THE PRINCIPLE OF SUPERPOSITION OF WAVES -

SUPERPOSITION PRINCIPLE

When two or more waves pass through the same medium at the same time, the net displacement at any point is equal to the vector sum of their individual displacement at that point.

i.e $\dpi{120} \fn_cm \vec{y}=\vec{y_1}+\vec{y_2}+\vec{y_3}+..........+\vec{y_n}$

NOTE:- Superposition is one of the major differences between a particle model and a wave model: Particles collide but wave superpose.

APPLICATION OF SUPERPOSITION OF WAVES

When two waves of the same frequency and having the same amplitude moving with the same speed in opposite directions superpose on each other, they give rise to stationary or standing waves.
When two weaves of slightly different frequency moving with the same speed in the same direction superpose on each other, they give rise to beat.

INTERFERENCE OF WAVES

Interference is what happens when two or more waves come together. Depending on how the peaks and troughs of the waves are matched up, the waves might add together or they can partially or even completely cancel each other.

Consider the superposition of two sinusoidal waves of the same angular frequency and wavenumber at a point. Let us assume that the two waves are travelling in the same direction with the same velocity. The equation of the two waves reaching a point can be written as

$\dpi{120} \fn_cm y_1=A_1 sin(kx-\omega t)$

and $\dpi{120} \fn_cm y_2=A_2 sin(kx-\omega t + \phi)$

Where $\dpi{120} \fn_cm \phi$ is the phase difference between two waves. According to the principle of superposition, the resultant wave is represented by

$\dpi{120} \fn_cm y=y_1 +y_2$

$\dpi{120} \fn_cm =A_1 sin(kx-\omega t)+A_2 sin(kx-\omega t+\phi)$

$\dpi{120} \fn_cm =A_1sin(kx-\omega t)+A_2 sin (kx-\omega t)cos \phi+ A_2 cos(kx-\omega t)sin \phi$

$\dpi{120} \fn_cm =(A_1 +A_2 cos \phi)sin (kx-\omega t)+A_2 sin \phi cos(kx-\omega t)$

Now,

Let A is the resultant amplitude and angle $\dpi{120} \fn_cm \theta$ represents the phase difference between the resulting wave and the first wave

Here $\dpi{120} \fn_cm A_1+A_2 cos\phi=A cos \theta$ ———————————-(1)

and $\dpi{120} \fn_cm A_2 sin \phi=A sin \theta$ ————————————–(2)

i.e $\dpi{120} \fn_cm y=A cos \theta sin(kx-\omega t)+A sin \theta cos(kx-\omega t)$

$\dpi{120} \fn_cm \left [ y=A sin(kx-\omega t +\theta) \right ]$

This is the equation of the resultant wave. The resultant wave is also a harmonic travelling wave in +ve x-axis with the same frequency and wavelength. However, its initial phase angle is $\dpi{120} \fn_cm \theta$ w.r.t first wave.

from equations (1) and (2) squaring both sides and adding, we get

$\dpi{120} \fn_cm A^2=(A_1 +A_2 cos \phi)^2+(A_2 sin \phi)^2$

$\dpi{120} \fn_cm =A_1^2+A_2^2+2A_1A_2 cos \phi$

Hence resultant amplitude is

$\dpi{120} \fn_cm \left [ A=\sqrt{A_1^2+A_2 ^2+2A_1A_2 cos \phi} \right ]$ ——————————————-(3)

And initial phase angle of the resultant is

$\dpi{120} \fn_cm \left [ \theta=\tan^{-1}\left ( \frac{A_2 sin\phi}{A_1 +A_2 cos \phi} \right ) \right ]$

If two waves meet at a point in the medium in such a way that the crest of two waves falls on each other(at the same phase). Then according to the superposition principle, the amplitude of the resultant wave is the sum of the amplitude of the individual wave. This phenomenon is known as the constructive interference of waves.

From equation (3), resultant amplitude A is maximum when

$\dpi{120} \fn_cm cos \phi=+1\;\;or\;\;\phi=0,2\pi,4\pi,......,2n\pi \;\;(where,n=0,1,2,...)$

Now, $\dpi{120} \fn_cm A_{max}=\sqrt{A_1^2+A_2^2+2A_1A_2}=\sqrt{(A_1+A_2)^2}$

i.e $\dpi{120} \fn_cm \left [ A_{max}=A_1+A_2 \right ]$

If they meet at a point in such a way that the crest of one wave falls on the trough of other waves (out of phase), then the resultant amplitude differs from the amplitudes of individual waves. This phenomenon is known as the destructive interference of waves.

From equation (3), resultant amplitude A is minimum when $\dpi{120} \fn_cm cos \phi=-1\;\;or\;\;\phi=\pi,3\pi,5\pi,......,(2n-1)\pi \;\;(where,n=1,2,3,...)$

Now, $\dpi{120} \fn_cm A_{min}=\sqrt{A_1^2+A_2^2-2A_1A_2}=\sqrt{(A_1-A_2)^2}$

i.e $\dpi{120} \fn_cm \left [ A_{min}=A_1-A_2 \right ]$

If both individual waves have the same amplitude. Let $\dpi{120} \fn_cm A_1=A_2=A_0$

Then $\dpi{120} \fn_cm A_{res}=\sqrt{A_0^2+A_0 ^2+2A_0A_0 cos \phi}=\sqrt{2A_0^2(1+cos\phi)}=\sqrt{2A_0^2.2cos^2\frac{\theta}{2}}$

$\dpi{120} \fn_cm A_{res}=2A_0cos\frac{\theta}{2}$

For constructive interference $\dpi{120} \fn_cm \theta=2n\pi\;\;\;and\;\;\;A_{res}=2A_0$

For destructive interference $\dpi{120} \fn_cm \theta=(2n-1)\pi\;\;\;and\;\;\;A_{res}=0$