• Formation of stationary waves - class-XI
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Formation of stationary waves - class-XI

Description: formation of stationary waves
Number of Questions: 18
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Tags: stationary waves waves oscillations and waves wave motion physics
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A standing wave is represented by an equation $y= 20 sin (50 \pi t )cos (10 \pi x)$. The frequency of the wave is

physics stationary waves formation of stationary waves stationary waves and its graphical representation standing waves in strings from moving to stationary stationary (or standing) waves

  1. 20 Hz

  2. 50 Hz

  3. 25 Hz

  4. 10 Hz

Show answer
Correct Option: C
Explanation:

The equation of a stationary wave is $y= 2A sin \omega t cos K x$. Comparing this equation with the equation given in the problem, we have
$2 \pi f = 50 \implies f = 25 Hz$

The correct option is (c)

Which of the following statement is incorrect during propagation of plane progressive mechanical wave?

physics stationary waves formation of stationary waves stationary waves and its graphical representation standing waves in strings from moving to stationary stationary (or standing) waves

  1. All the particles are vibrating in the same phase.

  2. Amplitude of all the particles is equal.

  3. Particles of the medium executes SHM.

  4. Wave velocity depends upon the nature of the medium.

Show answer
Correct Option: A
Explanation:

During propagation of a plane progressive mechanical wave all the particles are vibrating with different phases.
While all other statement are correct.

A standing wave is given by the equation $x=10 sin 5\pi t cos 3 x$. The amplitude of the wave will be

physics stationary waves formation of stationary waves stationary waves and its graphical representation standing waves in strings from moving to stationary stationary (or standing) waves

  1. a constant at all times

  2. will be increasing as t increases

  3. will be decreasing as t increases

  4. will fluctuate at x increases

Show answer
Correct Option: D
Explanation:

The amplitude of the stationary wave is $10 cos 3 x$ and this fluctuates as x increases

The correct option is (d)

A standing wave is represented by an equation $y= 10 sin 50 \pi t cos 10 \pi x$. The distance between adjacent nodes of the wave is

physics stationary waves formation of stationary waves stationary waves and its graphical representation standing waves in strings from moving to stationary stationary (or standing) waves

  1. 0.5 m

  2. 0.2 m

  3. 0.1 m

  4. 0.3 m

Show answer
Correct Option: C
Explanation:

At nodes, displacement $y=0$

For first node:
$\cos { \left( 10\pi x \right) =0\quad =\cos { \left( \cfrac { \pi  }{ 2 }  \right)  }  } \ \therefore \quad 10\pi x=\cfrac { \pi  }{ 2 } \ x=\cfrac { 1 }{ 20 } $
For second node:
$\cos { \left( 10\pi x \right) =0\quad =\cos { \left( \cfrac { 3\pi  }{ 2 }  \right)  }  } \ \therefore \quad 10\pi x=\cfrac { 3\pi  }{ 2 } \ x=\cfrac { 3 }{ 20 } $
$\therefore$ distance between them $=\cfrac { 3 }{ 20 } -\cfrac { 1 }{ 20 } \ \quad \quad =\cfrac { 1 }{ 10 } =0.1m$

The frequency of a sound wave is 250 Hz and its wavelength is 100 cm. The distance travelled by a sound wave in the time taken to produce 100 waves is ________.

physics stationary waves formation of stationary waves stationary waves and its graphical representation standing waves in strings from moving to stationary stationary (or standing) waves

  1. 100m

  2. 200m

  3. 300m

  4. 400m

Show answer
Correct Option: A
Explanation:

$v = 250 \times 1 = 250  m  s^{-1}$
$t = \displaystyle \frac{100}{250} s = 0.45$
Distance travelled by wave in the time taken to produce 100 waves $=$ v $\times$ time
$= $ 250 $\times$ 0.45
$=112.5m \approx 100m$

The waves in which the particles of the medium travel in the same direction as the waves are

physics stationary waves formation of stationary waves stationary waves and its graphical representation standing waves in strings from moving to stationary stationary (or standing) waves

  1. linear waves

  2. longitudinal waves

  3. transverse waves

  4. electromagnetic waves

Show answer
Correct Option: B
Explanation:

The waves in which the displacement of the medium is in the same direction as, or the opposite direction to the direction of the propagation of the wave, are called lognitudinal waves. Ex$:-$ Sound waves, sewmic waves

While $:-$ 
The wave in which the displacement of the medium are at right angles to the direction of propogation, are called trensverse waves.
Ex$:-$ Electromagnetic waves, ripple on water surface.
Option (B) is the correct answer

Which of the following functions represent a traveling wave ?

physics stationary waves formation of stationary waves stationary waves and its graphical representation standing waves in strings from moving to stationary stationary (or standing) waves

  1. $({ x-vt })^{ 2 }$

  2. $({ x-vt })^{ 3 }$

  3. ${ 2 }^{ -(x-vt)^{ 2 } }$

  4. ${ e }^{ (x-vt) }$

Show answer
Correct Option: D

The equation of the stationary wave is
$y=2A\quad sin(\cfrac { 2\pi ct }{ \lambda  } )cos(\cfrac { 2\pi x }{ \lambda  } )$
Which of the following statement (s) is wrong?

physics stationary waves formation of stationary waves stationary waves and its graphical representation standing waves in strings from moving to stationary stationary (or standing) waves

  1. The unit of ct is same as that of $\lambda$

  2. The unit of x is same as that of $\lambda$

  3. The unit of $2\pi t$ is same as that of $2\pi x/\lambda t$

  4. The unit of $c/\lambda$ is same as that of $x/\lambda$

Show answer
Correct Option: C,D
Explanation:

$y=2Asin(\dfrac{2\pi ct}{\lambda}). cos(\dfrac{2\pi x}{\lambda})$


The unit of $\lambda $ and $x$ is $m$


The unit of $ct$ is $m$

The unit of $2\pi t$ is $s$.

The unit of $2\pi x/\lambda t$ is $s^{-1}$

The unit of $\dfrac{c}{\lambda}$ is $s^{-1}$

The unit of $x/\lambda$ is unit less.

The wrong statement is C and D both.

In a plane progressive harmonic wave, $V _{P}$ is the maximum particle speed and $V$ is the wave speed. If amplitude of wave is less than $\lambda/ 2\pi$, then

physics stationary waves formation of stationary waves stationary waves and its graphical representation standing waves in strings from moving to stationary stationary (or standing) waves

  1. $V = V _{P}$

  2. $V > V _{P}$

  3. $V _{P} < V$

  4. Unpredictable

Show answer
Correct Option: C

Standing waves are produced in $10m$ long stretched wire. If wire vibrates in five segments and wave velocity is $20m/s$, then the frequency is $(in\ Hz)$

physics stationary waves formation of stationary waves stationary waves and its graphical representation standing waves in strings from moving to stationary stationary (or standing) waves

  1. $5$

  2. $10$

  3. $15$

  4. $20$

Show answer
Correct Option: A

The equation of a stationary wave is given by $ y= 5cos \frac { \pi x }{ 3 }  sin 40 \pi t $. where y and x are given cm and time t in second then the amplitude of the progressive wave is

physics stationary waves formation of stationary waves stationary waves and its graphical representation standing waves in strings from moving to stationary stationary (or standing) waves

  1. $2.5 cm$

  2. $10 cm$

  3. $5 cm$

  4. $7.5 cm$

Show answer
Correct Option: A

The equation of progressive wave travelling along positive direction of x-axis having an amplitude of $0.04\ m$, frequency $440\ Hz$ and wave velocity $330 m/s$ is

physics stationary waves formation of stationary waves stationary waves and its graphical representation standing waves in strings from moving to stationary stationary (or standing) waves

  1. $y = 0.04\sin 2\pi \left (440t - \dfrac {4x}{3}\right )$

  2. $y = 0.04\cos 2\pi \left (440t - \dfrac {3x}{4}\right )$

  3. $y = 0.04\sin 2\pi \left (440t + \dfrac {4x}{3}\right )$

  4. $y = 0.04\cos 2\pi \left (440t + \dfrac {4x}{3}\right )$

Show answer
Correct Option: C

In a stationary wave

physics stationary waves formation of stationary waves stationary waves and its graphical representation standing waves in strings from moving to stationary stationary (or standing) waves

  1. Strain is maximum at nodes

  2. amplitude is minimum at nodes

  3. Strain is maximum at antinodes

  4. Amplitude is zero at all points

Show answer
Correct Option: B

The frequency of plane progressive wave is $100$ Hz. After how much time the same point will be $90^o$ out of phase?

physics stationary waves formation of stationary waves stationary waves and its graphical representation standing waves in strings from moving to stationary stationary (or standing) waves

  1. $2.5\times 0^{-3}s$.

  2. $3.5\times 0^{-3}s$.

  3. $4.5\times 0^{-3}s$.

  4. $5.5\times 0^{-3}s$.

Show answer
Correct Option: A
Explanation:

$w=2\Pi f$

where f is frequency of wave
Phase angle, $\theta wt$
$90^{\circ}=2\Pi ft$

$t=\dfrac{\Pi }{2\times 2\Pi f}$

$t=\dfrac{1}{4\times 100} $sec

$t=2.5\times 10^{-3}$ sec

Progressive wave are waves originating from a source such that they never return to the source.

physics stationary waves formation of stationary waves stationary waves and its graphical representation standing waves in strings from moving to stationary stationary (or standing) waves

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

progressive waves are waves after generation from the source the keep on propagating on the direction of propagation .

so the answer is A.

The equation, $Y=0.02 sin (500 \pi t) cos(4.5x)$ represents

physics stationary waves formation of stationary waves stationary waves and its graphical representation standing waves in strings from moving to stationary stationary (or standing) waves

  1. progressive wave of frequency 250 Hz along x-axis

  2. a stationary wave of wavelength 1.4 m

  3. a transverse progressive wave of amplitude 0.02 m

  4. progressive wave of speed of about $350ms^{-1} $

Show answer
Correct Option: B
Explanation:

Comparing the given wave equation with standard standing wave equation
$y (x, t) = A \sin (\omega t)\cos (kx)$, 


we get, $k =4.5$

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

$\Rightarrow \lambda = \dfrac{2\pi}{k} =1.4$ $m$

The equation of a progressive wave is $y=4\,sin(4\pi t-0.04x+\dfrac{\pi}{3})$ where x is in metre and t is in second. The velocity of the wave is

physics stationary waves formation of stationary waves stationary waves and its graphical representation standing waves in strings from moving to stationary stationary (or standing) waves

  1. $100\pi\,m/s$

  2. $50\pi\,m/s$

  3. $25\pi\,m/s$

  4. $\pi\,m/s$

Show answer
Correct Option: A
Explanation:

The equation of the progressive wave is given as, $y=4\,sin(4\pi t-0.04x+\dfrac{\pi}{3})$.

The velocity of the wave would be equal to

$\dfrac{\omega}{k}=\dfrac{4\pi}{0.04}=100\pi\;m/s$

Which of the following statements are correct?

physics stationary waves formation of stationary waves stationary waves and its graphical representation standing waves in strings from moving to stationary stationary (or standing) waves

  1. A wave front is a locus of points vibratig in same phase

  2. Wavelength is separation between two consecutive points vibrating in same phase

  3. For two sources to be coherent their frequencies must be same

  4. All of the above statements are correct.

Show answer
Correct Option: D
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