Superposition of Waves

The superposition of waves is a fundamental idea in physics that explains what happens when two or more waves overlap. According to the superposition principle, the total effect at any point is simply the sum of the individual wave displacements. Depending on how the waves line up—whether they’re in phase or out of phase—this can result in constructive interference (where the waves add up) or destructive interference (where they cancel each other out). This principle applies to all types of waves, including mechanical waves like sound and electromagnetic waves like light. It's key to understanding many wave phenomena, such as interference and diffraction.

1.0Superposition Principle

• When two or more waves overlap at a point in space, the resultant displacement at that point is equal to the algebraic sum of the displacements due to each individual wave.

$y=y_1\pm y_2\pm y_3+\cdots$

2.0Types of Superposition

1.Constructive Superposition

• Constructive superposition occurs when two waves meet in phase (i.e., their crests and troughs align), resulting in a larger amplitude. The displacements of the individual waves add up, producing a wave with maximum intensity.

2.Destructive Superposition

• Destructive superposition happens when two waves meet out of phase (i.e., the crest of one wave aligns with the trough of another), causing their displacements to partially or completely cancel out, leading to a reduced or zero amplitude.

Relationship between Phase Difference and Path Difference

$$\begin{gathered} \text{Phase difference}=\frac{2\pi }{\text{wavelength}}\times \text{path difference} \\ \varphi =\frac{2\pi }{\lambda }\times \Delta x \end{gathered}$$

3.0Superposition of Two Sinusoidal Waves

$$\begin{gathered} \text{Resultant wave at P from }{S}_{1}and{S}_2\text{ is given as} \\ {y}_1=a\sin \omega t \\ {y}_2=b\sin (\omega t+\varphi ) \\ \text{Applying principle of superposition at point P we get resultant wave} \\ Y=y_1+y_2=a\sin \omega t+b\sin (\omega t+\varphi ) \\ =a\sin \omega t+b[\sin \omega t\cos \varphi +\cos \omega t\sin \varphi ] \\ =\sin \omega t(a+b\cos \varphi )+b\cos \omega t\sin \varphi \\ a+b\cos \varphi =R\cos \theta \dots \dots .(2) \\ b\sin \varphi =R\sin \theta \dots \dots .(3) \\ Y=R\sin \omega t\cos \theta +\cos \omega t\sin \theta \\ =R\sin (\omega t+\theta )\text{Resultant wave is a simple harmonic wave having amplitude R and phase difference}\theta \end{gathered}$$

Amplitude of Resultant Wave

$$\begin{gathered} \text{Squaring and adding equations (2) and (3)} \\ (a+b\cos \varphi {)}^2+b^2{\sin }^2\varphi =R^2{\cos }^2\theta +R^2{\sin }^2\theta \\ (a+b\cos \varphi {)}^2+b^2{\sin }^2\varphi =R^2({\cos }^2\theta +{\sin }^2\theta ) \\ {R}^2=a^2+b^2+2ab\cos \varphi \\ R=\sqrt{a^2+b^2+2ab\cos \varphi } \\ (Intensity\propto (amplitude{)}^2)\Rightarrow I_a\propto a^2,I_b\propto b^2 \\ {I}_R=K{R}^2=K(a^2+b^2+2ab\cos \varphi ) \\ {I}_R=I_a+I_b+2\sqrt{I_a{I}_b}\cos \varphi \text{(This is the Resultant Intensity Equation)} \\ \tan \theta =\frac{b\sin \varphi }{a+b\cos \varphi } \end{gathered}$$

4.0Condition for Constructive and Destructive Interference

1.Constructive Interference

It occurs when the amplitude and the resultant wave is maximum.

$$\begin{gathered} I_R=I_{\text{max}}\text{if }\cos \varphi =\text{maximum}=+1 \\ \varphi =0,2\pi ,4\pi \Rightarrow \varphi =n(2\pi )\text{where }n=0,1,2,3,\dots \\ \Delta x=\frac{\lambda }{2\pi }\times 2n\pi =n\lambda \\ {I}_{\text{max}}=(a+b{)}^2 \\ {I}_{\text{max}}={\left(\sqrt{I_a}+\sqrt{I_b}\right)}^2 \\ {R}_{\text{max}}=(a+b) \end{gathered}$$

2.Destructive Interference

$$\begin{gathered} \text{It occurs when the amplitude and the resultant wave is minimum.} \\ {I}_R=I_{\text{min}}\text{if }\cos \varphi =\text{minimum}=+1 \\ \varphi =0,2\pi ,4\pi \Rightarrow \varphi =n(2\pi )\text{where }n=0,1,2,3,\dots \\ \Delta x=\frac{\lambda }{2\pi }\times 2n\pi =n\lambda \\ {I}_{\text{min}}=(a+b{)}^2 \\ {I}_{\text{min}}={\left(\sqrt{I_a}+\sqrt{I_b}\right)}^2 \\ {R}_{\text{min}}=(a+b) \end{gathered}$$

5.0Ratio of Maximum And Minimum Intensity of Light

$$\begin{gathered} \frac{I_{\text{max}}}{I_{\text{min}}}={\left(\frac{\sqrt{I_a}+\sqrt{I_b}}{\sqrt{I_a}-\sqrt{I_b}}\right)}^2={\left(\frac{a+b}{a-b}\right)}^2={\left(\frac{\frac{a}{b}+1}{\frac{a}{b}-1}\right)}^2={\left(\frac{r+1}{r-1}\right)}^2 \\ \text{(since }\frac{a}{b}=r=\text{Amplitude Ratio)} \end{gathered}$$

6.0Relation Between Intensity and Width of Source

$$\begin{gathered} Let{w}_{a}and{w}_b\text{be the width of narrow sources}{s}_{1}and{s}_2 \\ {I}_a\propto w_a\Rightarrow I_a\propto a^2 \\ {I}_b\propto w_b\Rightarrow I_b\propto b^2 \\ \frac{a^2}{b^2}=\sqrt{\frac{w_a}{w_b}} \end{gathered}$$

Illustration-1:$S_{1}and{S}_2$ are two sources of light which produce individual disturbance at point P given by $E_1=3\sin \omega t,E_2=4\cos \omega t$. Assuming ${\vec{E}}_{1}and{\vec{E}}_2$ to be along the same line, find the resultant after their superposition.

Solution:

$$\begin{gathered} E=3\sin \omega t+4\sin \left(\omega t+\frac{\pi }{2}\right) \\ {A}^2=3^2+4^2=2(3)(4)\cos \frac{\pi }{2}=5^2 \\ \tan {\varphi }_0=\frac{4\sin \frac{\pi }{2}}{3+4\cos \frac{\pi }{2}}=\frac{4}{3}=53^{\circ } \\ E=5\sin (\omega t+53^{\circ }) \end{gathered}$$

Illustration-2:

$S_{1}and{S}_2$ are two coherent sources of frequency 'f ' each.

[Given: ${\theta }_1={\theta }_2=0^{\circ }and{v}_{\text{sound}}=330\text{m/s}.$] 

(i) So that constructive interference at 'p'

(ii) So that destructive interference at 'p'

Solution:

$$\begin{gathered} \text{For constructive interference:} \\ k\Delta x=2n\pi \\ \frac{2\pi }{\lambda }\times 2=2n\pi \\ \lambda =\frac{2}{n},V=\lambda f\Rightarrow V=\frac{2}{n}f \\ f=\frac{330}{2}\times n \\ \text{For destructive interference:} \\ k\Delta x=(2n+1)\pi \\ \frac{2\pi }{\lambda }\times 2=(2n+1)\pi \\ \frac{1}{\lambda }=\frac{(2n+1)}{4},f=\frac{V}{\lambda } \\ f=\frac{330\times (2n+1)}{4} \end{gathered}$$

Illustration-3:Two waves are given by the equations

$y_1=4\sin \omega tand{y}_2=3\sin \left(\omega t+\frac{\pi }{3}\right)$ When these two waves interfere at a point, what is the approximate amplitude of the resulting wave?

Solution:

$$\begin{gathered} A=\sqrt{a_1^2+a_2^2+2a_1{a}_2\cos \varphi } \\ A=\sqrt{(4{)}^2+(3{)}^2+2\cdot 4\cdot 3\cdot \cos \frac{\pi }{3}}=\sqrt{37}\approx 6 \end{gathered}$$

Illustration-4:Two waves with intensities 9I 9I and 4I 4I interfere at a point. If the path difference between them is11λ 11 \lambda , what will be the resultant intensity at that point?

Solution: Path difference, $\Delta x=\frac{\lambda }{2\pi }\times \varphi \Rightarrow \frac{2\pi }{\lambda }\times 11\lambda =22\pi ,$ ,i.e. At the same point, the superposition of the waves results in constructive interference.

$I_R={\left(\sqrt{I_1}+\sqrt{I_2}\right)}^2={\left(\sqrt{9I}+\sqrt{4I}\right)}^2=25I$

Illustration-5:In interference if $\frac{I_{\text{max}}}{I_{\text{min}}}=\frac{144}{81}$ .Determine the ratio of amplitudes of the two waves involved in the interference.

Solution:

$$\begin{gathered} \frac{a_1}{a_2}=\left(\sqrt{\frac{I_{\text{max}}}{I_{\text{min}}}}+1/\sqrt{\frac{I_{\text{max}}}{I_{\text{min}}}}-1\right) \\ =\left(\frac{\sqrt{\frac{144}{81}}+1}{\sqrt{\frac{144}{81}}-1}\right)=\left(\frac{\frac{12}{9}+1}{\frac{12}{9}-1}\right)=\frac{7}{1} \end{gathered}$$

Illustration-6:Two interfering waves with intensities x and y arrive at a point with a time delay of $\frac{3T}{2}$ . Determine the resultant intensity at that point.

Solution:

Time difference $\text{T.D. }\Delta x=\frac{T}{2\pi }\times \varphi \Rightarrow \frac{3T}{2}=\frac{T}{2\pi }\times \varphi \Rightarrow \varphi =3\pi$. This is the condition of constructive interference.

So resultant intensity,

$I_R={\left(\sqrt{I_1}+\sqrt{I_2}\right)}^2={\left(\sqrt{x}+\sqrt{y}\right)}^2=x+y+2\sqrt{xy}$

Illustration-7: If n identical waves, each having intensity $I_0$ , interfere at a point, what will be the maximum resultant intensity in the following cases?
(1) When the waves are coherent
(2) When the waves are incoherent

Solution:

$$\begin{gathered} \text{ In case of interference of two waves:} \\ I=I_1+I_2+2\sqrt{I_1{I}_2}\cos \varphi \\ \text{In case of coherent interference}\varphi \text{is unchanging with time and so I will be maximum when}\cos \varphi =\max =1 \\ (I_{\max }{)}_{\text{co}}=I_1+I_2+2\sqrt{I_1{I}_2}={\left(\sqrt{I_1}+\sqrt{I_2}\right)}^2 \\ \text{So for n n identical waves each of intensity} \\ {I}_0(I_{\max }{)}_{\text{co}}={\left(\sqrt{I_0}+\sqrt{I_0}+\dots \right)}^2={\left(n\sqrt{I_0}\right)}^2=n^2{I}_0 \\ \text{In case of incoherent interference at a given point,}\varphi \text{varies randomly with time, so } \\ {\left(\cos \varphi \right)}_{\text{av}}=0andhence(I_R{)}_{\text{inco}}=I_1+I_2 \\ \text{So, in case of n identical waves} \\ (I_R{)}_{\text{inco}}=I_0+I_0+\cdots =n{I}_0 \end{gathered}$$