Physics · Mechanics and Waves

Oscillations and Waves formulas for JEE

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

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QuestionState the formula — Acceleration in SHM

Reveal answer

Acceleration in SHM

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

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All 39 Oscillations and Waves formulas

Simple Harmonic Motion (SHM)

16 formulas

Acceleration in SHM

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 = 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 = \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 = -m\omega^2 x

Force law defining a linear simple harmonic oscillator.

applies whenSmall displacements yielding a linear restoring force.

oscillationsshmforce

Frequency-Period Relation

\nu = \frac{1}{T}

The relationship between frequency and time period.

applies whenValid for any periodic motion.

oscillationsfrequencyperiod

Kinetic Energy in SHM

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

\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 = 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 = 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 = \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 = 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) = 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 = 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 = 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) = -\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 = \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

Damped and Forced Oscillations

3 formulas

Damped Oscillation Displacement

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

\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 = \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

Wave Motion

9 formulas

Doppler Effect for Sound

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 = \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 = \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 = \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 = \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) = 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 = \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 = \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 = \frac{2\pi}{\lambda}

Propagation constant or angular wave number.

applies whenValid for any periodic wave.

waveskinematics

Superposition and Standing Waves

11 formulas

Position of Antinodes

x = \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

\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

\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 = \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

\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

\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

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

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) = (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

\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) = \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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