Oscillations and Waves – JEE Main Physics Formula Sheet & Class 11 Notes | SHM, Spring, Pendulum, Wave Equation, Standing Waves, Beats, Doppler Effect & All Formulas

Simple Harmonic Motion (SHM) — Displacement, Velocity, Acceleration, Energy

Why SHM Energy Formulas and v=ω√(A²–x²) Are the Most-Tested Oscillations JEE Main Results

SHM Definition and Condition (from Aakash PDF — Oscillations and Waves):

Simple Harmonic Motion is a special type of periodic motion in which the restoring force is directly proportional to the displacement from equilibrium and always directed toward equilibrium.

Defining equation: a = –ω²x (acceleration = –ω² × displacement)

The negative sign indicates the restoring nature (acceleration opposes displacement). ω = √(k/m) for spring-mass system.

Displacement, Velocity, Acceleration Equations (from Aakash PDF — Oscillations and Waves JEE Main):

x = A sin(ωt + φ) (displacement; φ = initial phase)

v = dx/dt = Aω cos(ωt + φ)

In terms of displacement: v = ω√(A²–x²)

a = dv/dt = –Aω² sin(ωt + φ) = –ω²x

v_max = Aω (when x=0, at equilibrium position)

a_max = Aω² (when x=±A, at extreme positions; v=0 at extremes)

v=0 at x=±A (amplitude); v=v_max at x=0 (equilibrium).

Time period: T = 2π/ω; Frequency: f = 1/T = ω/2π

Energy in SHM (from Aakash PDF — Oscillations and Waves JEE Main):

Kinetic Energy: KE = ½mv² = ½mω²(A²–x²)

Potential Energy: PE = ½kx² = ½mω²x²

Total Energy: E = KE + PE = ½mω²A² = ½kA²

E is constant (independent of time and position in ideal SHM).

KE = PE when ½mω²(A²–x²) = ½mω²x² → A²–x² = x² → x = ±A/√2

So KE = PE at x = ±A/√2 (at these positions, each equals E/2).

Average KE over one complete cycle = Average PE = E/2 = ½mω²A²/2 = ¼mω²A².

KE vs time: KE = ½mω²A²cos²(ωt+φ) → periodic with angular frequency 2ω.

PE vs time: PE = ½mω²A²sin²(ωt+φ) → periodic with angular frequency 2ω.

Download the Free PDF for Oscillations and Waves for all SHM energy examples for JEE Main.

Spring Systems — Series, Parallel, Two-Mass, Spring Cut

Why Spring Combination Formulas and Reduced Mass Are Essential Oscillations JEE Main Results

Spring-Mass System (from Aakash PDF — Oscillations and Waves JEE Main):

A mass m attached to a spring of constant k on a frictionless surface: restoring force F = –kx → ω = √(k/m).

T = 2π√(m/k)

Note: T is independent of amplitude A (fundamental property of SHM). T ∝ √m (heavier mass → longer period). T ∝ 1/√k (stiffer spring → shorter period).

Springs in Series (from Aakash PDF — Oscillations and Waves JEE Main):

Two springs k₁ and k₂ in series (end-to-end): same force on each spring; total extension = sum of individual extensions.

1/k_eff = 1/k₁ + 1/k₂ → k_eff = k₁k₂/(k₁+k₂)

k_eff < min(k₁, k₂) in series (softer combination).

T_series = 2π√(m/k_eff) = 2π√(m(k₁+k₂)/k₁k₂)

Springs in Parallel (from Aakash PDF — Oscillations and Waves JEE Main):

Two springs k₁ and k₂ in parallel (side-by-side): same extension; total force = sum of individual forces.

k_eff = k₁ + k₂

k_eff > max(k₁, k₂) in parallel (stiffer combination).

T_parallel = 2π√(m/(k₁+k₂))

Two-Mass Spring System (from Aakash PDF — Oscillations and Waves JEE Main):

Two masses m₁ and m₂ connected by a spring of constant k on frictionless surface, both free to move. The system oscillates with the reduced mass:

μ = m₁m₂/(m₁+m₂)

T = 2π√(μ/k) = 2π√(m₁m₂/k(m₁+m₂))

Spring Cutting (from Aakash PDF — Oscillations and Waves JEE Main):

If a spring of constant k and length l is cut into n equal parts: each part has length l/n and spring constant k' = nk.

General: if cut into ratio m:n, one piece has k' = k(m+n)/m and the other k'' = k(m+n)/n.

Download the Free PDF for Oscillations and Waves for all spring system examples for JEE Main.

Simple Pendulum, Compound Pendulum, and Other SHM Systems

Why Pendulum in Lift, Horizontal Acceleration, and Compound Pendulum Are Key Oscillations JEE Main Questions

Simple Pendulum (from Aakash PDF — Oscillations and Waves JEE Main):

A point mass m suspended by a massless inextensible string of length L, making small angle oscillations (θ<10°):

T = 2π√(L/g)

T is independent of mass m and amplitude (for small angles).

Seconds pendulum: T = 2s; L = gT²/4π² = 10×4/4π² = 10/π² ≈ 1 m at g=10 m/s².

Pendulum in a Lift (from Aakash PDF — Oscillations and Waves JEE Main):

(1) Lift accelerating UPWARD with a: effective g = g+a → T = 2π√(L/(g+a)) [T decreases, pendulum runs faster]

(2) Lift accelerating DOWNWARD with a: effective g = g–a → T = 2π√(L/(g–a)) [T increases, pendulum runs slower]

(3) Free fall (a=g downward): g_eff = 0 → T = ∞ (pendulum stops oscillating — weightlessness)

(4) Lift moving with constant velocity (up or down): g_eff = g → T unchanged.

Pendulum with Horizontal Acceleration (from Aakash PDF — Oscillations and Waves JEE Main):

If the support is given horizontal acceleration a: effective gravity is the vector resultant of g and pseudo force (ma backward):

g_eff = √(g²+a²)

T = 2π√(L/g_eff) = 2π√(L/√(g²+a²)) [T decreases since g_eff>g]

The pendulum tilts at angle θ = tan⁻¹(a/g) from vertical in its new equilibrium position.

Compound Pendulum (from Aakash PDF — Oscillations and Waves JEE Main):

A rigid body of mass M, pivoted at a point at distance l from CM, radius of gyration k about CM:

T = 2π√(I/Mgl) where I = Mk² + Ml² = M(k²+l²) [parallel axis theorem]

T = 2π√((k²+l²)/gl) = 2π√(L_eff/g) where L_eff = (k²+l²)/l = l + k²/l (effective length)

T is minimum when dT/dl = 0 → l = k (pivot distance from CM = radius of gyration). T_min = 2π√(2k/g).

Other SHM Systems (from Aakash PDF — Oscillations and Waves JEE Main):

Liquid in a U-tube (total liquid length L): T = 2π√(L/2g)

Ball in a bowl of radius R: T = 2π√(R/g) [like pendulum of length R]

Floating cylinder (length l, submerged depth h): T = 2π√(h/g)

Download the Free PDF for Oscillations and Waves for all pendulum examples for JEE Main.

Damped and Forced Oscillations, Wave Equation, and Wave Speed

Why Wave Speed Formulas and the Progressive Wave Equation Are Essential Oscillations and Waves JEE Main Results

Damped Oscillations (from Aakash PDF — Oscillations and Waves JEE Main):

When a resistive force F_d = –bv acts on an oscillator:

x = Ae^(–bt/2m) cos(ω't + φ)

where ω' = √(ω₀² – b²/4m²) (angular frequency of damped oscillation, slightly less than ω₀).

Amplitude decays exponentially: A(t) = A₀e^(–bt/2m).

Energy decays: E(t) = E₀e^(–bt/m).

Quality factor: Q = mω₀/b = ω₀/2γ (where γ = b/2m = decay constant).

Critical damping: b = 2mω₀ → system returns to equilibrium fastest without oscillating.

Over damping: b > 2mω₀ → system returns slowly, no oscillation. Under damping: b < 2mω₀ → oscillates with decreasing amplitude.

Forced Oscillations and Resonance (from Aakash PDF — Oscillations and Waves JEE Main):

When external periodic force F = F₀cos(ωdt) is applied: the system eventually oscillates at the driver frequency ωd (not natural ω₀).

At resonance (ωd = ω₀): amplitude is maximum; energy transfer from driver to oscillator is maximum.

Wave Equation (from Aakash PDF — Oscillations and Waves JEE Main):

A progressive transverse wave traveling in +x direction:

y(x,t) = A sin(kx – ωt + φ)

where A = amplitude, k = 2π/λ (wave number), ω = 2πf = 2π/T, φ = initial phase.

Wave speed: v = ω/k = fλ = λ/T

Particle velocity (at fixed x): v_p = ∂y/∂t = –Aω cos(kx–ωt)

Particle acceleration: a_p = ∂²y/∂t² = –Aω²sin(kx–ωt) = –ω²y

Relation: v_p = –(ω/k)×(∂y/∂x) = –v×slope (particle velocity = –wave speed × slope of wave profile)

Wave Speed in Different Media (from Aakash PDF — Oscillations and Waves JEE Main):

Speed on a string/wire: v = √(T/μ) where T = tension (N), μ = mass per unit length (kg/m)

Speed in a solid rod: v = √(Y/ρ) where Y = Young's modulus, ρ = density

Speed in a liquid: v = √(B/ρ) where B = Bulk modulus

Speed of sound in a gas (adiabatic): v = √(γP/ρ) = √(γRT/M)

At 0°C: v = 331 m/s for air. Temperature dependence: v ≈ 331 + 0.61T m/s (T in °C)

Newton's formula (isothermal, γ=1): v = √(P/ρ) = 280 m/s [WRONG → Laplace's correction γ→5/3 for air→343 m/s]

Download the Free PDF for Oscillations and Waves for all wave speed examples for JEE Main.

Standing Waves — Open Pipe, Closed Pipe, Vibrating String

Why Open and Closed Pipe Harmonics Are the Most-Tested Standing Waves JEE Main Questions

Formation of Standing Waves (from Aakash PDF — Oscillations and Waves JEE Main):

When two identical waves traveling in opposite directions superpose: y₁ = Asin(kx–ωt) and y₂ = Asin(kx+ωt):

y = y₁+y₂ = 2A sin(kx) cos(ωt) → y = 2A cos(kx) sin(ωt) (or with sin(kx) depending on boundary)

Standing wave: amplitude 2A|sin(kx)| varies with position; all points oscillate in phase or antiphase.

Nodes (zero displacement always): kx = nπ → x = nλ/2 (spacing = λ/2 between successive nodes)

Antinodes (max displacement 2A): kx = (2n+1)π/2 → x = (2n+1)λ/4 (spacing = λ/2; antinodes midway between nodes)

Vibrating String Fixed at Both Ends (from Aakash PDF — Oscillations and Waves JEE Main):

Nodes at both ends. Condition: l = nλ/2 → λ = 2l/n.

fₙ = nv/2l (n = 1, 2, 3, …) [all harmonics present]

f₁ = v/2l (fundamental); f₂ = 2f₁ (second harmonic = first overtone); f₃ = 3f₁ etc.

v = √(T/μ) for string. So fₙ = (n/2l)√(T/μ)

Open Organ Pipe (open at both ends) (from Aakash PDF — Oscillations and Waves JEE Main):

Antinodes at both open ends. Condition: l = nλ/2 → same as string.

fₙ = nv/2l (n = 1, 2, 3, …) [ALL harmonics present]

f₁ = v/2l (fundamental). All harmonics (integer multiples of f₁) are present.

With end correction e (radius r, e ≈ 0.6r): effective length l_eff = l + 2e; fₙ = nv/(2(l+2e))

Closed Organ Pipe (closed at one end) (from Aakash PDF — Oscillations and Waves JEE Main):

Node at closed end; antinode at open end. Condition: l = (2n–1)λ/4 → λ = 4l/(2n–1).

fₙ = (2n–1)v/4l (n = 1, 2, 3, …) [ONLY ODD harmonics present]

f₁ = v/4l (fundamental); f₃ = 3v/4l (3rd harmonic = 1st overtone); f₅ = 5v/4l etc.

The fundamental of a closed pipe = half the fundamental of an open pipe of same length.

With end correction: l_eff = l + e (one end only); fₙ = (2n–1)v/(4(l+e)).

Download the Free PDF for Oscillations and Waves for all standing wave examples for JEE Main.

Beats, Doppler Effect, Sound Intensity — All Formulas

Why Doppler Effect All Four Cases and Beats Frequency Are the Most-Tested Waves JEE Main Questions

Beats (from Aakash PDF — Oscillations and Waves JEE Main):

When two sound waves of nearly equal frequencies f₁ and f₂ (|f₁–f₂| small) superpose, the resultant sound has alternating loud (constructive) and soft (destructive) regions = beats.

Beat frequency: f_beat = |f₁–f₂|

Beat period: T_beat = 1/f_beat = 1/|f₁–f₂|

Resultant amplitude: 2A cos(π(f₁–f₂)t). Envelope oscillates at frequency |f₁–f₂|.

Hearing limit for beats: human ear can detect beats up to ~10 beats/second; above this, sounds merge.

Applications: tuning musical instruments (tune until beat frequency→0); identifying unknown frequency (load/unload tuning fork near known frequency).

Doppler Effect (from Aakash PDF — Oscillations and Waves JEE Main):

The apparent change in frequency of sound when there is relative motion between source and observer.

General formula: f' = f × (v ± v_o)/(v ∓ v_s)

where v = speed of sound in medium, v_o = speed of observer, v_s = speed of source.

Sign convention: Use + for numerator (v+v_o) when observer moves TOWARD source; – when observer moves AWAY. Use – for denominator (v–v_s) when source moves TOWARD observer; + when source moves AWAY.

Four Standard Cases (from Aakash PDF — Oscillations and Waves JEE Main):

(1) Source moving toward stationary observer: f' = fv/(v–v_s) [f' > f]

(2) Source moving away from stationary observer: f' = fv/(v+v_s) [f' < f]

(3) Observer moving toward stationary source: f' = f(v+v_o)/v [f' > f]

(4) Observer moving away from stationary source: f' = f(v–v_o)/v [f' < f]

General: source AND observer both moving: f' = f(v+v_o)/(v–v_s) [when both moving toward each other]

Doppler shift: Δf = f'–f. For v_s << v: Δf ≈ ±fv_s/v (for source motion) and Δf ≈ ±fv_o/v (for observer motion).

Sound Intensity and Level (from Aakash PDF — Oscillations and Waves JEE Main):

Intensity I = Power/Area = P/4πr² for point source (inverse square law).

I ∝ A² (proportional to square of amplitude); I ∝ f²A² (for same medium).

Sound level: β = 10 log₁₀(I/I₀) dB where I₀ = 10⁻¹² W/m² (threshold of hearing).

Doubling I → β increases by 10 log₁₀(2) ≈ 3 dB. 10× I → β increases by 10 dB. Download the Free PDF for Oscillations and Waves for all Doppler and beats examples for JEE Main.

Download Free PDF — Oscillations and Waves JEE Main Formula Sheet

All Oscillations and Waves formulas from the Aakash Rapid Revision PDF: SHM defining equation a=–ω²x; x=Asin(ωt+φ); v=ω√(A²–x²); v_max=Aω; a=–ω²x; a_max=Aω²; T=2π/ω; f=ω/2π; KE=½mω²(A²–x²); PE=½mω²x²; E=½kA²=½mω²A²; KE=PE at x=±A/√2; avg KE=avg PE=E/2; spring-mass T=2π√(m/k); series 1/k_eff=1/k₁+1/k₂; parallel k_eff=k₁+k₂; two-mass μ=m₁m₂/(m₁+m₂) T=2π√(μ/k); spring cut into n parts k'=nk; simple pendulum T=2π√(L/g); seconds pendulum T=2s L≈1m; lift up T=2π√(L/(g+a)); lift down T=2π√(L/(g–a)); free fall T→∞; horizontal a g_eff=√(g²+a²); compound T=2π√(I/Mgl) min at l=k; U-tube T=2π√(L/2g); damped x=Ae^(–bt/2m)cos(ω't) ω'=√(ω₀²–b²/4m²); y=Asin(kx–ωt); k=2π/λ; v=ω/k=fλ; v_p=∂y/∂t; string v=√(T/μ); solid v=√(Y/ρ); liquid v=√(B/ρ); gas v=√(γP/ρ)=√(γRT/M); sound at 0°C 331 m/s; v=331+0.61T; standing y=2Asin(kx)cos(ωt); nodes nλ/2; antinodes (2n+1)λ/4; open pipe fₙ=nv/2l (all harmonics); closed pipe fₙ=(2n–1)v/4l (odd only); string fₙ=nv/2l=(n/2l)√(T/μ); end correction e≈0.6r; beats f_beat=|f₁–f₂|; Doppler f'=f(v+v_o)/(v–v_s); all 4 cases; I=P/4πr²; β=10log(I/I₀) dB.

Why Oscillations and Waves Is the Highest-Frequency JEE Main Chapter

SHM energy formulas KE=½mω²(A²–x²), PE=½mω²x², and E=½kA² are the most-tested Oscillations results in JEE Main. The total energy is constant and equals ½kA²; KE is maximum at equilibrium (x=0), PE is maximum at extremes (x=±A). The crossover point where KE=PE occurs at x=±A/√2. These relations are tested both directly ("find PE when v=v_max/2") and graphically ("which graph shows KE vs x for SHM"). The key insight: both KE and PE oscillate with double the frequency of the displacement (period = T/2).

Open vs closed pipe distinction (fₙ=nv/2l vs fₙ=(2n–1)v/4l) is tested in every few JEE Main sessions. The critical difference: open pipe has ALL harmonics (n=1,2,3...); closed pipe has ONLY ODD harmonics (n=1,3,5...). The fundamental of a closed pipe (v/4l) equals half the fundamental of an open pipe of the same length (v/2l). This means a closed pipe produces a lower fundamental frequency — it "sounds deeper" for the same physical length.

Doppler effect formula f'=f(v±v_o)/(v∓v_s) with correct sign convention is the most calculation-intensive Waves JEE Main question. The sign rule: numerator +v_o when observer moves toward source (frequency increases); denominator –v_s when source moves toward observer (frequency increases). Both toward each other → f'=f(v+v_o)/(v–v_s) (maximum increase). Both moving apart → f'=f(v–v_o)/(v+v_s) (maximum decrease). Download the Free PDF for Oscillations and Waves to have all formulas ready.

Who Should Use This Oscillations and Waves Formula Sheet?

Learning Outcomes After Completing Oscillations and Waves

After working through Oscillations and Waves using this formula sheet, a student should confidently accomplish: On SHM: identify SHM from a=–ω²x condition; write x=Asin(ωt+φ) with correct initial conditions; compute v=ω√(A²–x²); find v_max=Aω and a_max=Aω²; apply KE=½mω²(A²–x²), PE=½mω²x², E=½kA²; state KE=PE at x=A/√2; compute average KE and PE.

On spring systems: compute T=2π√(m/k) for spring-mass; apply series (1/k=1/k₁+1/k₂) and parallel (k=k₁+k₂) formulas; use reduced mass for two-mass system (μ=m₁m₂/(m₁+m₂), T=2π√(μ/k)); find k after spring cutting (k'=nk for n parts).

On pendulums: apply T=2π√(L/g); identify seconds pendulum (T=2s, L≈1m); modify T for lift (g→g±a); compute g_eff=√(g²+a²) for horizontal acceleration; apply compound pendulum T=2π√(I/Mgl) and find minimum T at l=k.

On waves: write y=Asin(kx–ωt); compute v=ω/k=fλ; apply v=√(T/μ) for string; apply v=√(γRT/M) for sound; use v=331+0.61T for air. On standing waves: write harmonic frequencies for open pipe (all harmonics nv/2l) and closed pipe (odd harmonics (2n–1)v/4l); apply fₙ=(n/2l)√(T/μ) for string; apply end correction. On beats and Doppler: compute f_beat=|f₁–f₂|; apply Doppler f'=f(v±v_o)/(v∓v_s) for all four cases correctly. Download the Free PDF for Oscillations and Waves to test all outcomes before your JEE Main exam.

Get the Free PDF for Oscillations and Waves — JEE Main Quick Revision

The Aakash Rapid Revision & Formula Bank PDF for Oscillations and Waves contains all SHM formulas, spring combination results, pendulum variations, damped oscillation equation, wave equation and speed formulas, complete standing wave harmonics for open/closed pipes and string, beats frequency formula, and Doppler effect all cases in one structured JEE Main physics reference.

Conclusion — Oscillations and Waves: The Periodic Motion Language of JEE Main Physics

Oscillations and Waves establishes the mathematical framework of periodic motion and wave propagation — two concepts that permeate all of physics. SHM is the prototype of oscillatory behaviour: its defining equation a=–ω²x generates sinusoidal solutions that describe everything from spring-mass systems to LC circuits to quantum mechanical wavefunctions. Wave motion is SHM propagating through a medium, carrying energy without transporting matter.

The five most JEE Main-tested results: (1) E=½kA²=½mω²A² (total SHM energy, constant); (2) v=ω√(A²–x²) (speed at displacement x); (3) T=2π√(L/g) with all lift/acceleration variants; (4) Open pipe fₙ=nv/2l (all harmonics) vs closed pipe fₙ=(2n–1)v/4l (odd only); (5) Doppler f'=f(v+v_o)/(v–v_s) with sign convention. Use this page and the Free PDF Download for Oscillations and Waves as your complete JEE Main revision foundation.

Frequently Asked Questions — Oscillations and Waves Formulas

Related Formula Sheets — JEE Main Physics

This is a free educational resource for JEE Main 2025–2026, JEE Advanced, NEET, Class 11 CBSE, and BITSAT Physics preparation. The PDF formula sheet for Oscillations and Waves is sourced from Aakash's Rapid Revision & Formula Bank. All Oscillations and Waves formulas and content are for academic reference only.