Oscillations and Waves Formula Sheet — JEE Main Physics

Q: What are the most important Oscillations and Waves formulas for JEE Main?

This Oscillations and Waves formula sheet covers all the high-yield Physics formulas, definitions and theorems you need for JEE Main, across Oscillations and periodic motion, Time period, frequency, Displacement as a function of time, Periodic functions, Simple harmonic motion (S.H.M.) — each shown with the key result and, where useful, a worked example.

Q: Is this Oscillations and Waves formula sheet free?

Yes — the full chapter formula sheet is free to read online, no login or payment required.

Q: How should I revise Oscillations and Waves formulas?

Blurt the Oscillations and Waves formulas from memory, then check against this sheet to find your gaps — and practise a few previous-year questions on the chapter to make sure you can apply them under time pressure.

Every key Oscillations and Waves formula, definition and theorem for JEE Main Physics in one place — with common examiner traps and worked examples. Free to read; blurt from memory, then check your gaps.

Syllabus — topics coveredNTA · 18 sub-topics

Oscillations and periodic motion
Time period, frequency
Displacement as a function of time
Periodic functions
Simple harmonic motion (S.H.M.)
Phase
Oscillations of a spring
Energy in S.H.M.
Simple pendulum
Wave motion
Longitudinal and transverse waves
Speed of travelling wave
Displacement relation for progressive wave
Principle of superposition of waves
Reflection of waves
Standing waves in strings and organ pipes
Fundamental mode and harmonics
Beats

SHM — Kinematics & Equation

Defining condition

F = - k x \Rightarrow \frac{d ^{2} x}{d t ^{2}} = - ω^{2} x, ω = \frac{k}{m}

restoring force

\propto

displacement, toward mean position;

ω =

angular frequency (rad/s)

Displacement

x = A sin (ω t + ϕ)

A =

amplitude,

ϕ =

phase constant;

T = \frac{2 π}{ω}

f = \frac{1}{T}

ω = 2 π f

Velocity & acceleration

v = A ω cos (ω t + ϕ) = \pm ω A^{2} - x^{2}, a = - ω^{2} x

v_{ma x} = A ω

(at mean);

a_{ma x} = A ω^{2}

(at extremes)

x, v and a versus time for SHM: velocity leads displacement by 90 degrees, acceleration is opposite to displacement — v leads x by $9 0^{\circ}$ ; a is anti-phase to x.

Comparative: at mean vs extreme position

Quantity	Mean ( $x = 0$ )	Extreme ( $x = A$ )
Displacement	$0$	$\pm A$
Velocity	$\pm A ω$ (max)	$0$
Acceleration	$0$	$\mp A ω^{2}$ (max)
KE / PE	KE max, PE min	KE min, PE max

⚡ Shortcut · SHM = projection of circular motion

SHM is the projection of uniform circular motion (radius A, angular speed

ω

) on a diameter. v–x is an ellipse; a–x is a straight line of slope

- ω^{2}

— read

ω

straight off it.

🚫 Examiner Trap · SHM kinematics

(1) A motion is SHM

only

a = - ω^{2} x

(linear in x, opposite sign) —

a \propto - x^{3}

is NOT SHM. (2)

ω

angular

frequency (rad/s), not f (Hz). (3) Period is

independent of amplitude

. (4) v and a are max at

different

points — v peaks at the mean, a at the extremes.

SHM Energy, Springs & Pendulum

Energy in SHM

K = \frac{1}{2} m ω^{2} (A^{2} - x^{2}), U = \frac{1}{2} k x^{2}, E = \frac{1}{2} k A^{2} = \frac{1}{2} m ω^{2} A^{2}

E constant (

\propto A^{2}

); K max at mean, U max at extremes;

K = U

x = A / 2

Kinetic and potential energy parabolas versus displacement summing to a constant total energy — $K + U = E$ ; each of $K, U$ varies at frequency $2 f$ .

Comparative: spring combinations

	Series	Parallel
Effective $k$	$\frac{1}{k} = \sum \frac{1}{k _{i}}$ (softer)	$k = \sum k_{i}$ (stiffer)
Period	longer	shorter

Spring–mass

T = 2 π \frac{m}{k}

k =

spring constant;

k \propto 1/ length

, so a cut spring is

stiffer

; T independent of g

Simple pendulum

T = 2 π \frac{L}{g}

small angles only; seconds pendulum

T = 2

s; in a lift use

g_{e ff} = g \pm a

; T independent of mass

Physical pendulum

T = 2 π \frac{I}{m g d}

I =

moment of inertia about the pivot,

d =

pivot-to-CM distance

A horizontal spring-mass system and a simple pendulum with their time-period formulas — Two canonical SHM systems.

🚫 Examiner Trap · Energy, springs & pendulum

(1) K and U oscillate at

double

the SHM frequency (

2 f

); total E is constant. (2) Spring period depends on m,k only —

not

on g or amplitude; pendulum on L,g only —

not

on mass. (3) Cutting a spring into n pieces makes each

k \to nk

(stiffer, shorter T). (4)

T = 2 π L / g

holds for

small

angles (

θ ≲ 1 5^{\circ}

) only.

Damped & Forced Oscillations, Resonance

Damped SHM

m \overset{x}{¨} + b \overset{x}{˙} + k x = 0 \Rightarrow x = A e^{- b t /2 m} cos (ω^{'} t + ϕ)

b =

damping constant; amplitude decays as

e^{- b t /2 m}

Damped frequency & energy

ω^{'} = \frac{k}{m} - \frac{b ^{2}}{4 m ^{2}}; E (t) = \frac{1}{2} k A^{2} e^{- b t / m}

ω^{'} < ω_{0}

; for small b,

ω^{'} \approx ω_{0}

; energy decays twice as fast as amplitude

Left: damped oscillation decaying within an exponential envelope. Right: resonance amplitude peaks when driving frequency equals natural frequency, sharper for smaller damping — Decay envelope (left); resonance peak near $ω_{d} = ω_{0}$ (right).

Forced (steady-state) amplitude

A = \frac{F _{0}}{m ^{2} ( ω _{0}^{2} - ω _{d}^{2} ) ^{2} + ω _{d}^{2} b ^{2}}

ω_{d} =

driving frequency,

ω_{0} =

natural frequency; steady state oscillates at

ω_{d}

, not

ω_{0}

Comparative: oscillation types

Type	Amplitude	Frequency
Free (ideal)	constant	$ω_{0}$
Damped	decays $e^{- b t /2 m}$	$ω^{'} < ω_{0}$
Forced	constant (steady)	$ω_{d}$ (driver)

🚫 Examiner Trap · Damping & resonance

(1) At steady state the body oscillates at the

driving

frequency

ω_{d}

, NOT its natural

ω_{0}

. (2) Resonance (max amplitude) is at

ω_{d} \approx ω_{0}

— strictly just below

ω_{0}

, but

\to ω_{0}

for small damping. (3) Smaller damping

\Rightarrow

taller,

sharper

peak; zero damping

\Rightarrow

amplitude

\to \infty

(ideal). (4) Damped frequency

ω^{'} < ω_{0}

— don't equate them.

Wave Motion & Travelling Waves

Comparative: transverse vs longitudinal

	Transverse	Longitudinal
Particle motion	$⊥$ to wave	$∥$ to wave
Form	crests / troughs	compressions / rarefactions
Media	strings, solids, EM	sound; all media
Polarisable?	yes	no

Progressive wave

y = A sin (ω t - k x); v = f λ = \frac{ω}{k}

k = \frac{2 π}{λ}

(wave number),

ω = 2 π f

;

(ω t - k x) \Rightarrow

travels in

+ x

Transverse wave (sine curve, particle motion perpendicular) and longitudinal wave (compressions and rarefactions, particle motion parallel) — Transverse vs longitudinal.

Speed on a string

v = \frac{T}{μ}

T =

tension,

μ =

linear mass density (kg/m)

Speed of sound

v = \frac{E}{ρ} : solid \frac{Y}{ρ}, liquid \frac{B}{ρ}, gas \frac{γ P}{ρ} = \frac{γ RT}{M}

Laplace (adiabatic) for gases;

\approx 331

m/s in air at

0^{\circ}

Sound-speed facts

▸ $v_{so u n d} \propto T (K)$ ; rises $\approx 0.61$ m/s per $^{\circ}$ C; $independent of pressure$ .
▸Particle velocity $v_{p} = - v \frac{\partial y}{\partial x}$ (wave speed $\times$ slope) — not the wave speed.

🚫 Examiner Trap · Travelling waves

(1) Wave speed v (constant, set by the medium)

\neq =

particle speed

v_{p} = A ω

(varies). (2) Sound speed

\propto T

and is

independent of pressure

(

P / ρ

fixed at constant T). (3) Newton's isothermal

P / ρ

was wrong by

\sim 15%

; Laplace's adiabatic

γ P / ρ

is correct. (4)

(ω t - k x)

moves

+ x

;

(ω t + k x)

moves

- x

Superposition & Standing Waves

★ Remember · Superposition & reflection

Overlapping waves add:

y = y_{1} + y_{2}

. Reflection at a

fixed

(rigid) end gives a

π

phase reversal; at a

free

end, no phase change.

Standing wave

y = 2 A cos (k x) sin (ω t)

two identical waves travelling opposite ways; amplitude

2 A cos (k x)

depends on position x

Nodes & antinodes

▸Node–node (or antinode–antinode) spacing $= \frac{λ}{2}$ ; node–antinode $= \frac{λ}{4}$ .
▸At a $node$ : displacement 0, pressure max; at an $antinode$ : displacement max, pressure min.

String harmonics (fixed both ends)

f_{n} = \frac{n}{2 L} \frac{T}{μ}, n = 1, 2, 3, \dots

all

harmonics present; fundamental

f_{1} = \frac{1}{2 L} T / μ

; n loops

=

nth harmonic

First three harmonics of a string fixed at both ends with their nodes — String fixed both ends: $n$ loops $=$ $n$ th harmonic.

Power & intensity (travelling wave)

P = \frac{1}{2} μ ω^{2} A^{2} v, I = \frac{1}{2} ρ ω^{2} A^{2} v

both

\propto A^{2}

and

\propto ω^{2}

🚫 Examiner Trap · Standing waves

(1) A standing wave transports

no net energy

(unlike a travelling wave) — energy stays trapped between nodes. (2) All particles between two nodes are

in phase

; they cross zero together. (3) A

displacement

node is a

pressure antinode

(and vice-versa). (4) Fixed end

=

node, free end

=

antinode — getting this wrong flips every harmonic.

Organ Pipes & Beats

Comparative: closed vs open pipe

	Closed pipe	Open pipe
Fundamental	$\frac{v}{4 L}$	$\frac{v}{2 L}$
Harmonics	odd only ( $1, 3, 5$ )	all ( $1, 2, 3$ )
$n$ th overtone	$f_{n} = \frac{( 2 n - 1 ) v}{4 L}$	$f_{n} = \frac{n v}{2 L}$
Ends	node + antinode	antinode both

Closed organ pipe

f = \frac{( 2 n - 1 ) v}{4 L}, n = 1, 2, 3, \dots

only

odd

harmonics; closed end

=

displacement node, open end

=

antinode

Open organ pipe

f = \frac{n v}{2 L}, n = 1, 2, 3, \dots

all

harmonics; both ends

=

displacement antinodes

Fundamental modes of a closed pipe (node at closed end) and an open pipe (antinodes at both ends) — Closed (odd only) vs open (all harmonics).

Resonance column

λ = 2 (L_{2} - L_{1}), v = 2 f (L_{2} - L_{1})

L_{1}, L_{2} =

1st/2nd resonance lengths; end correction

e = \frac{L _{2} - 3 L _{1}}{2} \approx 0.6 r

Beats

f_{b e a t} = ∣ f_{1} - f_{2} ∣

envelope amplitude

= 2 a cos (π Δ f t)

; audible for

Δ f ≲ 10

Beats: superposition of two close frequencies producing a slow amplitude envelope — Beat frequency $=$ difference of the two frequencies.

🚫 Examiner Trap · Pipes & beats

(1) A closed pipe gives

odd

harmonics only; its fundamental (

v /4 L

) is HALF an open pipe's of the same length. (2) Apply the end correction (

+ 0.6 r

per open end) before equating lengths. (3) Beat frequency is

∣ f_{1} - f_{2} ∣

, and there are

∣ f_{1} - f_{2} ∣

beats per second

. (4) Loading (wax)

lowers

a fork's f, filing

raises

it — use the change in beat count to identify the unknown fork.

Doppler Effect

General formula

f^{'} = f \frac{v \pm v _{o}}{v \mp v _{s}}

v =

speed of sound,

v_{o} =

observer speed,

v_{s} =

source speed; top signs

=

motion that shortens the gap

Comparative: the four cases

Motion	$f^{'}$	Pitch
Source $\to$ observer	$f \frac{v}{v - v _{s}}$	higher
Source receding	$f \frac{v}{v + v _{s}}$	lower
Observer $\to$ source	$f \frac{v + v _{o}}{v}$	higher
Observer receding	$f \frac{v - v _{o}}{v}$	lower

A moving sound source compresses wavefronts ahead (higher frequency) and spreads them behind (lower frequency) — Approaching $\Rightarrow$ higher pitch; receding $\Rightarrow$ lower.

Rules

▸Sign rule: a velocity that $reduces$ the source–observer distance $raises$ f'.
▸No shift if motion is $⊥$ to the line of sight, or if source & observer move identically.
▸Source on a circle with observer at the centre (or vice-versa): $no$ shift.
▸Use medium-frame speeds; for wind, add wind speed to $v$ along the line.

🚫 Examiner Trap · Doppler effect

(1) Sound Doppler is

NOT

symmetric: source-moving and observer-moving by the same speed give different f' (the medium frame matters). (2) Get the

\pm / \mp

from 'does it shorten the gap?', not from memorised signs. (3) No shift for

perpendicular

motion. (4) For

light

use the relativistic formula — it depends only on relative velocity, unlike sound.

More JEE Main Physics formula sheets

Frequently Asked Questions

What are the most important Oscillations and Waves formulas for JEE Main?

This Oscillations and Waves formula sheet covers all the high-yield Physics formulas, definitions and theorems you need for JEE Main, across Oscillations and periodic motion, Time period, frequency, Displacement as a function of time, Periodic functions, Simple harmonic motion (S.H.M.) — each shown with the key result and, where useful, a worked example.

Is this Oscillations and Waves formula sheet free?

Yes — the full chapter formula sheet is free to read online, no login or payment required.

How should I revise Oscillations and Waves formulas?

Blurt the Oscillations and Waves formulas from memory, then check against this sheet to find your gaps — and practise a few previous-year questions on the chapter to make sure you can apply them under time pressure.

Also useful: all formula sheets · JEE Main previous-year papers · most important chapters.