Class 11 Physics · NCERT notes for NEET

Waves: NCERT Notes, Formulas and Tricks for NEET

Travelling waves, the speed of string waves and sound, superposition, standing waves on strings and in pipes, and beats. NEET usually asks a question from here, most often on harmonics of strings and pipes, wave equations or beats.

• 1. Wave motion and speed of waves
• 2. Superposition and standing waves 🔒
• 3. Strings and organ pipes 🔒
• 4. Beats 🔒

Topic 1 is open for everyone. The full chapter, quick revision and practice open with a plan, or start free with 5 full chapters.

Open the interactive version (simulations and tap-to-answer quick checks).

Topic 1

Wave motion and speed of waves

In a stadium Mexican wave the 'wave' runs around the whole ground, yet every fan just stands up and sits down in their own seat. That is exactly how a wave carries energy without carrying matter.

• A wave is a travelling pattern of disturbance. It carries energy and information, but the medium as a whole does not travel; each particle only oscillates about its mean position.
• Mechanical waves need a medium and depend on its elasticity and inertia (string waves, sound). Electromagnetic waves need no medium and all travel at $c\approx 3\times 10^8{\text{m s}}^{-1}$ in vacuum. Matter waves go with moving electrons, protons, atoms.
• Transverse: particles move perpendicular to the direction of travel (string). Needs shear rigidity, so mechanical transverse waves travel in solids, not through the bulk of fluids.
• Longitudinal: particles move along the direction of travel, forming compressions and rarefactions (sound). These can travel in solids, liquids and gases.
• Progressive wave: $y(x,t)=a\sin (kx-\omega t+\varphi )$ moves along $+x$; $(kx+\omega t)$ means it moves along $-x$. If $y=A\sin (kx-\omega t)+B\cos (kx-\omega t)$, the amplitude is $\sqrt{A^2+B^2}$.
• Wavelength $\lambda$: least distance between two points in the same phase. $k=2\pi /\lambda$ (rad m${}^{-1}$), $T=2\pi /\omega$, $\nu =1/T$.
• Wave speed $v=\omega /k=\nu \lambda$. The medium fixes $v$; the source fixes $\nu$.
• String: $v=\sqrt{T/\mu }$ ($\mu$ = mass per unit length). It does not depend on $\nu$ or $\lambda$.
• Sound: $v=\sqrt{B/\rho }$ in a fluid, $v=\sqrt{Y/\rho }$ in a solid rod. Usually $v_{\text{solid}}>v_{\text{liquid}}>v_{\text{gas}}$.
• Newton (isothermal): $v=\sqrt{P/\rho }\approx 280{\text{m s}}^{-1}$ in air at STP, too low. Laplace (adiabatic): $v=\sqrt{\gamma P/\rho }\approx 331{\text{m s}}^{-1}$ with $\gamma =1.4$. In a gas $v=\sqrt{\gamma RT/M}$: $v\propto \sqrt{T}$, independent of pressure.

Formulas

• $y=a\sin (kx-\omega t+\varphi )$
  Wave along $+x$; $a$ amplitude, $k$ angular wave number, $\omega$ angular frequency
• $v=\nu \lambda =\frac{\omega }{k}$
  Wave speed in m/s
• $v=\sqrt{\frac{T}{\mu }}$
  String: $T$ tension (N), $\mu$ mass per length (kg/m)
• $v=\sqrt{\frac{\gamma P}{\rho }}=\sqrt{\frac{\gamma RT}{M}}$
  Sound in a gas (Laplace); $M$ in kg/mol
• $v=\sqrt{\frac{B}{\rho }},v=\sqrt{\frac{Y}{\rho }}$
  Fluid (bulk modulus) and solid rod (Young's modulus)
• $\Delta \varphi =\frac{2\pi }{\lambda }\Delta x$
  Phase difference between two points $\Delta x$ apart
• $v_{p,\max }=a\omega$
  Maximum particle speed (not the wave speed)

Tips, tricks and hacks

Common mistakes

NEET traps

NEET may write $y=a\sin (\omega t-kx)$: this also moves along $+x$ (same as $-\sin (kx-\omega t)$). The sign between the terms decides direction, not which term comes first.

'Sound travels faster in humid air' is TRUE: moist air is less dense than dry air at the same $T$ and $P$.

Newton used isothermal modulus $P$; Laplace used adiabatic $\gamma P$. An option with $\sqrt{P/\rho }$ is Newton's (wrong) value.

Practise Waves questions

Topic-wise practice from Level 1 to Level 3, NCERT-based questions and NEET PYQs, with a full solution after every answer.