Physics · Mechanics and Waves

Oscillations and Waves formulas for JEE

Every Oscillations and Waves formula you need for JEE, grouped by concept.

FormulasRevision notes
39 formulas4 concepts
01Simple Harmonic Motion (SHM)02Damped and Forced Oscillations03Wave Motion04Superposition and Standing Waves
01

Simple Harmonic Motion (SHM)

16 formulas

Acceleration in SHM

a(t)=−ω2Acos⁡(ωt+ϕ)=−ω2x(t)a(t) = -\omega^2 A \cos(\omega t + \phi) = -\omega^2 x(t)

Instantaneous acceleration of a particle in SHM.

applies whenStandard SHM, directly proportional and opposite to displacement.
oscillationsshmacceleration

Period of a Floating Cork

T=2πhρρlgT = 2\pi\sqrt{\frac{h\rho}{\rho_l g}}

Oscillation period of a cork of density rho and height h floating in liquid of density rho_l.

applies whenSmall vertical displacements, negligible fluid viscosity/damping.
oscillationsshmfluids

Total Energy in SHM

E=12kA2=12mω2A2E = \frac{1}{2}kA^2 = \frac{1}{2}m\omega^2 A^2

Total mechanical energy of a harmonic oscillator.

applies whenNo damping or dissipative forces.
oscillationsshmenergy

Restoring Force in SHM

F=−kx=−mω2xF = -kx = -m\omega^2 x

Force law defining a linear simple harmonic oscillator.

applies whenSmall displacements yielding a linear restoring force.
oscillationsshmforce

Frequency-Period Relation

ν=1T\nu = \frac{1}{T}

The relationship between frequency and time period.

applies whenValid for any periodic motion.
oscillationsfrequencyperiod

Kinetic Energy in SHM

K=12mω2A2sin⁡2(ωt+ϕ)=12k(A2−x2)K = \frac{1}{2}m\omega^2 A^2 \sin^2(\omega t + \phi) = \frac{1}{2}k(A^2 - x^2)

Kinetic energy of a harmonic oscillator.

applies whenStandard SHM.
oscillationsshmenergykinetic

Angular Frequency

ω=2πT=2πν\omega = \frac{2\pi}{T} = 2\pi\nu

Angular frequency related to period and linear frequency.

applies whenValid for periodic and oscillatory motions.
oscillationsfrequency

Pendulum in Accelerating Frame

T=2πLgeffT = 2\pi\sqrt{\frac{L}{g_{eff}}}

Time period of a simple pendulum in a non-inertial accelerating frame.

applies whenEffective gravity g_{eff} must be calculated based on frame acceleration.
oscillationsshmpendulum

Period of a Simple Pendulum

T=2πLgT = 2\pi\sqrt{\frac{L}{g}}

Time period of a simple pendulum.

applies whenSmall angular displacements (theta < 20 degrees).
oscillationsshmpendulum

Potential Energy in SHM

U=12kx2=12mω2A2cos⁡2(ωt+ϕ)U = \frac{1}{2}kx^2 = \frac{1}{2}m\omega^2 A^2 \cos^2(\omega t + \phi)

Potential energy of a conservative harmonic oscillator.

applies whenAssuming U=0 at the mean position (x=0).
oscillationsshmenergypotential

Period of a Physical Pendulum

T=2πImgdT = 2\pi\sqrt{\frac{I}{mgd}}

Time period of a rigid body oscillating about a pivot.

applies whenSmall angular displacements, I is moment of inertia about pivot, d is distance from pivot to COM.
oscillationsshmpendulumjee-advanced

Displacement in SHM

x(t)=Acos⁡(ωt+ϕ)x(t) = A \cos(\omega t + \phi)

Displacement of a particle executing simple harmonic motion as a function of time.

applies whenStandard SHM with amplitude A, angular frequency omega, and initial phase phi.
oscillationsshmdisplacement

Period of a Spring-Mass System

T=2πmkT = 2\pi\sqrt{\frac{m}{k}}

Time period of a linear harmonic oscillator.

applies whenMassless spring, constant spring constant k.
oscillationsshmspring

Period of a Torsional Pendulum

T=2πICT = 2\pi\sqrt{\frac{I}{C}}

Time period of torsional oscillations.

applies whenC is the torsional constant (restoring torque per unit angle).
oscillationsshmpendulumjee-advanced

Velocity in SHM

v(t)=−ωAsin⁡(ωt+ϕ)v(t) = -\omega A \sin(\omega t + \phi)

Instantaneous velocity of a particle in SHM.

applies whenObtained by differentiating the displacement equation.
oscillationsshmvelocity

Velocity-Position Relation in SHM

v=±ωA2−x2v = \pm\omega\sqrt{A^2 - x^2}

Velocity of an oscillator as a function of its displacement from the mean position.

applies whenStandard SHM.
oscillationsshmvelocityjee-advanced
02

Damped and Forced Oscillations

3 formulas

Damped Oscillation Displacement

x(t)=Ae−bt2mcos⁡(ω′t+ϕ)x(t) = A e^{-\frac{bt}{2m}} \cos(\omega' t + \phi)

Displacement of a damped harmonic oscillator.

applies whenUnderdamped system where damping force F_d = -bv.
oscillationsdampingjee-advanced

Damped Angular Frequency

ω′=km−b24m2\omega' = \sqrt{\frac{k}{m} - \frac{b^2}{4m^2}}

Angular frequency of an underdamped oscillator.

applies whenUnderdamped condition: k/m > b^2/(4m^2).
oscillationsdampingjee-advanced

Forced Oscillation Amplitude

A=F0m2(ω02−ωd2)2+ωd2b2A = \frac{F_0}{\sqrt{m^2(\omega_0^2 - \omega_d^2)^2 + \omega_d^2 b^2}}

Steady-state amplitude of a forced, damped oscillator.

applies whenDriving force F(t) = F_0 cos(omega_d t).
oscillationsforcedresonancejee-advanced
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03

Wave Motion

9 formulas

Doppler Effect for Sound

f′=f(v±vov∓vs)f' = f \left( \frac{v \pm v_o}{v \mp v_s} \right)

Apparent frequency heard by an observer due to relative motion of source and observer.

applies whenv_o is observer velocity, v_s is source velocity, v is sound speed. Signs depend on approach/recession.
wavessounddopplerjee-advanced

Intensity of a Wave

I=12ρω2a2vI = \frac{1}{2} \rho \omega^2 a^2 v

Average power transmitted per unit area by a progressive wave.

applies whenStandard progressive harmonic wave in a uniform medium.
wavesenergyintensityjee-advanced

Speed of Sound in Ideal Gas (Laplace)

v=γPρ=γRTMv = \sqrt{\frac{\gamma P}{\rho}} = \sqrt{\frac{\gamma RT}{M}}

Adiabatic speed of sound in an ideal gas with Laplace correction.

applies whenIdeal gas behavior, adiabatic compressions/rarefactions.
wavessoundgasspeed

Speed of Sound in Fluid

v=Bρv = \sqrt{\frac{B}{\rho}}

Speed of a longitudinal wave in a fluid.

applies whenFluid with bulk modulus B and density rho.
wavessoundfluidspeed

Speed of Sound in Solid Bar

v=Yρv = \sqrt{\frac{Y}{\rho}}

Speed of a longitudinal wave in a thin solid bar.

applies whenLinear medium where lateral expansion is negligible (Young's modulus Y).
wavessoundsolidspeed

Progressive Wave Equation

y(x,t)=asin⁡(kx∓ωt+ϕ)y(x,t) = a \sin(kx \mp \omega t + \phi)

Displacement relation for a one-dimensional progressive harmonic wave.

applies whenConstant amplitude, single frequency. Minus sign for +x direction, plus sign for -x direction.
wavesprogressive

Wave Speed

v=ωk=λνv = \frac{\omega}{k} = \lambda\nu

Phase speed of a travelling wave.

applies whenNon-dispersive or specific frequency wave.
waveskinematicsspeed

Speed of Wave on a String

v=Tμv = \sqrt{\frac{T}{\mu}}

Speed of a transverse wave on a stretched string.

applies whenSmall amplitude waves, uniform linear mass density mu, constant tension T.
wavesstringspeed

Angular Wave Number

k=2πλk = \frac{2\pi}{\lambda}

Propagation constant or angular wave number.

applies whenValid for any periodic wave.
waveskinematics
04

Superposition and Standing Waves

11 formulas

Position of Antinodes

x=(n+12)λ2x = \left(n + \frac{1}{2}\right)\frac{\lambda}{2}

Locations of maximum amplitude in a standing wave.

applies whenn = 0, 1, 2, 3... Origin is at a node.
wavesstandingantinodes

Beat Frequency

νbeat=∣ν1−ν2∣\nu_{beat} = |\nu_1 - \nu_2|

Frequency of amplitude modulation (beats) when two close frequencies are superposed.

applies whenWaves of comparable amplitude with slightly different frequencies.
wavesinterferencebeats

Harmonics of Closed Organ Pipe

νn=(2n+1)v4L\nu_n = \frac{(2n+1)v}{4L}

Natural frequencies of an air column closed at one end and open at the other.

applies whenn = 0, 1, 2, 3... Generates only odd harmonics.
wavesstandingpipeharmonics

Position of Nodes

x=nλ2x = \frac{n\lambda}{2}

Locations of zero amplitude in a standing wave.

applies whenn = 0, 1, 2, 3... Origin is at a node.
wavesstandingnodes

Harmonics of Open Organ Pipe

νn=nv2L\nu_n = \frac{nv}{2L}

Natural frequencies of an air column open at both ends.

applies whenn = 1, 2, 3... Generates all harmonics.
wavesstandingpipeharmonics

Phase Difference and Path Difference

Δϕ=2πλΔx\Delta\phi = \frac{2\pi}{\lambda}\Delta x

Relation between phase difference and path difference for two overlapping waves.

applies whenWaves with identical wavelength.
wavessuperpositioninterferencejee-advanced

Reflection at Open Boundary

yr(x,t)=asin⁡(kx+ωt)y_r(x,t) = a \sin(kx + \omega t)

Reflected wave from an open (free) boundary undergoes zero phase change.

applies whenIncident wave y = a sin(kx - wt). Free boundary.
wavesreflection

Reflection at Rigid Boundary

yr(x,t)=−asin⁡(kx+ωt)y_r(x,t) = -a \sin(kx + \omega t)

Reflected wave from a rigid boundary undergoes a pi phase change.

applies whenIncident wave y = a sin(kx - wt). Rigid constraint at boundary.
wavesreflection

Standing Wave Equation

y(x,t)=(2asin⁡kx)cos⁡ωty(x,t) = (2a \sin kx) \cos \omega t

Equation of a standing (stationary) wave on a string.

applies whenSuperposition of two identical waves travelling in opposite directions.
wavesstanding

Harmonics of Stretched String

νn=nv2L=n2LTμ\nu_n = \frac{nv}{2L} = \frac{n}{2L}\sqrt{\frac{T}{\mu}}

Natural frequencies (normal modes) of a string fixed at both ends.

applies whenn = 1, 2, 3... Generates all harmonics.
wavesstandingstringharmonics

Superposition of Two Waves

y(x,t)=[2acos⁡(ϕ2)]sin⁡(kx−ωt+ϕ2)y(x,t) = \left[2a \cos\left(\frac{\phi}{2}\right)\right] \sin\left(kx - \omega t + \frac{\phi}{2}\right)

Resultant of two harmonic waves of equal amplitude and frequency differing by phase phi.

applies whenWaves travelling in the same direction with same w and k.
wavessuperpositioninterference
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