Tag: superposition of waves-1: interference and beats

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Questions Related to superposition of waves-1: interference and beats

Multiple choice physics superposition of waves-1: interference and beats distinction between interference and beats beats and its applications beats in sound waves

$y _1 = A cos (2f _1t)$     and $y _2 = A cos (2 f _2t),$, then $y _{total} $ is

  1. $y _{total} = y _1 + y _2 = A {cos (2 f _1t) + cos (2 f _2t)}$
  2. $y _{total} = y _1 - y _2 = A {cos (2 f _1t) - cos (2 f _2t)}$
  3. $y _{total} =\dfrac{ y _1}{ y _2} = A\dfrac{{cos (2 f _1t)}}{{cos (2 f _2t)}}$
  4. $y _{total} = y _1 \times y _2 = A {cos (2 f _1t) \times cos (2 f _2t)}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Resultant  $(y _{total})$ of the two waves is equal to the superposition of the waves and is given by,
$y _{total}  = y _1+y _2$
$\therefore$  $y _{total} = A \ cos(2f _1t)+ A \ cos(2f _2t)$

Multiple choice physics superposition of waves-1: interference and beats distinction between interference and beats beats and its applications beats in sound waves

Two waves of wavelengths 99 cm and 100 cm both travelling with velocity 396 m/s are made of interfere. The number of beats produced by them per second are

  1. $1$
  2. $2$
  3. $4$
  4. $8$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\begin{array}{l} Velocity\, \, of\, \, wave\, \, V=n\lambda  \ Where\, \, n=frequency\, \, of\, \, wave\, \,  \ \Rightarrow n=\frac { v }{ \lambda  }  \ { n _{ 2 } }=\frac { { { v _{ 2 } } } }{ { { \lambda _{ 2 } } } } =\frac { { 396 } }{ { 100\times { { 10 }^{ -2 } } } } =396Hz \ no.\, \, of\, \, beats\, \, ={ n _{ 1 } }-n _2\, =4 \end{array}$

Multiple choice physics superposition of waves-1: interference and beats distinction between interference and beats beats and its applications beats in sound waves

If 2 waves of same frequency and same amplitude on superposition, produce a resultant disturbance of the same amplitude, the waves differ in phase by

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

Let the amplitude of each wave be $A$ i.e.  $A _1 = A _2 = A$
Thus amplitude of resultant wave  $A _R =A$
Using   $A^2 _R =  A _1^2 + A _2^2 + 2A _1A _2 \cos\theta$
$\therefore$  $A^2 =  A^2 + A^2 + 2A^2 \cos\theta$
Or  $\cos\theta = -0.5$
$\implies  \ \theta = \dfrac{2\pi}{3}$

Multiple choice physics superposition of waves-1: interference and beats distinction between interference and beats beats and its applications beats in sound waves

Two identical wires are stretched by the same tension of $101$ N & each emits a note of frequency $202$ Hz. If the tension in one wire is increased by $1N$ , then the beat frequency is:

  1. $2 Hz$
  2. $\frac{1}{2}Hz$
  3. $1 Hz$

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Frequency f is proportional to sqrt(T). f1 = 202 Hz at T1 = 101 N. New tension T2 = 102 N. f2 = f1 * sqrt(T2 / T1) = 202 * sqrt(102 / 101) = 202 * sqrt(1 + 1/101) approx 202 * (1 + 1/202) = 202 + 1 = 203 Hz. Beat frequency = 203 - 202 = 1 Hz.

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 difference between the frequencies of two waves is 10 Hz then time interval between successive maximum intensity is:

  1. $10 s$
  2. $1 s$
  3. $0.1 s$
  4. $0.01 s$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The beat frequency is the difference between the frequencies of two waves. The time interval between successive maximum intensities (beats) is the reciprocal of the beat frequency: T = 1/10 = 0.1 s.

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 intensity of the sound gets reduced by $10$% on passing through a slab. The reduction in  intensity on passing through two consecutive slab, would be 

  1. $20$%
  2. $50$%
  3. $19$%
  4. $5$%
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

If intensity is reduced by 10 percent, 90 percent (0.9) remains. After two slabs, the remaining intensity is 0.9 * 0.9 = 0.81. The total reduction is 1 - 0.81 = 0.19, which is 19 percent.

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