JEE Mains Physics · Waves

Wave Equation and Particle Motion

A travelling wave y = A sin(ωt − kx) carries its frequency in ω = 2πf, its wavelength in k = 2π/λ and its speed in ω/k; each particle only oscillates in place.

All Waves notes JEE Mains Physics strategy

Why this matters

Nineteen PYQs, five of them asking for a number, and three from 2026. Ten read a speed, frequency, wavelength or phase difference off a given equation; four build the equation from a description or ask which function is a travelling wave; five are about how fast the particles move, or how intensity falls away from a point source. The reading is quick once the coefficients are pulled out cleanly; marks go on units and on the direction of travel.

Concept 1 of 3: Reading speed, frequency and wavelength from a wave equation

y = A\sin(\omega t - kx)

the number in front of t says how fast each point oscillates, and the number in front of x says how fast the phase changes along the string. Their ratio is the speed. Expand any common factor such as

\pi

2\pi

first, and put x and t in the units you want the answer in.

Definition

$\omega = 2\pi f$ , $k = \dfrac{2\pi}{\lambda}$ , and the wave speed is $v = \dfrac{\omega}{k} = f\lambda$ .
Direction: opposite signs on the x and t terms, as in $\omega t - kx$ , mean travel along +x. The same sign, as in $\omega t + kx$ , means travel along −x.
In the form $A\sin\dfrac{2\pi}{\lambda}(vt - x)$ the speed is the number in front of t inside the bracket.
In $A\sin 2\pi(ft - x/\lambda)$ the frequency and wavelength can be read directly.
A constant added inside the bracket is only a starting phase; it does not change f, λ or v.
Phase difference between two points $\Delta x$ apart: $\Delta\phi = k\,\Delta x = \dfrac{2\pi}{\lambda}\Delta x$ .
If x is in cm, k is in cm⁻¹, and $\omega/k$ comes out in cm/s. Divide by 100 for m/s; multiply m/s by 18/5 for km/h.

Wave speed from the equation

v = \frac{\omega}{k} = f\lambda \qquad \omega = 2\pi f \qquad k = \frac{2\pi}{\lambda} \qquad \Delta\phi = \frac{2\pi}{\lambda}\,\Delta x

Worked example

A wave is given by

y = 2\sin\pi\left(100t - \dfrac{x}{40}\right)

, with x and y in cm and t in s. Find its frequency, wavelength, speed in m/s and direction of travel.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 2 Apr 2026 Shift 1 · Q10Moderate

Example 1 · Waves · Wave Equation and Particle Motion

The equation of a plane progressive wave is given by

y= 5\cos\pi\left( 200t-\frac{x}{150} \right)

where

x

and

y

are in cm and

t

is in second. The velocity of the wave is

\_\_\_\_

m/s

k in cm⁻¹ gives a speed in cm/s

When x is measured in cm, ω/k is in cm/s. Answer options in m/s are then 100 times smaller. Convert k to m⁻¹, or convert the speed at the end.

Expand the common factor before reading

In y = A sin π(300t − x/60) the coefficient of t is 300π, not 300, and that of x is π/60. Here the π cancels in ω/k, but in the frequency f = ω/2π it does not: f is 150 Hz, not 300/2π.

Same signs mean travel along −x

In y = A sin(kx + ωt) the wave moves towards −x, so a velocity asked with its sign is negative. Opposite signs mean +x.

Drill 9 more on reading speed, frequency and wavelength from a wave equation

Concept 2 of 3: Writing the equation of a travelling wave

Any shape that slides along without changing form is a function of

x - vt

(moving +x) or

x + vt

(moving −x). For a sine wave, four facts fix the equation: the amplitude, ω, k, and where a particle is at t = 0. A crest at the origin at t = 0 calls for a cosine; a particle at its mean position calls for a sine.

Definition

Amplitude = half the total to-and-fro distance of a particle.
$\omega = 2\pi f$ and $k = \dfrac{\omega}{v} = \dfrac{2\pi}{\lambda}$ .
Travelling along +x: $y = A\cos(kx - \omega t)$ or $A\sin(\omega t - kx)$ . Along −x: change the sign between the two terms.
Crest ( $y = +A$ ) at x = 0, t = 0: use a cosine. Mean position at x = 0, t = 0: use a sine, with the sign set by which way that particle moves next.
A pulse $y = f(x)$ at t = 0 that becomes $f(x - d)$ at time t has moved a distance d along +x: $v = d/t$ . If it becomes $f(x + d)$ , it moved along −x.
$A\sin kx\cos\omega t$ is a standing wave, not a travelling one: x and t are in separate factors. A function of $x^{2}$ and t separately, such as $e^{-x^{2}}\cos t$ , is not travelling either.

A travelling wave

y = f(x \mp vt) \qquad y = A\sin(\omega t - kx + \phi_0) \qquad k = \frac{\omega}{v}

Worked example

A wave of frequency 50 Hz travels along −x at

20\ \text{m/s}

. Each particle moves through a total distance of 8 mm to and fro. At t = 0 the particle at x = 0 is at its mean position, moving in the +y direction. Write y(x, t) in metres.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2025 · 2 Apr 2025 · Q81Moderate

Example 2 · Waves · Wave Equation and Particle Motion

A sinusoidal wave of wavelength 7.5 cm travels a distance of 1.2 cm along the

x

- direction in 0.3 sec . The crest

P

is at

x= 0

t= 0sec

and maximum displacement of the wave is 2 cm . Which equation correctly represents this wave ?

The to-and-fro distance is twice the amplitude

A particle that moves through a total of 6 cm swings 3 cm each side of its mean position. The amplitude is 3 cm, not 6 cm.

A crest at the origin is a cosine

At t = 0 a sine is zero at x = 0, so it cannot describe a crest there. When the stem puts a crest at the origin, the correct option is a cosine.

Drill 3 more on writing the equation of a travelling wave

Concept 3 of 3: Particle velocity and intensity of a wave

The wave moves along the string, but each particle only moves up and down. Its speed changes all the time and peaks at Aω as it passes the mean position. That peak, compared with the wave speed ω/k, is just Ak. For a point source, the energy spreads over a sphere, so the intensity falls as the square of the distance.

Definition

Particle velocity $v_p = \dfrac{\partial y}{\partial t}$ ; its maximum is $A\omega$ .
$\dfrac{\partial y}{\partial t} = -v\,\dfrac{\partial y}{\partial x}$ : the particle velocity is minus the wave speed times the slope of the string.
$\dfrac{v_{p,\max}}{v} = \dfrac{A\omega}{\omega/k} = Ak = \dfrac{2\pi A}{\lambda}$ .
Two points $\lambda/2$ apart always move with equal speeds in opposite directions.
Intensity $I \propto A^{2}\omega^{2}$ . From a point source, $I = \dfrac{P}{4\pi r^{2}}$ , so $I \propto \dfrac{1}{r^{2}}$ .
For a sphere around the source: area $\propto r^{2}$ and volume $\propto r^{3}$ . Find r first, then I.

Particle speed and intensity

v_{p,\max} = A\omega \qquad \frac{v_{p,\max}}{v} = Ak = \frac{2\pi A}{\lambda} \qquad I = \frac{P}{4\pi r^{2}}

Worked example

A wave is

y = 3\sin(60t - 0.5x)

, with x and y in cm and t in s. Find the wave speed, the maximum particle speed and their ratio.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2022 · 26 July 2022 · Q103Moderate

Example 3 · Waves · Wave Equation and Particle Motion

A transverse wave is represented by

y = 2\sin(\omega t - kx)\ cm

. The value of wavelength (in

cm

) for which the wave velocity becomes equal to the maximum particle velocity, will be;

Particle velocity is not wave velocity

The wave speed ω/k is the same everywhere. The particle velocity ∂y/∂t changes from zero to Aω in every cycle. A question on 'maximum particle velocity' wants Aω.

Scale the radius, not the area or volume

Intensity goes as 1/r². If a sphere's surface area grows 9 times, r grows 3 times and I falls to one ninth, not to one eighty-first.

Drill 4 more on particle velocity and intensity of a wave

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Reading speed, frequency and wavelength from a wave equation
Wave speed from the equation
$v = \frac{\omega}{k} = f\lambda \qquad \omega = 2\pi f \qquad k = \frac{2\pi}{\lambda} \qquad \Delta\phi = \frac{2\pi}{\lambda}\,\Delta x$
Writing the equation of a travelling wave
A travelling wave
$y = f(x \mp vt) \qquad y = A\sin(\omega t - kx + \phi_0) \qquad k = \frac{\omega}{v}$
Particle velocity and intensity of a wave
Particle speed and intensity
$v_{p,\max} = A\omega \qquad \frac{v_{p,\max}}{v} = Ak = \frac{2\pi A}{\lambda} \qquad I = \frac{P}{4\pi r^{2}}$

Watch out for (7)

k in cm⁻¹ gives a speed in cm/s
→ Reading speed, frequency and wavelength from a wave equation
Expand the common factor before reading
→ Reading speed, frequency and wavelength from a wave equation
Same signs mean travel along −x
→ Reading speed, frequency and wavelength from a wave equation
The to-and-fro distance is twice the amplitude
→ Writing the equation of a travelling wave
A crest at the origin is a cosine
→ Writing the equation of a travelling wave
Particle velocity is not wave velocity
→ Particle velocity and intensity of a wave
Scale the radius, not the area or volume
→ Particle velocity and intensity of a wave

Test yourself on Waves

20 past JEE Mains questions from this chapter, timed at 48 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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