Interference — constructive, destructive conditions

1.6.17 · Physics › Oscillations & Waves

1. Interference kya hai?

HUM kya add karte hain: displacements, na ki intensities. Intensity sirf end mein aati hai, jab humein resultant amplitude pata chal jaaye.

2. Resultant nikalna — first principles se

Do waves ko point $P$ pe aate huye lo, same frequency $\omega$, same amplitude $a$, lekin phase difference $\varphi$ hai:

$y_1=a\sin (\omega t),y_2=a\sin (\omega t+\varphi )$

Superpose karo (bas add karo): $y=y_1+y_2=a\sin \omega t+a\sin (\omega t+\varphi )$

$\sin A+\sin B=2\sin \frac{A+B}{2}\cos \frac{A-B}{2}$ use karo jahan $A=\omega t,B=\omega t+\varphi$:

$y=\underset{\text{resultant amplitude }A}{\underbrace{2a\cos \frac{\varphi }{2}}}\sin \left(\omega t+\frac{\varphi }{2}\right)$

Yeh step kyun? Same frequency ke do sines ka sum abhi bhi same frequency ka sine hi hai — sirf uski amplitude aur phase change hoti hai. Amplitude woh bracketed factor hai.

$I\propto A^2$ kyun? Ek wave ki energy (aur brightness/loudness) amplitude ke square ke saath scale hoti hai — yeh ek universal wave fact hai.

3. Conditions (80/20 core)

Poora subtopic isi mein hai — ${\cos }^2(\varphi /2)$ kab maximum hai aur kab zero.

Constructive — waves in step hain, amplitudes add hoti hain: $\varphi =2\pi n⟹\cos \frac{\varphi }{2}=\pm 1⟹A_{\max }=2a,I_{\max }=4I_0$ $\boxed{\Delta x=n\lambda }n=0,\pm 1,\pm 2,\dots$

Destructive — waves opposite hain, cancel ho jaati hain: $\varphi =(2n+1)\pi ⟹\cos \frac{\varphi }{2}=0⟹A_{\min }=0,I_{\min }=0$ $\boxed{\Delta x=\left(n+\frac{1}{2}\right)\lambda }n=0,\pm 1,\dots$

4. Worked examples

5. Common mistakes

6. Active recall

7. Forecast-then-Verify

Connections

• Principle of Superposition — woh parent rule jis par interference tika hai.
• Young's Double Slit Experiment — $\Delta x=d\sin \theta$ ko in conditions par apply karta hai.
• Standing Waves — opposite directions mein travel karti waves ka interference.
• Beats — alag frequencies ka time mein interference.
• Coherence and Path Difference — kyun steady $\varphi$ zaroori hai.
• Energy in Waves — kyun $I\propto A^2$ aur fringes mein energy conservation.

Concept Map

Deep Dive

• D1 · Foundations — har symbol zero se
• D2 · Visual walkthrough — derivation pictures mein
• D3 · Worked examples — har scenario
• D4 · Exercises — graded, full solutions
• D5 · Question bank — concept traps