1. Principle of Superposition 🌊➕🌊

When two wave pulses travel toward each other, nothing dramatic happens—they simply glide through one another. While they overlap, every particle shifts by the straight-forward sum of the individual shifts. For two waves this reads:
\( y(x,t)=y_1(x,t)+y_2(x,t) \)   (Eq. 14.25) :contentReference[oaicite:0]{index=0}

If more than two waves meet, just keep adding their functions:
\( y=\displaystyle\sum_{i=1}^{n}f_i(x-vt) \)   (Eq. 14.26) :contentReference[oaicite:1]{index=1}

2. Two Harmonic Waves on a String 🎸

  • Same angular frequency \( \omega \) and wave-number \( k \) ⇒ same wavelength.
  • Equal amplitude \( a \), both heading rightward.
  • Only difference is their phase shift \( \phi \).

Individual waves:
\( y_1(x,t)=a\sin(kx-\omega t) \)   (Eq. 14.27) :contentReference[oaicite:2]{index=2}
\( y_2(x,t)=a\sin(kx-\omega t+\phi) \)   (Eq. 14.28) :contentReference[oaicite:3]{index=3}

3. Adding Them Up ✨

Superposition gives the total shift:
\( y(x,t)=a\sin(kx-\omega t)+a\sin(kx-\omega t+\phi) \)   (Eq. 14.29) :contentReference[oaicite:4]{index=4}

After a quick sine-addition trick, it becomes:
\( y(x,t)=2a\cos\!\Bigl(\dfrac{\phi}{2}\Bigr)\sin\!\Bigl(kx-\omega t+\dfrac{\phi}{2}\Bigr) \)   (Eq. 14.31) :contentReference[oaicite:5]{index=5}

The resultant amplitude is therefore
\( A(\phi)=2a\cos\!\Bigl(\dfrac{\phi}{2}\Bigr) \)   (Eq. 14.32) :contentReference[oaicite:6]{index=6}

4. Interference Patterns 🎯

  • Constructive interference (waves in-phase, \( \phi=0 \)) ⬆️
    Amplitude doubles to \( 2a \); the wave keeps its original sine shape:
    \( y=2a\sin(kx-\omega t) \). :contentReference[oaicite:7]{index=7}
  • Destructive interference (waves half a cycle out of phase, \( \phi=\pi \)) ❌
    Cosine factor goes to zero ⇒ the string stays at rest with \( y=0 \) everywhere. :contentReference[oaicite:8]{index=8}

5. Key Takeaways 📚

  1. Superposition lets us predict multi-wave behavior by simply adding displacements.
  2. The phase difference \( \phi \) is the master control for the combined amplitude.
  3. 0-phase shift → waves boost each other (constructive). π-phase shift → waves cancel (destructive).
  4. The handy formula \( A(\phi)=2a\cos(\phi/2) \) is your quick amplitude calculator.
  5. Each wave continues its journey unchanged after they cross—waves “ghost” through one another! 👻

High-Yield NEET Nuggets 💡

  1. The core statement \( y=y_1+y_2 \) means just add displacements—no extra fuss. :contentReference[oaicite:9]{index=9}
  2. Resultant amplitude formula \( A(\phi)=2a\cos(\phi/2) \) quickly predicts maxima & minima. :contentReference[oaicite:10]{index=10}
  3. Constructive condition \( \phi=0 \) (or multiples of \( 2\pi \)) and destructive condition \( \phi=\pi \). :contentReference[oaicite:11]{index=11}
  4. Waves retain their identity after overlapping—perfect example of non-destructive interference. :contentReference[oaicite:12]{index=12}