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Mechanical Waves revision notes

Every idea in the Mechanical Waves chapter for JEE Main, with the usual mistakes and the fast methods. Topic 1 is open to everyone; sign in free to read all 22 and download the PDF.

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How we count: 7× asked means 7 past JEE Main questions in the PrepWiser bank needed that concept, and "last" is the latest year one appeared. ★ marks ideas asked 8 or more times.

Wave motion and the wave equation

Mechanical waves
1× asked, last 2022
waveparticles move up–downtransverselongitudinalcompressionrarefaction
Transverse: particles move across the travel. Longitudinal: along it, in compressions and rarefactions
  • Particles only oscillate about their mean positions; energy and momentum travel. Transverse: oscillation perpendicular to travel (string). Longitudinal: along travel (sound), by compressions and rarefactions.
  • Transverse mechanical waves need rigidity, so they cannot travel inside liquids or gases; longitudinal waves travel in all three. The equations look the same: for a longitudinal wave the displacement is along xx.
Travelling wave equation
6× asked, last 2025
y=Asin⁡(kx−ωt+ϕ),y=f(x−vt)y = A\sin(kx-\omega t+\phi),\qquad y=f(x-vt)
kxkx and ωt\omega t with opposite signs: moves along +x+x; same signs: along −x-x. ϕ\phi = phase at x=0x=0, t=0t=0.
xxyyv Δtv\,\Delta tat tt (dashed), at t+Δtt+\Delta t
kx−ωtkx-\omega t: the whole shape slides along +x+x at v=ω/kv=\omega/k
  • Any y=f(ax±bt)y=f(ax\pm bt) is a travelling wave of speed b/ab/a, shape unchanged. A pulse f(x)f(x) at t=0t=0 moving along +x+x becomes f(x−vt)f(x-vt) (along −x-x: f(x+vt)f(x+vt)); read vv from the peak shift. sin⁡kxcos⁡ωt\sin kx\cos\omega t is a standing wave.
  • Not +v+v when kxkx and ωt\omega t carry the same sign. That wave runs towards −x-x, so its velocity is −ω/k-\omega/k. A minus sign in front of the whole sine only shifts the phase by π\pi.
  • Quick method. Crest at x=0x=0 when t=0t=0: use y=Acos⁡(kx−ωt)y=A\cos(kx-\omega t). Zero displacement there: use sine. Check any option by putting x=0x=0, t=0t=0.
Wave parameters
11× asked, last 2026
k=2πλ,ω=2πf=2πT,v=fλ=ωkk=\frac{2\pi}{\lambda},\qquad \omega=2\pi f=\frac{2\pi}{T},\qquad v=f\lambda=\frac{\omega}{k}
kk in rad m⁻¹, ω\omega in rad s⁻¹. Speed is set by the medium, frequency by the source.
xxyyλ\lambda (crest to crest)AA
λ=2π/k\lambda=2\pi/k is set by the medium and source together; ff by the source alone
  • Not the bare number beside xx or tt. Read kk and ω\omega with all their factors (π\pi, 2π2\pi; x/bx/b means k=π/bk=\pi/b inside π(… )\pi(\dots)). The speed needs both: v=ω/kv=\omega/k.
  • Not changed by the phase constant. ϕ\phi shifts the wave, but λ=2π/k\lambda=2\pi/k and f=ω/2πf=\omega/2\pi stay the same. Crest-to-crest distance is always λ\lambda.
  • Quick method. Speed = (coefficient of tt)/(coefficient of xx) in any form; in sin⁡2π(at−bx)\sin2\pi(at-bx), v=a/bv=a/b as common factors cancel. Then convert units: cm/s to m/s, and m/s × 3.6 gives km/h.
Phase difference
3× asked, last 2025
Δϕ=2πλ Δx,Δϕ=2πT Δt\Delta\phi=\frac{2\pi}{\lambda}\,\Delta x,\qquad \Delta\phi=\frac{2\pi}{T}\,\Delta t
Δx\Delta x = gap along the wave at one instant; Δt\Delta t = time gap at one point. 60∘=π/360^\circ=\pi/3 means λ/6\lambda/6.
  • Crest to any later crest: nλn\lambda. Crest to any trough: an odd multiple of λ/2\lambda/2. When both distances are given, λ\lambda must fit both, so several wavelengths can be possible.
  • Points λ\lambda apart move identically. Points λ/2\lambda/2 apart always have equal speed and opposite velocity; no closer pair always shares a speed.
Particle velocity and acceleration
3× asked, last 2025
vp=∂y∂t=−v ∂y∂x,ap=−ω2y,vp,max⁡=Aωv_p=\frac{\partial y}{\partial t}=-v\,\frac{\partial y}{\partial x},\qquad a_p=-\omega^2 y,\qquad v_{p,\max}=A\omega
Wave moving along +x+x. vpv_p changes with time; wave speed v=ω/kv=\omega/k is constant. vpv_p is largest at the mean position.
wave
Wave moving right: vp=−v dy/dxv_p=-v\,dy/dx, so particles on a falling slope move up, on a rising slope down
  • vp,max⁡/v=Ak=2πA/λv_{p,\max}/v=Ak=2\pi A/\lambda. Maximum particle speed equals wave speed when λ=2πA\lambda=2\pi A, and is nn times the wave speed when λ=2πA/n\lambda=2\pi A/n.
  • Not the wave speed. vpv_p is how fast a particle moves, vv how fast the shape moves. In sin⁡2π(nt−x/λ)\sin2\pi(nt-x/\lambda), ω=2πn\omega=2\pi n, not nn, so vp,max⁡=2πnAv_{p,\max}=2\pi nA.
  • Quick method. Slope gives the motion: vp=−v×v_p=-v\times(slope). For a wave moving right, particles on a rising slope move down and those on a falling slope move up.

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Practise Mechanical Waves

17 more concepts in 6 topics

Speed of a transverse wave on a string, Energy and intensity, Superposition and interference, Reflection and standing waves and Standing waves on strings, plus ideas from sound. Read the first part free. Sign in free with Google to read the rest and download the PDF.

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Speed of a transverse wave on a string

  • Wave speed on a stretched string7×
  • Wire of given density and radius3×
  • Hanging rope1×
  • Load hanging in a liquid

Energy and intensity

  • Power carried along a string
  • Intensity from a source1×

Superposition and interference

  • Principle of superposition
  • Two waves of the same frequency5×
  • Intensity of coherent waves1×
  • Path difference from two sources2×

Reflection and standing waves

  • Reflection and transmission
  • Standing wave equation2×
  • Nodes and antinodes2×

Standing waves on strings

  • String fixed at both ends10×
  • Laws of a vibrating string (sonometer)4×
  • String fixed at one end
  • Organ pipes (in brief)14×

Also asked here: sound

  • Key points2

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About 60 past JEE Main questions from this chapter, with instant answer checking and step-by-step solutions where available. Free, and you can start without signing up.

Practise Mechanical Waves

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