Physics

Wave Motion

536 Questions

Wave motion questions cover the principles of traveling and stationary waves, including their equations and intensities. The topics explore interference patterns, phase differences, and electromagnetic radiation speeds. Mastery of these concepts is vital for physics sections in engineering and civil services examinations.

Wave interferenceStanding wavesPhase differenceElectromagnetic radiationWave equations

Wave Motion Questions

Multiple choice physics superposition of waves-1: interference and beats interference of sound waves superposition and interference of sound waves properties of sound waves

Statement-1:
Two longitudinal waves given by equations; ${ y } _{ 1 }$(x,t) = 2a $\sin { \left( \omega t-kx \right)  } $ and ${ y } _{ 2 }\left( x,t \right) $ = a $\sin { \left( 2\omega t-2kx \right)  } $  will have equal intensity.

Statement-2:
Intensity of waves of given frequency in same medium is proportional to square of amplitude only.

  1. Statement-1 is true, statement -2 is true; statement-2 is not correct explanation of statement -1.

  2. Statement-1 is false, statement -2 is true.

  3. Statement-1 is true, statement -2 is false.

  4. Statement -1 is true, statement-2 true; statement-2 is the correct explanation of statement-1.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Intensity is proportional to the square of the amplitude and the square of the frequency (I proportional to A^2 * f^2). Wave 1 has amplitude 2a and frequency f, while Wave 2 has amplitude a and frequency 2f. Thus, intensities are proportional to (2a)^2 * f^2 = 4a^2f^2 and a^2 * (2f)^2 = 4a^2f^2, making them equal. Statement 2 is false because intensity depends on both amplitude and frequency.

Multiple choice physics superposition of waves-1: interference and beats interference of sound waves superposition and interference of sound waves properties of sound waves

Two coherent sources of different intensities send waves which interfere. If the ratio of maximum and minimum intensity in the interference pattern is $25$ then find ratio of intensity of source :

  1. $25 : 1$
  2. $5 : 1$
  3. $9 : 4$
  4. $25 : 16$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\cfrac { { I } _{ max } }{ { I } _{ min } } ={ \left[ \cfrac { \sqrt { { I } _{ 1 } } +\sqrt { { I } _{ 2 } }  }{ \sqrt { { I } _{ 1 } } -\sqrt { { I } _{ 2 } }  }  \right]  }^{ 2 }$

Where and  are intensities of two waves 

given 

$\cfrac { { I } _{ max } }{ { I } _{ min } } =\cfrac { 25 }{ 1 } \\ \therefore \cfrac { \sqrt { { I } _{ 1 } } +\sqrt { { I } _{ 2 } }  }{ \sqrt { { I } _{ 1 } } -\sqrt { { I } _{ 2 } }  } =\cfrac { 5 }{ 1 } $

use componendo and dividendo

we get

$\cfrac { \sqrt { { I } _{ 1 } } +\sqrt { { I } _{ 2 } } +\sqrt { { I } _{ 1 } } -\sqrt { { I } _{ 2 } }  }{ \sqrt { { I } _{ 1 } } +\sqrt { { I } _{ 2 } } -\sqrt { { I } _{ 1 } } +\sqrt { { I } _{ 2 } }  } =\cfrac { 5+1 }{ 5-1 } \\ or,\quad \cfrac { \sqrt { { I } _{ 1 } }  }{ \sqrt { { I } _{ 2 } }  } =\cfrac { 3 }{ 2 } \\ or,\quad \cfrac { { I } _{ 1 } }{ { I } _{ 2 } } =\cfrac { 9 }{ 4 } $

 

 

Multiple choice physics superposition of waves-1: interference and beats interference of sound waves superposition and interference of sound waves properties of sound waves

Statement -1:
Two longitudinal waves given by equation $y _{1}$(x,t) = 2a sin $(\omega - kx)$ and $y _{2}$(x,t) = a sin $(2\omega - 2kx)$ will have equal intensity.
Statement -2:
Intensity of waves of given frequency in the same medium is proportional to the square of amplitude only.

  1. Statement -1 is true, statement -2 is true; statement -2 is not correct explanation of statement-1.

  2. Statement -1 is false, statement -2 is true.

  3. Statement -1 is true, statement -2 is false

  4. Statement -1 is true, statement -2 true; statement -2 is the correct explanation of statement -1
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Intensity of waves of given frequency in same medium is not only proportional to square of amplitude. It depends on other factors also

Multiple choice physics superposition of waves-1: interference and beats interference of sound waves superposition and interference of sound waves properties of sound waves

If two waves maintain constant phase difference or same phase at any two points on a wave is known as spatial coherence.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Two wave sources are said to be coherent if the waves either have same phase or constant phase difference at any two points on a wave and the phenomenon is known as spatial coherence.

Multiple choice physics superposition of waves-1: interference and beats interference of sound waves superposition and interference of sound waves properties of sound waves

Intensity (I) is related to amplitude (A) as:

  1. $I \propto A$
  2. $I \propto {A^2}$
  3. $I \propto {A^4}$
  4. $I \propto {A^{-1}}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Intensity of a wave is directly proportional to the square of amplitude of the wave.
i.e.  $I \propto A^2$
More is the amplitude of wave, more will be the intensity.
So, we write  $I = kA^2$
where $k$ is a proportionality constant.

Multiple choice physics superposition of waves-1: interference and beats interference of sound waves superposition and interference of sound waves properties of sound waves

The resultant intensity for two identical waves of intensity I with a phase difference of $\pi/3$ is 

  1. $R=2\sqrt{3I}$
  2. $R=\sqrt{3I}$
  3. $R=4\sqrt{3I}$
  4. $R=3\sqrt{3I}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The resultant of two waves is given by $R^2=A^2+B^2+2AB cos \theta; \theta$ is the phase difference and A and B are the amplitudes of the individual waves.

Since both the waves are identical, their amplitudes are equal

Thus, $R^2 = A^2+A^2+A^2=3A^2 \implies R=\sqrt{3}A$

Since intensity is proportional to $\sqrt{A}$, we can write the resultant amplitude $R = \sqrt{3I}$

The correct option is (b)

Multiple choice physics superposition of waves-1: interference and beats interference of sound waves superposition and interference of sound waves properties of sound waves

If the sum of the intensities of two component waves are 5I units and upon superposition with a phase difference of $\pi $ radians, their resultant is 2I, what are the intensities of component waves

  1. $I _1=3I, I _2=I$
  2. $I _1=3.5I, I _2=1.5I$
  3. $I _1=2I, I _2=3I$
  4. $I _1=I, I _2=4I$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

If $I _1$ and $I _2$ are intensities of two waves, then $I _1+I _2=5I$ and $(I _1-I _2)=2I$. Solving, we get,$I _1=3.5I$ and $I _2=1.5I$

The correct option is (b)

Multiple choice physics superposition of waves-1: interference and beats interference of sound waves superposition and interference of sound waves properties of sound waves

Waves from two sources superpose on each other at a particular point amplitude and frequency of both the waves are equal. The ratio of intensities when both waves reach in the same phase and they reach with the phase difference of $90^{\circ}$ will be

  1. $1 : 1$
  2. $\sqrt {2} : 1$
  3. $2 : 1$
  4. $4 : 1$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
Relation between intensity $f$
Amplitude
$I= kA^{2} \cos^{2} \left( \dfrac{|theta}{2} \right)$
When both the waves reach in the same phase, then 
$\theta= 0^{o}$ 
$I _{1}= kA^{2} \cos^{2} \left( \dfrac{0^{o}}{2} \right)$
$I _{1} = kA^{2} [\cos 0^{o}=1]$ ------- $(1)$
When both the waves reach with phase difference $90^{o}$.
$\theta = 90^{o}$
$I _{2}= kA^{2} \cos^{2} \left( \dfrac{90^{o}}{2} \right)$
$I _{2} = kA^{2} \cos^{2} (45^{o})$
$I _{2}= kA^{2} \left( \dfrac{1}{2} \right)$ ----- $(2)$
$\dfrac{I _{1}}{I _{2}}= \dfrac{kA^{2}}{kA^{2} (1/2)} =\dfrac{2}{1}$
Multiple choice physics superposition of waves-1: interference and beats interference of sound waves superposition and interference of sound waves properties of sound waves

Suppose displacement produced at some point $P$ by a wave is $y _1=a cos \omega t$ and by another wave is $y _2=a cos \omega t$.Let $I _0$ represents intensity produced by each one of individual wave, then resultant intensity due to overlapping of both wave is

  1. $I _0$
  2. $2I _0$
  3. $\dfrac{I _0}{2}$
  4. $4I _0$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$y _{res}$ at point $P=y _1+y _2\=a\cos \omega t+a\cos\omega t\=2\cos\omega t$

Now amplitude $=2a$
Since  intensity $\propto$(amplitude)$^2$
So, $\cfrac{I _{new}}{I _0}=\cfrac{4a^2}{a^2}\\implies I _{new}=4I _0$

Multiple choice physics superposition of waves-1: interference and beats interference of sound waves superposition and interference of sound waves properties of sound waves

Two waves $Y _{1}=a\sin \omega t$ and $Y _{2}=a\sin (\omega t+\delta)$ are producing interference, then resultant intensity is-

  1. $a^{2}\cos^{2}\delta/2$
  2. $2a^{2}\cos^{2}\delta/2$
  3. $3a^{2}\cos^{2}\delta/2$
  4. $4a^{2}\cos^{2}\delta/2$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The resultant intensity of two waves with equal amplitude 'a' and phase difference 'delta' is I = I1 + I2 + 2*sqrt(I1*I2)*cos(delta). With I1 = I2 = k*a^2, I = 2*k*a^2(1 + cos(delta)) = 4*k*a^2*cos^2(delta/2). Assuming k=1, the result is 4*a^2*cos^2(delta/2).

Multiple choice physics superposition of waves-1: interference and beats interference of sound waves superposition and interference of sound waves properties of sound waves

Path difference between two waves from a coherent sources is 5 nm at a  point P. Wavelength of these waves is 100 $\mathring { A } $. Resultant intensity at point P if intensity of sources is $l _0$ and $4l _0$

  1. Zero
  2. $l _0$
  3. $5l _0$
  4. $3l _0$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\begin{array}{l} \Delta x=5\times { 10^{ -9 } }m \ \Delta \phi =\frac { { 2\pi  } }{ \lambda  } \Delta x \ =\frac { { 2\pi \times 5\times { { 10 }^{ -9 } } } }{ { 100\times { { 10 }^{ -10 } } } }  \ =\pi  \ Now, \ I={ I _{ 0 } }+4{ I _{ 0 } }+2\sqrt { { I _{ 0 } } } \sqrt { 4{ I _{ 0 } } } \cos  \phi  \ =5{ I _{ 0 } }+4{ I _{ 0 } }\left[ { \cos  \pi  } \right]  \ ={ I _{ 0 } } \ \therefore resula\tan  t\, \, \, { { intensity } }\, ={ I _{ 0 } } \ Hence,\, option\, \, B\, \, is\, the\, correct\, answer. \end{array}$

Multiple choice physics superposition of waves-1: interference and beats interference of sound waves superposition and interference of sound waves properties of sound waves

Two waves of intensities $I$ and $4I$ superimpose. The minimum and maximum intensities will respectively be

  1. $I,\space 9I$
  2. $3I,\space 5I$
  3. $I,\space 5I$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
The Intensity of the wave is directly proportional to the square of its amplitude.
$I \propto A^{2}$;
$I = cA$$^2$;   '$c$' is an arbitrary constant.
So, if a wave with Intensity $I$ has an amplitude of $A(A{ _{1}}$)
A wave with an Intensity of $4I$ would respectively have an amplitude of $2A$($A{ _{2}}$)

If 2 waves with amplitudes $A{ _{1}}$,$A{ _{2}}$ are superimposed, the resultants would be
Maximum of $A{ _{1}}$ + $A{ _{2}} = 3A$ (Constructive Interference)
Minimum of $|A{ _{1}}$ - $A{ _{2}} |   = A $ (Destructive Interference)
The wave of amplitude $3A$ would have an Intensity of $9I$
The wave of amplitude $A$ would have an Intensity of $I$
Multiple choice physics superposition of waves-1: interference and beats interference of sound waves superposition and interference of sound waves properties of sound waves

For a wave displacement amplitude is $10^{-8} m$ density of air $1.3 kg m^{-3}$ velocity in air $340 ms^{-1}$ and frequency is 2000 Hz.The average intensity of wave is

  1. $5.3\times 10^{-4} Wm^{-2}$
  2. $5.3\times 10^{-6} Wm^{-2}$
  3. $3.5\times 10^{-8} Wm^{-2}$
  4. $3.5\times 10^{-6} Wm^{-2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Intensity I = 2 * pi^2 * f^2 * A^2 * rho * v. Plugging in values: I = 2 * (3.14)^2 * (2000)^2 * (10^-8)^2 * 1.3 * 340. I = 2 * 9.86 * 4 * 10^6 * 10^-16 * 1.3 * 340 = 5.3 * 10^-4 W/m^2.

Multiple choice physics superposition of waves-1: interference and beats interference of sound waves superposition and interference of sound waves properties of sound waves

Two sound waves of equal intensity $I$ superimpose at point $P$ in $90^{\small\circ}$ out of phase. The resultant intensity at point $P$ will be

  1. $4I$
  2. $\sqrt2I$
  3. $2I$
  4. $I$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Amplitude of the resultant wave is:
$A _R=\sqrt{A^2 _1+A^2 _2+2A _1A _2\cos(\theta)}$.

Here, $A _1=A _2=A \text{ and } \theta = \pi/2$

So, $A _R=A\sqrt{2(1+\cos(\theta))}=2A\cos(\theta/2)=2A\cos(\pi/4)=\sqrt{2}A$ 

$\Rightarrow I _R=|A _R|^2=2A^2=2I$