Superposition of waves

1. Principle of superposition

$\to$ When two or more waves overlap, the resultant displacement at any point and at any instant is given by adding the instantaneous displacements that would be produced at any point by the individual waves if each were present alone.

i.e. $y = y_1 + y_2 + ..... + y_n$

2. Superposition of waves is off four types

i. Interference

ii. Beats

iii. Stationary waves

iv. Lissajous figures

3. $\textbf{Interference of sound:}$

$\to$If two sound waves of equal frequency and amplitude move in the same direction, then at points where they meet in same phase, intensity of sound become maximum (constructive interferences) and at points where they superimpose in becomes minimum (destructive interference). This variation of intensity of sound is called interference of sound.

-Let, $y_1 = a_1 cos \omega t$ and $y_2 = a_2 cos ( \omega t$ + $\phi )$ superimpose. Then,

i) Phase difference = $\phi$

ii) Resultant amplitude , $a = \sqrt{a_1^2 + a_2^2 + 2 a_1 a_2 cos \phi}$

-$a_{max} = a_1 + a_2 (for \phi = 0^0)$

-$a_{min} = a_1 - a_2 (for \phi = 180^0)$

iii) Resultant intensity , $I = I_1 + I_2+ 2 \sqrt{I_1} \sqrt{I_ 2} cos \phi$

-$I_{max} = (I_1 + I_2)^2 (for \phi = 0^0)$

-$I_{min} = (I_1 - I_2 )^2 (for \phi = 180^0)$

4. $Condition$ $for$ $constructive$ $interference:$

-When crest of one wave falls on crest of another wave.

-Phase difference (∅) between two waves = even multiple of π

i.e., ∅ = 0, 2π, 4 π, 6 π, 8 π ……

or, ∅ = 2nπ, where n = 0, 1, 2, 3, 4, ……...

and path difference between the waves,

$∆∅ = \frac{2π}{λ} ∆x$

∆x = integral multiple of λ

so, ∆x=nλ, where n = 0, 1, 2, 3, 4, 5, 6, 7, 8, ……...

-$a_{max} = a_1 + a_2$

-$I_{max} = (I_1 + I_2)^2$

5. $Condition$ $for$ $destructive$ $interference:$

-When crest of one wall on trough on wave.

∆∅=(2n-1)π

n = 1, 2, 3 ……

∆∅ = π, 3 π, 5 π …….. (odd multiple of π)

$∆x = \frac{(2n -1) λ}{2}$

∆x = λ/2, 3λ/2 , 5λ/2, …….

-$a_{min} = a_1 - a_2$

-$I_{min} = (I_1 - I_2 )^2$

Beats

1. When two notes (or sound waves) of nearly equal frequency travelling in same direction superpose each other at a given point, then the intensity of the resulting sound rises and falls periodically. The periodic rise and falls in intensity of sound at a given point is called beats

2. Time interval between successive maxima or minima is $T = \frac{1}{(f_1 – f_2)}$

3. The number of maxima or minima per sec or beat frequency =$f_b$ = $f_1 – f_2$

4. No. of beats / sec is called beat frequency and is given by $f_1 – f_2$. [$f_1 – f_2$ should not be more than 10 Hz]

5. Beat:

$y =y_1+y_2$

= a sin 2π$f_1$t + a sin 2π$f_2$t

= 2 a sin 2π ($f_1$ + $f_2$) t / 2 cos 2π ($f_1$ – $f_2$) t / 2

so, y = 2a cos ($f_1$ – $f_2$)πt. Sin2πft ($f_1$≈ $f_2$ = f)

Or, y = A sin 2πft

Where, A = 2a cos π ($f_1$