PiTheory

PiTheory JEE Physics Practice & Revision Pack · Class 11 · Chapter 14 · JEE Main

Waves

Chapter-wise practice and revision for JEE Main: concise notes on the wave equation, wave speed, standing waves on strings, organ pipes, beats and the Doppler effect, concept figures, and 50 selected previous-year JEE Main questions covering the major concepts and question patterns, each with a step-by-step solution.

50 selected questions6 topics5 concept figuresPreview · first 3 of 50 questions

How to study this chapter

  1. Look at the topic's figures, then read its notes (5 to 10 minutes). They give the major equations and methods the questions use.
  2. Solve the topic's questions with a timer: 2 minutes for Easy, 3 to 4 for Medium and Hard.
  3. Check the solution. Read the Point to note line even if you were right: it is the idea you will reuse.
  4. Mark each question Got it or Revise. Come back to the Revise list after two days.

Where the marks come from

Main idea of each of the 114 previous-year JEE questions in the PiTheory bank

Strings24
Organ pipes22
Wave equation21
Doppler21
Wave speed15
Beats11

Contents

  1. 1Travelling waves and the wave equationQ1–Q8
  2. 2Speed of waves on strings, in solids and in gasesQ9–Q16
  3. 3Standing waves and vibrating stringsQ17–Q25
  4. 4Organ pipes and the resonance tubeQ26–Q34
  5. 5BeatsQ35–Q41
  6. 6Doppler effectQ42–Q50
  7. ✓Last-day checklist
01

Topic 1 · Q1–Q8

Travelling waves and the wave equation

21 of the 114 JEE PYQs in the bank test this as their main idea

Look first

xλAv = ω/ky = A sin(ωt − kx)k = 2π/λ, ω = 2πf
Wavelength, amplitude and wave speed on one snapshot.

Quick notes

$$y=A\sin(\omega t-kx+\phi),\qquad k=\frac{2\pi}{\lambda},\quad \omega=2\pi f,\quad v=f\lambda=\frac{\omega}{k}$$

$\omega t-kx$: moves towards $+x$; $\omega t+kx$: towards $-x$. Any function of $(x\mp vt)$ alone is a travelling wave; $\sin kx\cos\omega t$ is a standing wave.

QuantityFormula
Particle velocity$\dfrac{\partial y}{\partial t}=-v\dfrac{\partial y}{\partial x}$ (wave towards $+x$); max $A\omega$
Phase difference for path $\Delta x$$\Delta\phi=\dfrac{2\pi}{\lambda}\Delta x$
Superposition, same $\omega$, $k$$R^2=A_1^2+A_2^2+2A_1A_2\cos\delta$
Intensity$I\propto A^2\omega^2$; point source $I=\dfrac{P}{4\pi r^2}$
  • Wave speed $\ne$ particle speed: equal when $A\omega=\omega/k$, i.e. $\lambda=2\pi A$.
  • Points $\lambda/2$ apart always have equal speeds and opposite velocities; $\lambda$ apart move identically.
  • Crest to crest: $n\lambda$; crest to trough: $(n+\tfrac12)\lambda$.
  • On refraction, frequency is fixed by the source; $v$ and $\lambda$ change together.
  • Read units: $k$ in rad cm$^{-1}$ gives $v$ in cm s$^{-1}$.
Traps. Sign of direction. Forgetting $2\pi$ when the equation is written as $\sin2\pi(ft-x/\lambda)$.

Practice: first 3 of 8 questions

Q1JEE Main 2019 (12 Jan, Shift 1)Easy

A travelling harmonic wave is represented by the equation $y(x, t) = 10^{-3}\sin(50t + 2x)$, where $x$ and $y$ are in meter and $t$ is in seconds. Which of the following is a correct statement about the wave?

  1. AThe wave is propagating along the positive $x$-axis with speed $25\ \text{m s}^{-1}$
  2. BThe wave is propagating along the positive $x$-axis with speed $100\ \text{m s}^{-1}$
  3. CThe wave is propagating along the negative $x$-axis with speed $25\ \text{m s}^{-1}$
  4. DThe wave is propagating along the negative $x$-axis with speed $100\ \text{m s}^{-1}$
Answer and solution

Answer: Option C: The wave is propagating along the negative $x$-axis with speed $25\ \text{m s}^{-1}$

Concept. In $y=A\sin(\omega t\pm kx)$ the wave speed is $v=\dfrac\omega k$. The sign tells the direction: $\omega t-kx$ moves towards $+x$, $\omega t+kx$ towards $-x$ (the phase stays constant when $x$ decreases as $t$ grows).

Step 1. Speed. $\omega=50$, $k=2$: $v=25$ m s$^{-1}$.

Step 2. Direction. $50t+2x$: negative $x$.

Answer. Option C.

Point to note. $+kx$: towards $-x$.

Q2JEE Main 2025 (23 Jan, Shift 2)Easy

The equation of a transverse wave travelling along a string is $y(x,t) = 4.0\sin\left[20\times10^{-3}x + 600t\right]\ \text{mm}$, where $x$ is in mm and $t$ is in seconds. The velocity of the wave is

  1. A$-60\ \text{m/s}$
  2. B$-30\ \text{m/s}$
  3. C$+30\ \text{m/s}$
  4. D$+60\ \text{m/s}$
Answer and solution

Answer: Option B: $-30\ \text{m/s}$

Concept. $v=\dfrac\omega k$ in whatever units $x$ and $t$ carry; then convert. The $+$ sign between the $x$ and $t$ terms means motion towards $-x$.

Step 1. Speed. $\dfrac{600}{20\times10^{-3}}=3\times10^4$ mm s$^{-1}=30$ m s$^{-1}$.

Step 2. Direction. Negative $x$.

Answer. $-30$ m s$^{-1}$ (Option B).

Point to note. $\omega/k$, then units.

Q3JEE Main 2023 (25 Jan, Shift 1)Easy

The distance between two consecutive points with phase difference of $60^\circ$ in a wave of frequency $500\ \text{Hz}$ is $6.0\ \text{m}$. The velocity with which wave is travelling is ______ $\text{km s}^{-1}$.

Numerical answer type

Answer and solution

Answer: 18

Concept. A full wavelength corresponds to a phase difference of $360^\circ$, so path difference $=\dfrac{\phi}{2\pi}\lambda$.

Step 1. Wavelength. $60^\circ$ is $\dfrac16$ of a cycle: $\dfrac\lambda6=6$ m, $\lambda=36$ m.

Step 2. Speed. $v=f\lambda=500\times36=18\,000$ m s$^{-1}=18$ km s$^{-1}$.

Answer. $18$.

Point to note. $60^\circ=\lambda/6$.

47 more questions and 5 more topics in the full chapter

  1. 2Speed of waves on strings, in solids and in gases8 questions with solutions
  2. 3Standing waves and vibrating strings9 questions with solutions
  3. 4Organ pipes and the resonance tube9 questions with solutions
  4. 5Beats7 questions with solutions
  5. 6Doppler effect9 questions with solutions
  6. ✓Last-day checklist

The complete Waves chapter is in the Class 11 Pack and the Complete Pack, as a PDF on WhatsApp. Want to see a whole chapter first? Laws of Motion and Current Electricity are free in full.

Class 11 Pack

All 14 Class 11 chapters: kinematics to waves.

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  • 651 selected JEE Main questions (up to 50 per chapter)
  • Concept notes, equations and methods
  • Step-by-step solutions, answer keys, difficulty levels, checklists
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  • Concept notes, equations and methods
  • Step-by-step solutions, answer keys, difficulty levels, checklists
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