Tag: waves

Questions Related to waves

The equation of a traveling and stationary wave are ${ y } _{ 1 }=a sin(\omega t-kx)$ and ${ y } _{ 2 }=a \sin kx  \cos \omega t$. The phase difference between two point ${ x } _{ 1 }=\dfrac { \pi  }{ 4k }$ and $ { x } _{ 2 }=\dfrac { 4\pi  }{ 3k } $ are ${ \phi  } _{ 1 }$ and ${ \phi  } _{ 2 }$ respectively for two waves where k is the wave number, the ratio of ${ \phi  } _{ 1 }/{ \phi  } _{ 2 }$ 

standing waves waves physics

  1. 6/7

  2. 16/3

  3. 12/13

  4. 13/12

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Correct Option: D

A standing wave pattern is formed on a string. One of the waves is given by equation  $Y _ { 1 } a \cos ( \omega t - K X + \pi / 3 )$  then the equation of the other wave such at  $X = 0$  a noode is formal

standing waves waves physics

  1. $y _{ 2 } = \operatorname { a sin } \left( \omega t + K X + \dfrac { \pi } { 3 } \right)$

  2. $y _ { 2 } = a \cos \left( \omega t + K X + \dfrac { \pi } { 3 } \right)$

  3. $y _ { 2 } = a \cos \left( \omega t + K X + \dfrac { 2 \pi } { 3 } \right)$

  4. $y _ { 2 } = a \cos \left( \omega t + K X + \dfrac { 4 \pi } { 3 } \right)$

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Correct Option: D

Two simple harmonic waves of amplitude 5 cm and 3 cm and of the same frequency travelling with the same speed in opposite directions superpose to produce stationary waves. The ration of the amplitude at a node to that at an antinode in the resultant wave is

standing waves waves physics

  1. zero

  2. infinity

  3. 5:3

  4. 1:4

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Correct Option: A

The equation of stationary wave is given by $y=5\, cos (\pi x/3)\, sin 40 \pi t$ where y and x are given in cm and time t in second. Then a node occurs at the following distance 

standing waves waves physics

  1. 3 cm

  2. 10 cm

  3. 5 cm

  4. 1.5 cm

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Correct Option: A

A $string$ is stretched between fixed points separated by $75.0\ cm$. It is observed to have resonant frequencies of $420\ Hz$ and $315\ Hz$. There are no other resonant frequencies between these two.
Then, the lowest resonant frequency for this string is :

standing waves waves physics

  1. $1.05$Hz

  2. $1050$Hz

  3. $10.5$Hz

  4. $105$Hz

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Correct Option: D
Explanation:

Given,  $\dfrac { nV }{ 2L } =315\quad \longrightarrow (1)$

     &     $\dfrac { \left( n+1 \right) V }{ 2L } =420\quad \longrightarrow (2)$
equation (2) $-$ equation (1), we get
$\dfrac { \left( n+1 \right) V }{ 2L } -\dfrac { nV }{ 2L } =420-315$
$\Rightarrow \quad \left[ \dfrac { V }{ 2L } =105\quad { H } _{ 3 } \right] \rightarrow $  Lowest possible resonant frequency

$\therefore $  Option (D) is correct.

A wave represented by $y=2 cos (4x-\pi t)$ is superposed with another wave to form a stationary wave such that the point x= 0 is a node. The equation of other wave is:

standing waves waves physics

  1. $2 sin(4x+\pi t)$

  2. $-2 cos (4x -\pi t)$

  3. $-2 cos (4x +\pi t)$

  4. $-2 sin (4x -\pi t)$

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Correct Option: C
Explanation:

According to options

If $y _2=-2\cos(4x+\pi t)$
Then, when superimposed,
$y=y _1+y _2\ \quad 2\cos(4x-\pi t)-2\cos(4x+\pi t)\ =2[2\sin(\cfrac{(4x-\pi t)+(4x+\pi t)}{2})\sin(\cfrac{(4x-\pi t)-(4x+\pi t)}{2})]\ \quad=2[2\sin(4x)\sin(-\pi t)]\y=-4\sin(4x)\sin(\pi t)$
at $y=0\Rightarrow y=0$ (i.e node)

If the energy density and velocity of a wave are $u$ and $c$ respectively then the energy propagating per second per unit area will be

energy in wave motion oscillation and waves waves physics

  1. $u/c$

  2. $c^2u$

  3. $uc$

  4. $c/u$

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Correct Option: C
Explanation:

If the energy density and velocity of a wave are $u$ and $c$ respectively then the energy propagating per second per unit area will be $uc.$

The kinetic energy per unit length for a wave on a string is the positional coordinate

energy in wave motion oscillation and waves waves physics

  1. True

  2. False

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Correct Option: B
Explanation:

The kinetic energy for a particle is given by $(\mu \Delta x/ 2) (\dfrac{dy}{dt})^2$

Thus, it depends only on the time variable and not on the position variable

A travelling wave has an equation of the form $A(x,t)=f(x+vt)$. The relation connecting positional derivative with time derivative of the function is:

energy in wave motion oscillation and waves waves physics

  1. $\dfrac{dA}{dt}=\pm v^2 \dfrac {dA}{dx}$

  2. $\dfrac{dA}{dt}=\pm v \dfrac {dA}{dx}$

  3. $\dfrac{dA}{dt}=\pm \sqrt(v) \dfrac {dA}{dx}$

  4. $\dfrac{dA}{dt}=(2 \pi v/\lambda) \dfrac {dA}{dx}$

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Correct Option: B
Explanation:

Positional derivative and time derivative of a function f is $\dfrac{dA}{dt}=\pm v \dfrac {dA}{dx}$

The correct option is (b)

Kinetic energy per unit length for a particle in a standing wave is zero at:

energy in wave motion oscillation and waves waves physics

  1. nodes

  2. antinodes

  3. mid-way between a node and an antinode

  4. None of the above

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Correct Option: B
Explanation:

Particle at antinodes is momentarily at rest and hence has zero kinetic energy. Its speed comes down to zero at this point and all energy is stored in the form of potential energy.