Mechanical Waves
Learn how disturbances travel as progressive waves, form standing patterns on strings and air columns, and create beats and Doppler shifts in JEE Main problems.
The central idea of this chapter
A wave is a travelling disturbance that carries energy without bulk transport of matter. The same relation v = fλ connects speed, frequency and wavelength for all mechanical waves.
What should you understand first?
Waves connects the mathematical form of a travelling disturbance with physical ideas like speed on a string, standing wave patterns, beats and Doppler shift.
Core concepts
- Transverse and longitudinal waves High
- Progressive wave equation High
- Wave speed and medium High
- Standing waves and harmonics High
- Beats & Doppler effect Medium
Wave summary
- Wave relation v = fλ
- Progressive wave y = A sin(kx − ωt)
- String speed v = √(T/μ)
- Sound in gas v = √(γRT/M)
- Standing wave y = 2A sin(kx) cos(ωt)
Waves formula sheet for JEE Main
Focus on how v, f, λ and medium properties appear together. Write the wave equation carefully with signs for direction of propagation.
| Topic | Formula / Relation | Meaning or use |
|---|---|---|
| Progressive wave (right) | y(x,t) = A sin(kx − ωt + φ) | Sinusoidal wave travelling in +x direction |
| Wave speed | v = fλ = ω/k | Relation between speed, frequency and wavelength |
| Speed on stretched string | v = √(T/μ) | T is tension, μ is mass per unit length |
| Speed of sound in gas | v = √(γRT/M) | γ is ratio of specific heats, M is molar mass |
| Standing wave on string | y = 2A sin(kx) cos(ωt) | Nodes at sin(kx) = 0, antinodes at sin(kx) = ±1 |
| String (both ends fixed) | fₙ = n v/(2L) | All harmonics present (n = 1,2,3,…) |
| Open pipe | fₙ = n v/(2L) | All harmonics present; antinode at both ends |
| Closed pipe | fₙ = n v/(4L), n = 1,3,5,… | Only odd harmonics present; node at closed end |
| Beats | fbeat = |f₁ − f₂| | Slow variation of intensity when two close frequencies interfere |
| Doppler effect (sound) | f′ = f₀ (v ± vo)/(v ∓ vs) | Use plus sign when observer/source moves towards, minus when moves away |
How to choose the correct relation
Waves questions usually fall into speed-on-string, standing wave pattern, beats or Doppler categories. Identify the pattern before writing equations.
Progressive & standing waves
- Direction of travel? Sign in kx ± ωt
- Given tension & mass? Use v = √(T/μ)
- Nodes / antinodes asked? Use standing wave form
- String or air column length? Use L with harmonics
- Overtones vs harmonics? Map n carefully
Beats & Doppler
- Two close frequencies? Use fbeat
- Source / observer moving? Use Doppler
- Approach vs recede? Sign choice
- Medium moving (wind)? Modify v
- Graph-based question? Read λ, A from graph
How to prepare Waves
Start with basic v = fλ and progressive wave form, then practise string/pipe harmonics, beats and Doppler effect.
What to do
- Write the standard wave equation and identify A, λ, f, v
- Derive string and pipe harmonics once, then memorise results
- Practise node–antinode pattern questions with diagrams
- Solve numerical problems on beats and tuning
- Finish with Doppler effect examples in different cases
Common mistakes
- Mixing up node and antinode positions in pipes and strings
- Forgetting that only odd harmonics exist in closed pipe
- Using wrong sign in Doppler formula for approach/away
- Ignoring that wave speed on string depends on T and μ, not on A
- Confusing harmonics with overtones in numbering
Ready to practise Waves?
Revise the waves formula sheet, then attempt JEE Main questions on standing waves, beats and Doppler effect.
Waves FAQ
Short answers to frequently tested wave ideas.
A progressive wave is a disturbance that travels through a medium carrying energy from one point to another without permanently transporting matter. A sinusoidal wave can be written as y(x,t) = A sin(kx − ωt + φ).
For any wave, speed v equals the product of frequency and wavelength: v = fλ. For a sinusoidal wave, v is also equal to ω/k.
A standing wave is formed by the superposition of two identical progressive waves travelling in opposite directions, producing fixed nodes and antinodes. Its displacement can be written as y = 2A sin(kx) cos(ωt).