Oscillations and Waves formulas for JEE Main & Advanced

23 formula entries for Oscillations and Waves, in the order the chapter builds them. Each shows the formula, when to use it and the first trap to avoid.

Simple Harmonic Motion (SHM)

F = - k x

When to use. Use when analyzing oscillating systems like springs, pendulums, or any system with linear restoring force.

Trap. Confusing SHM with uniform circular motion; SHM is projection of UCM.

Differential Equation of SHM

\frac{d^{2} x}{d t^{2}} + ω^{2} x = 0

When to use. Use to derive time period, angular frequency, or general solution for SHM systems.

Trap. Forgetting the $ω^{2}$ term; it's not $ω$ .

General Solution of SHM

x = A \sin (ω t + ϕ)

When to use. Use to find displacement, velocity, acceleration at any time in SHM.

Trap. Forgetting the phase constant $ϕ$ ; initial conditions determine it.

Velocity and Acceleration in SHM

v = ω \sqrt{A^{2} - x^{2}}

a = - ω^{2} x

When to use. Use to compute velocity or acceleration at a given displacement in SHM.

Trap. Using $v = ω A$ for all positions; it's only maximum at $x = 0$ .

Time Period of Spring-Mass System

T = 2 π \sqrt{\frac{m}{k}}

When to use. Use for any spring-mass system oscillating horizontally or vertically (vertical adds static extension but same $T$ ).

Trap. Including gravity; spring-mass system's $T$ is independent of $g$ .

Time Period of Simple Pendulum

T = 2 π \sqrt{\frac{L}{g}}

When to use. Use for small-angle oscillations of a simple pendulum.

Trap. Using for large amplitudes; formula valid only for $θ < 10^{\circ}$ .

Energy in SHM

E = \frac{1}{2} k A^{2} = \frac{1}{2} m ω^{2} A^{2}

When to use. Use to find amplitude from energy or vice versa, or to find speed at a point.

Trap. Forgetting total energy is constant; only KE and PE change.

Finding Time Period of Compound Systems

T = 2 π \sqrt{\frac{I}{m g d}}

(physical pendulum)

When to use. Use for rigid bodies oscillating about a pivot (e.g., rod, disc).

Trap. Using simple pendulum formula for physical pendulum; need moment of inertia.

Damped Oscillations

x = A e^{- b t / 2 m} \cos (ω^{'} t + ϕ)

When to use. Use when friction or resistance is present in oscillatory systems.

Trap. Forgetting exponential decay factor; amplitude is not constant.

Forced Oscillations and Resonance

A = \frac{F_{0}}{m \sqrt{(ω_{0}^{2} - ω^{2})^{2} + (b ω / m)^{2}}}

When to use. Use when analyzing systems with external periodic driving force.

Trap. Confusing resonance with natural frequency; resonance occurs at driving frequency near natural frequency.

Wave Motion

y = f (x \pm v t)

When to use. Use to describe propagation of disturbances in strings, sound, etc.

Trap. Thinking particles move with wave speed; they oscillate about mean position.

Wave Speed on a String

v = \sqrt{\frac{T}{μ}}

When to use. Use for transverse waves on a stretched string.

Trap. Using $T$ as tension in N, not including weight of string.

Speed of Sound in a Gas

v = \sqrt{\frac{γ P}{ρ}} = \sqrt{\frac{γ R T}{M}}

When to use. Use for speed of sound in gases, e.g., air.

Trap. Using isothermal bulk modulus; sound propagation is adiabatic.

Principle of Superposition

y_{net} = y_{1} + y_{2}

When to use. Use whenever waves overlap, leading to interference, beats, or standing waves.

Trap. Assuming superposition only for same frequency; it works for all waves.

Interference of Waves

Δ x = n λ

(constructive),

Δ x = (n + 1 / 2) λ

(destructive)

When to use. Use for two-source interference problems (e.g., Young's double slit, sound waves).

Trap. Forgetting phase difference also matters; if sources are out of phase, conditions swap.

Beats

f_{beat} = ∣ f_{1} - f_{2} ∣

When to use. Use when two sources of nearly equal frequency produce periodic loudness variation.

Trap. Confusing beat frequency with average frequency; beat is difference.

Standing Waves

y = 2 A \cos (k x) \sin (ω t)

When to use. Use for waves on strings fixed at both ends or in pipes.

Trap. Assuming all particles oscillate with the same amplitude; amplitude varies as $∣ 2 A \cos k x ∣$ (zero at nodes, maximum at antinodes).

Harmonics in a Stretched String

f_{n} = \frac{n}{2 L} \sqrt{\frac{T}{μ}}

n = 1, 2, 3, \dots

When to use. Use for string instruments or any string fixed at both ends.

Trap. Forgetting $n$ starts at 1; fundamental is $n = 1$ .

Harmonics in Organ Pipes

Open:

f_{n} = \frac{n v}{2 L}

n = 1, 2, 3, \dots

; Closed:

f_{n} = \frac{n v}{4 L}

n = 1, 3, 5, \dots

When to use. Use for sound waves in pipes (e.g., flute, organ).

Trap. Using open pipe formula for closed pipe; closed pipe has $4 L$ in denominator.

Doppler Effect in Sound

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

When to use. Use when source or observer is moving relative to medium.

Trap. Sign convention: numerator: + if observer moves toward source; denominator: - if source moves toward observer.

Phase Difference and Path Difference

Δ ϕ = \frac{2 π}{λ} Δ x

When to use. Use to relate path difference to phase difference in interference problems.

Trap. Forgetting $Δ ϕ$ can also come from initial phase difference.

Intensity in Interference

I = I_{1} + I_{2} + 2 \sqrt{I_{1} I_{2}} \cos Δ ϕ

When to use. Use to find intensity at a point in interference pattern.

Trap. Forgetting the cross term; intensity is not simply sum.

Lissajous Figures

x = A \sin (ω_{1} t + ϕ_{1})

y = B \sin (ω_{2} t + ϕ_{2})

When to use. Use in problems involving two perpendicular oscillations (e.g., oscilloscope).

Trap. Assuming same frequency always gives straight line; phase difference matters.

Values to remember

$g = 9.8 {m/s}^{2}$ Time Period of Simple Pendulum
$γ_{air} = 1.4$ Speed of Sound in a Gas
$R = 8.314 J/mol K$ Speed of Sound in a Gas
$v_{sound} = 330 m/s$ (approx) Doppler Effect in Sound

Read the full chapter: Oscillations and Waves for JEE, with what JEE asks and the common traps.