📐 Formula Bank

JEENEET

Every formula for Waves — with dimensional analysis, units, conditions, and exam relevance. Searchable. Not just a dump — each formula comes with the "when to use" context.

All Formulas Wave Basics Sound Speed Standing Waves Pipes & Strings Beats Doppler Intensity

Wave Speed Relation

v = fλ = λ/T = ω/k

[M⁰L¹T⁻¹]

v = speed (m/s), f = frequency (Hz), λ = wavelength (m), T = time period (s), ω = angular frequency (rad/s), k = wave number (rad/m).

CBSENEETJEE Main

Wave Number

k = 2π/λ

[M⁰L⁻¹T⁰]

k is the spatial frequency. Analogue of ω for time. Units: rad/m.

JEE MainJEE Adv

Angular Frequency

ω = 2πf = 2π/T

[M⁰L⁰T⁻¹]

ω in rad/s. Temporal frequency. Note: ω is NOT the same as f. ω = 2πf.

CBSENEET

Progressive Wave Equation

y(x,t) = A sin(kx − ωt + φ)

[M⁰L¹T⁰]

Displacement y at position x, time t. +kx−ωt: wave in +x direction. −kx−ωt: wave in −x direction. φ = initial phase.

JEE MainJEE Adv

Particle Velocity (max)

v_max = Aω = A(2πf)

[M⁰L¹T⁻¹]

Maximum transverse velocity of a medium particle. Not the wave speed. This is a classic confusion in JEE.

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Speed of Sound in Gas (Newton)

v = √(P/ρ)

[M⁰L¹T⁻¹]

Newton's formula (isothermal). Gives ~280 m/s for air at STP. Wrong! Because compression/rarefaction is adiabatic, not isothermal.

CBSE

Speed of Sound in Gas (Laplace)

v = √(γP/ρ) = √(γRT/M)

[M⁰L¹T⁻¹]

Laplace's correction: B = γP (adiabatic). For air (γ=1.4, T=300K): v ≈ 347 m/s. v ∝ √T, v ∝ 1/√M. Doesn't depend on pressure at constant T.

CBSENEETJEE Main

Speed of Sound in Solid

v = √(Y/ρ)

[M⁰L¹T⁻¹]

Y = Young's modulus, ρ = density. Fastest in solids. v_solid > v_liquid > v_gas (generally).

CBSENEET

Speed of Sound — Temperature Dependence

v_t = v₀ (1 + t/546) ≈ 332 + 0.6t m/s

[M⁰L¹T⁻¹]

t in °C. For every 1°C rise, v increases by ~0.6 m/s. At 0°C: v₀ = 332 m/s. Exact: v ∝ √T (Kelvin).

CBSENEET

Standing Wave Equation

y = 2A sin(kx) cos(ωt)

[M⁰L¹T⁰]

Formed by superposition of two identical waves in opposite directions. Amplitude = 2A sin(kx) — position dependent, not constant. NO energy transport.

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Node Positions

x = nλ/2 (n = 0, 1, 2...)

[L]

Where sin(kx) = 0. Distance between consecutive nodes = λ/2.

CBSENEET

Antinode Positions

x = (2n+1)λ/4 (n = 0, 1, 2...)

[L]

Where |sin(kx)| = 1. Max amplitude = 2A. Node-to-antinode distance = λ/4.

CBSENEET

String — Frequency of nth Harmonic

fₙ = (n/2L) √(T/μ)

[M⁰L⁰T⁻¹]

L = length, T = tension, μ = mass per unit length. n = 1: fundamental. All harmonics present. f₁:f₂:f₃ = 1:2:3...

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Open Organ Pipe

fₙ = nv/(2L) n = 1,2,3...

[T⁻¹]

Both ends open: antinodes at both ends. All harmonics present. Same series as strings. Ratio: 1:2:3:4:...

CBSENEETJEE Main

Closed Organ Pipe

fₙ = (2n−1)v/(4L) n = 1,2,3...

[T⁻¹]

One end closed: node at closed, antinode at open. Only ODD harmonics. Ratio: 1:3:5:7... Fundamental = v/4L (half of open pipe).

CBSENEETJEE Main

End Correction

e ≈ 0.6r (one end) 2e ≈ 1.2r (both ends)

[L]

r = radius of pipe. Effective length = L + e (closed) or L + 2e (open). Eliminates if two resonance points are used: v = 2f(L₂−L₁).

CBSE

Beat Frequency

f_beat = |f₁ − f₂|

[T⁻¹]

Number of beats per second. Human ear perceives beats for |f₁−f₂| ≤ 10 Hz. Beats = waxing & waning of sound intensity.

CBSENEETJEE Main

Resultant Amplitude (Beats)

y = 2A cos(π|f₁−f₂|t) · sin(π(f₁+f₂)t)

[L]

Average frequency: (f₁+f₂)/2. Beat (amplitude variation) frequency: |f₁−f₂|. Intensity completes one cycle in 1/|f₁−f₂| seconds.

JEE Adv

Doppler Effect (General)

f' = f₀ · (v ± v_o) / (v ∓ v_s)

[T⁻¹]

v = speed of sound, v_o = observer speed, v_s = source speed. + numerator when observer moves toward source. − denominator when source moves toward observer. Use sign that gives f increase when approaching.

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Doppler — Both Moving Same Direction

f' = f₀ · (v − v_o) / (v − v_s) [source & obs in same dir]

[T⁻¹]

If both move in same direction (source chasing observer or vice versa), the shifts partially cancel. If v_s = v_o → no Doppler shift (no relative motion).

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Intensity of Sound Wave

I = 2π²f²A²ρv

[M¹L⁰T⁻³]

I ∝ A² ∝ f². Also I = P/4πr² (point source). Units: W/m². I₀ = 10⁻¹² W/m² (threshold of hearing).

CBSENEET

Sound Level (Decibels)

β = 10 log₁₀(I/I₀) dB

dimensionless

I₀ = 10⁻¹² W/m². Every 10 dB: I multiplies by 10. Threshold of pain ≈ 120 dB. Normal conversation ≈ 60 dB.

CBSENEET

Intensity — Interference

I = I₁ + I₂ + 2√(I₁I₂) cosφ

[MT⁻³]

I_max = (√I₁+√I₂)² when φ=2nπ. I_min = (√I₁−√I₂)² when φ=(2n+1)π. Ratio: I_max/I_min = ((√I₁+√I₂)/(√I₁−√I₂))².

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Dimensional Analysis

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Exam Insight — Dimensional Analysis Traps

JEE asks: "Which formula is dimensionally correct?" Learn to check quickly. Key: [frequency] = T⁻¹, [wave number k] = L⁻¹, [tension] = MLT⁻², [μ] = ML⁻¹, [intensity] = MT⁻³.

Quantity	Symbol	Dimensions	SI Unit	Note
Frequency	f	[T⁻¹]	Hz	f = 1/T
Angular Frequency	ω	[T⁻¹]	rad/s	ω = 2πf
Wave Number	k	[L⁻¹]	rad/m	k = 2π/λ
Wave Speed	v	[LT⁻¹]	m/s	v = ω/k = fλ
Tension in String	T	[MLT⁻²]	N	Same as force
Linear Mass Density	μ	[ML⁻¹]	kg/m	μ = m/L
Bulk Modulus	B	[ML⁻¹T⁻²]	Pa	B = γP for gas
Intensity	I	[MT⁻³]	W/m²	Power per area
Amplitude	A	[L]	m	Max displacement
Phase Difference	φ	dimensionless	rad	φ = (2π/λ)Δx
Path Difference	Δx	[L]	m	Δx = φλ/2π
Decibel Level	β	dimensionless	dB	Logarithmic scale
Young's Modulus	Y	[ML⁻¹T⁻²]	Pa	For solid rod
Particle Velocity	v_p	[LT⁻¹]	m/s	v_p = ∂y/∂t ≠ wave speed

Interactive Calculators

🚗 Doppler Effect Calculator

Speed of sound v (m/s)

Source frequency f₀ (Hz)

Source speed vs (m/s)

Source direction

Observer speed vo (m/s)

Observer direction

Observed Frequency f'

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Strategy Tip

Use this calculator to verify answers. The formula: f' = f₀(v±v_o)/(v∓v_s). Remember — numerator handles OBSERVER, denominator handles SOURCE. "Toward" always increases f'.

🎵 Beat Frequency Calculator

Frequency f₁ (Hz)

Frequency f₂ (Hz)

Beat Frequency

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❌

Common Mistake Alert

If a fork gives 4 beats with a 256 Hz fork, the unknown is EITHER 260 Hz or 252 Hz. Loading with wax decreases frequency. If beats increase → unknown was 252 Hz. If beats decrease → unknown was 260 Hz.

🎸 Vibrating String Calculator

Length L (m)

Tension T (N)

Linear density μ (kg/m)

Harmonic n

Frequency fₙ

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🪗 Organ Pipe Calculator

Pipe length L (m)

Speed of sound v (m/s)

Harmonic n

Pipe type

Frequency fₙ

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Exam Insight

A closed pipe of length L has the same fundamental frequency as an open pipe of length 2L. Also: an open pipe produces RICHER sound (more harmonics) than a closed pipe. This is tested in CBSE theory questions.

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