🌊 Coherent & Incoherent Addition of Waves 🔔

When two waves meet, they add “tip-for-tip” thanks to the superposition principle: the total displacement at any point equals the vector sum of their individual displacements :contentReference[oaicite:0]{index=0}. Whether that sum blooms into bright peaks or fades into silence depends on coherence and path difference.

1. What does “coherent” mean? 🤝

  • Two sources stay locked in phase when the gap between their wave peaks stays fixed in time :contentReference[oaicite:1]{index=1}.
  • Think of twin needles bobbing up and down together on water: their crests & troughs line up, creating a stable pattern of bright (antinodal) and dark (nodal) lines :contentReference[oaicite:2]{index=2}.

2. Adding two coherent waves ✨

Let the displacements at point P be

\( y_1 = a\cos\omega t \quad\text{and}\quad y_2 = a\cos(\omega t + \phi)\, .\)

The total wave becomes

\( y = y_1 + y_2 = 2a\cos\!\left(\dfrac{\phi}{2}\right)\cos\!\left(\omega t + \dfrac{\phi}{2}\right). \)

The intensity (brightness or loudness) is proportional to the square of the amplitude, so

\( I = 4 I_0 \cos^2\!\left(\dfrac{\phi}{2}\right) \) :contentReference[oaicite:3]{index=3},

with \(I_0\) the intensity of either single wave.

  • Constructive interference 🎆 happens when crests meet crests:
    Path difference \( \Delta S = n\lambda\;(n = 0,1,2,\dots)\)
    Phase difference \( \phi = 2n\pi\)
    Peak intensity \( I_{\text{max}} = 4 I_0 \) :contentReference[oaicite:4]{index=4}.
  • Destructive interference 🕳️ occurs when crests meet troughs:
    Path difference \( \Delta S = \bigl(n+\tfrac12\bigr)\lambda\)
    Phase difference \( \phi = (2n+1)\pi\)
    Intensity drops to zero :contentReference[oaicite:5]{index=5}.

3. Incoherent addition: why most bulbs don’t make fringes 💡

  • If the phase difference \(\phi\) jitters quickly, the bright–dark pattern smears out. All you see is the time-averaged intensity

\( I_{\text{avg}} = 2 I_0 \) :contentReference[oaicite:6]{index=6}.

That is why two separate lamps on a wall just give a uniform glow, not stripes :contentReference[oaicite:7]{index=7}.

4. Making light waves behave: Young’s clever trick 🎩

Thomas Young drilled a single bright pinhole to illuminate two neighboring pinholes \(S_1\) and \(S_2\). Because both secondary pinholes inherit waves from the same parent wavefront, they stay perfectly in step and act as coherent sources. On a distant screen, their waves weave a crisp fringe pattern of alternating bright and dark bands :contentReference[oaicite:8]{index=8}.

🎯 High-Yield Ideas for NEET

  1. Conditions for constructive (\(\Delta S = n\lambda\)) and destructive (\(\Delta S = (n+\tfrac12)\lambda\)) interference :contentReference[oaicite:9]{index=9}.
  2. Resultant intensity formula \( I = 4I_0\cos^2(\phi/2) \) and its extreme values :contentReference[oaicite:10]{index=10}.
  3. Concept of coherence — constant phase difference leads to stable fringes :contentReference[oaicite:11]{index=11}.
  4. For incoherent sources the average intensity doubles instead of showing fringes (\(I_{\text{avg}} = 2I_0\)) :contentReference[oaicite:12]{index=12}.
  5. Young’s double-pinhole setup as the classic method to produce coherent light beams :contentReference[oaicite:13]{index=13}.

Keep these gems handy, practice visualizing the crest-trough dance, and you’ll nail every interference question with confidence! 🚀