x-direction

Q. When two waves with same frequency and constant phase difference interfere,

1. there is a gain of energy
2. there is a loss of energy
3. the energy is redistributed and the distribution changes with time
4. the energy is redistributed and the distribution remains constant in time

Q. Two travelling sinusoidal waves are described by the wave functions
$y_1=10\sin [\pi (8x-1400t)]$
and $y_2=10\sin [\pi (8x-1400t-0.5)]$ Where $x$, $y_1$, and $y_2$ are in metres and $t$ is in seconds . which of the following options is/are correct ?

1. Amplitude of the resultant wave is $10\sqrt{2}m$
2. Frequency of the resultant wave is $700$ $Hz$
3. Time period of the resultant wave is $0.1$ $s$
4. None of these

Q. A wire of linear density $9.8\times {10}^{-3}\text{kg/m}$ passes over a frictionless light pulley fixed on the top of a frictionless inclined plane which makes an angle ${30}^0$ with the horizontal. Masses $m$ and $M$ are tied at the two ends of the wire such that $m$ rests on the plane and $M$ hangs freely vertically downwards. The entire system is in equilibrium and a transverse wave propagates along the wire with a velocity of $100\text{m/s}$. Then:

1. $m=20\text{kg}$
2. $M=5\text{kg}$
3. $\frac{m}{M}=\frac{1}{2}$
4. $\frac{m}{M}$=2

Q. When we clap our hands, the sound produced is best described by

1. $p=p_{0}sin(kx-\omega t)$
2. $p=p_{0}sinkxcos\omega t$
3. $p=p_{0}coskxsin\omega t$
4. $p=\sum p_{on}sin(k_{n}x-{\omega }_{n}t)$

Q. A sinusoidal wave travels along a taut string of linear mass density $0.1\text{g/cm}$. The particles oscillate along $y-$ direction and the wave moves in the positive $x-$direction. The amplitude and frequency of oscillation are $2\text{mm}$ and $50\text{Hz}$ respectively. The minimum distance between two particles oscillating in the same phase is $4\text{m}$. The tension in the string is

1. $4000\text{N}$
2. $400\text{N}$
3. $45\text{N}$
4. $250\text{N}$

Q. Sources separated by $20\text{m}$ vibrate according to the equation $y_1=0.06\sin \pi t$ and $y_2=0.02\sin \pi t.$ They send out waves along a rod with speed $3\text{m/s}$. What is the equation of motion of a particle $12\text{m}$ from the first source and $8\text{m}$ from the second? Given $y_1,y_2$ are in metres.

1. $0.0173\sin \pi t-0.05\cos \pi t$
2. $0.0173\sin \pi t-0.017\cos \pi t$
3. $0.05\sin \pi t-0.0173\cos \pi t$
4. $0.05\sin \pi t-0.5\cos \pi t$

Q. If two waves represented by
$y_1=4\sin \omega t$ , and
$y_2=3\sin (\omega t+\frac{\pi }{3})$
(where $y_1,y_2$ are in $\text{cm}$ and $t$ is in $\text{sec}$),
interfere at a point, the resultant amplitude of that point will be about,

1. $7\text{cm}$
2. $6\text{cm}$
3. $5\text{cm}$
4. $3.5\text{cm}$

Q. The resultant amplitude, when two waves of same frequency but with amplitudes $a_1$ and $a_2$ superimpose with a phase difference of $\pi /2$ will be

1. $\sqrt{a_1^2+a_2^2}$
2. $a_1^2+a_2^2$
3. $a_1+a_2$
4. $a_1-a_2$

Q. The displacement due to a wave moving in the positive $x$-direction is given by $y=\frac{1}{(1+x^2)}$ at time $t=0$ and by $y=\frac{1}{[1+(x-1{)}^2]}$ at $t=2$ seconds, where $x$ and $y$ are in metres. The velocity of the wave in $\text{m/s}$ is:

1. $0.5$
2. $1$
3. $2$
4. $4$

Q. The displacement equation of a particle is
$x=\left[3\sin \right(4t)+4\sin (4t\left)\right]cm$ . The amplitude and maximum speed of the particle during oscillation will be

1. $4cm,2cm{s}^{-1}$
2. $2cm,4cm{s}^{-1}$
3. $10cm,5cm{s}^{-1}$
4. $5cm,20cm{s}^{-1}$

Q. A simple harmonic motion is given by the equation $y=5(\sin 3\pi t+\sqrt{3}\cos 3\pi t)$. What is the amplitude of motion if y is in m ?

1. 100 cm
2. 5 m
3. 200 cm
4. 1000 cm

Q. A travelling wave represented by $y=A\sin (\Omega \text{t-Kx)}$ is superimposed on another wave represented by $y=A\sin (\Omega \text{t+Kx)}$. The resultant is

1. A standing wave having nodes at
   $x=(n+\frac{1}{2})\frac{\lambda }{2},n=0,1,2$
2. A standing wave having nodes at $\frac{n\lambda }{2};n=0,1,2$
3. A wave travelling along $+x$ direction
   
4. A wave travelling along $-\text{x}$ direction
   

Q. In an experimental setup shown below


A block of mass $m=10\text{kg}$ is hanging. If mass of string is $0.01\text{kg}$ and length of string is $1\text{m}$. Find the number of loop formed if a standing wave is set on the string. Frequency of the wave is given as $50\text{Hz}$.

1. $1$
2. $2$
3. $4$
4. $3$

Q. A wave travelling along a string is described by the equation
$y=A\sin (\omega t-kx)$
The maximum velocity of a particle of the wave is

1. $A\omega$
2. $\omega /k$
3. $d\omega /dk$
4. $x/l$

Q. The graph shows three waves travelling with same velocity, that are separately sent along a string, which is under a certain tension along $x-$axis. If ${\omega }_1,{\omega }_2$ and ${\omega }_3$ are their angular frequencies respectively, then,

1. ${\omega }_1={\omega }_3>{\omega }_2$
2. ${\omega }_1>{\omega }_2>{\omega }_3$
3. ${\omega }_2>{\omega }_1={\omega }_3$
4. ${\omega }_1={\omega }_2={\omega }_3$

Q.

Two harmonic waves are represented in SI units by, $y_1(x,t)=0.2sin(x-3.0t)$ and $y_2(x,t)=0.2sin(x-3.0t+\varphi )$. What is the expression for the sum $y=y_1+y_{2}for\varphi =\frac{\pi }{2}$.

1. y = 8 sin (x - 3.0t)

2. y = 0.28 sin (x - 3.0t+$\frac{\pi }{4}$)

3. y = 0.2 sin (x - 3.0t+$\frac{\pi }{4}$)

4. y = 0.28 cos (x - 3.0t+$\frac{\pi }{4}$)

Q. The displacement y of a wave travelling in the x-direction is given by $y={10}^{-4}\sin (600t-2x+\frac{\pi }{2})$ metres. Where x is expressed in metre and t in second. The speed of the wave motion is

1. 300 m/s
2. 200 m/s
3. 600 m/s
4. 1200 m/s

Q. A rope, under a tension of $200N$ and fixed at both ends, oscillates in a second-harmonic standing wave pattern. The displacement of the rope is given by : $y=0.1\sin \left(\frac{\pi x}{2}\right)\sin 12\pi t$.
Where $x=0$ at one end of the rope, $x$ is in meters and $t$ is in seconds. The length of the rope is

Q. If the voltage in an ac circuit is represented by the equation, $V=220\sqrt{2}\sin (314t-\varphi )$, calculate RMS value of the voltage.

1. $220\sqrt{2}V$
2. $220V$
3. $314V$
4. $220/\sqrt{2}V$

Q.

Two harmonic waves are represented in SI units by, $y_1(x,t)=0.2sin(x-3.0t)$ and $y_2(x,t)=0.2sin(x-3.0t+\varphi )$. What is the expression for the sum $y=y_1+y_{2}for\varphi =\frac{\pi }{2}$.

1. y = 8 sin (x - 3.0t)

2. y = 0.28 sin (x - 3.0t+$\frac{\pi }{4}$)

3. y = 0.2 sin (x - 3.0t+$\frac{\pi }{4}$)

4. y = 0.28 cos (x - 3.0t+$\frac{\pi }{4}$)

Q. The transverse displacement of a string clamped at its both ends in given by
$y(x,T)=2sin\left(\frac{2\pi }{3}x\right)cos\left(100\pi t\right)$ Where $x$ and $y$ are in cm and $t$ is in $s$. Which of the following statement is correct?

1. All the points on the string between two consecutive nodes vibrate with same frequency, phase and amplitude.
2. All the points on the string between two consecutive nodes vibrate with same frequency, phase but different amplitude.
3. All the points on the string between two consecutive nodes vibrate with different frequency and phase but same amplitude.
4. All the points on the string between two consecutive nodes vibrate with different frequency, phase and amplitude.

Q.

Two harmonic waves are represented in SI units by, $y_1(x,t)=0.2sin(x-3.0t)$ and $y_2(x,t)=0.2sin(x-3.0t+\varphi )$. What is the expression for the sum $y=y_1+y_{2}for\varphi =\frac{\pi }{2}$.

1. y = 8 sin (x - 3.0t)

2. 

3. 

4. none of these

Q. Two travelling sinusoidal waves are described by the wave functions $y_1=10\sin [\pi (8x-1400t)]$
and $y_2=10\sin [\pi (8x-1400t-0.5)]$ where $x$, $y_1$, and $y_2$ are in metres and $t$ is in seconds. Which of the following option(s) is/are correct ?

1. Amplitude of the resultant wave is $10\sqrt{2}m$
2. Frequency of the resultant wave is $700$ $Hz$
3. Time period of the resultant wave is $0.1$ $s$
4. None of these

Q. The equation $y=5\sin \left(3x/50\right)\cos \left(450t\right)$ represents the stationary wave setup in a vibrating sonometer wire, where $x,y$ are in $cm$ and $t$ in second. The velocity of one of the two progressive waves in that stationary wave is

1. $2.7ms-1$
2. $27m{s}^{-1}$
3. $75m{s}^{-1}$
4. $7.5m{s}^{-1}$

Q. A standing wave pattern on a string is described by
$y(x,t)=0.040\left(\sin 5\pi x\right)\left(\cos 40\pi t\right)$
where $x$ and $y$ are in meters and $t$ is in seconds. For $x\ge 0,$ What is the speed of the two traveling waves that interfere to produce this wave?

Q. What will be the equation of AC of frequency $75Hz$ if its rms value is $20A$?

Q. Assertion :Two waves $y_1$ = A sin $(\omega t+kx)$ and $y_1$ = A cos $(\omega t-kx)$ are superimposed, then x = o, becomes a node. Reason: At node net displacement due to two waves should be zero.

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Assertion is incorrect but Reason is correct