Superposition & Interference 🎶

The disturbance in a medium adds up algebraically. For many individual disturbances \(y_1,\,y_2,\dots,y_n\) travelling with the same speed \(v\), the combined wave is

$$y(x,t)=\sum_{i=1}^{n}f_i\!\bigl(x-vt\bigr) \tag{14.26}$$ :contentReference[oaicite:0]{index=0}

With just two harmonic waves of equal amplitude \(a\) and wavelength \(\lambda\), but a phase gap \(\phi\), the net displacement becomes

$$y(x,t)=2a\cos\!\Bigl(\tfrac{\phi}{2}\Bigr)\,\sin(kx-\omega t+\tfrac{\phi}{2}) \tag{14.31}$$ :contentReference[oaicite:1]{index=1}

  • Constructive interference 🌟 when \(\phi=0\): amplitude doubles to \(2a\). :contentReference[oaicite:2]{index=2}
  • Destructive interference 😶‍ when \(\phi=\pi\): amplitude drops to zero everywhere. :contentReference[oaicite:3]{index=3}

Reflection of Waves 🔄

When a travelling wave meets a boundary:

  • Rigid boundary (e.g., fixed string end) → the reflected wave flips its phase by \(\pi\) (180°):
    \(y_r(x,t)=a\sin(kx-\omega t+\pi)=-a\sin(kx-\omega t)\)   (14.35) :contentReference[oaicite:4]{index=4}
  • Free boundary (e.g., loose ring, organ-pipe open end) → no phase change:
    \(y_r(x,t)=a\sin(kx-\omega t)\)   (14.36) :contentReference[oaicite:5]{index=5}

Standing Waves & Normal Modes 🪄

On a Stretched String (both ends fixed)

Superposing waves travelling left and right gives a stationary pattern

$$y(x,t)=2a\sin kx\,\cos\omega t \tag{14.37}$$ :contentReference[oaicite:6]{index=6}

Key features:

  • Nodes (no motion): \(\sin kx=0\Rightarrow x=n\frac{\lambda}{2}\)  (14.38) :contentReference[oaicite:7]{index=7}
  • Antinodes (max motion): \(x=(n+\tfrac12)\frac{\lambda}{2}\)  (14.39) :contentReference[oaicite:8]{index=8}
  • For length \(L\) with nodes at both ends:
    \(L=n\frac{\lambda}{2}\) → allowed wavelengths \(\lambda=\frac{2L}{n}\) (14.41),
    frequencies \(\displaystyle \nu_n=\frac{nv}{2L}\)  (14.42) :contentReference[oaicite:9]{index=9}
  • First harmonic (fundamental) \(n=1\): \(\nu_1=\dfrac{v}{2L}\). Higher harmonics follow integer multiples. :contentReference[oaicite:10]{index=10}

Air Columns 🌬️

One end closed, one end open (node at water surface, antinode at open end):

  • Allowed wavelengths: \(\displaystyle \lambda=\frac{4L}{2n+1}\)  (14.43) :contentReference[oaicite:11]{index=11}
  • Frequencies: \(\displaystyle \nu=\frac{(2n+1)\,v}{4L}\) (only odd harmonics)  (14.44) :contentReference[oaicite:12]{index=12}

Both ends open behave like the string—all harmonics are possible with \(\nu_n=\dfrac{nv}{2L}\). :contentReference[oaicite:13]{index=13}

Beats 🎧

Two sound waves with close frequencies \(\nu_1\) and \(\nu_2\) produce a pleasant “wax-and-wane” in loudness.

Writing \(s_1=a\cos\omega_1 t,\; s_2=a\cos\omega_2 t\) and adding, we find

$$s=2a\cos\bigl(\omega_b t\bigr)\,\cos\bigl(\omega_a t\bigr) \tag{14.47}$$ :contentReference[oaicite:14]{index=14}

with \(\omega_a=\tfrac{\omega_1+\omega_2}{2}\) (average pitch) and \(\omega_b=\tfrac{\lvert\omega_1-\omega_2\rvert}{2}\). The ear perceives a beat frequency \(f_{\text{beat}}=\lvert\nu_1-\nu_2\rvert\) 🎵—handy for tuning instruments!

NEET High-Yield Pointers 🚀

  • Phase flip rule: Reflection at a rigid end adds a phase of \(\pi\); a free end does not. :contentReference[oaicite:15]{index=15}
  • Node spacing in a standing wave is \(\lambda/2\); positions follow \(x=n\lambda/2\). :contentReference[oaicite:16]{index=16}
  • String frequencies: \(\displaystyle \nu_n=\dfrac{nv}{2L}\). Fundamental \(=\) first harmonic. :contentReference[oaicite:17]{index=17}
  • Closed–open pipe: only odd harmonics, \(\displaystyle \nu=(2n+1)\dfrac{v}{4L}\). :contentReference[oaicite:18]{index=18}
  • Beat frequency: \(f_{\text{beat}}=\lvert f_1-f_2\rvert\). :contentReference[oaicite:19]{index=19}

Keep these gems in your toolkit, and wave questions on exam day will feel like a breeze! 💪😊