⚡ Quick Revision

CBSENEETJEE

Everything for Waves in one place. Formula dump, flashcards, memory tricks, one-line summaries. The night-before-exam page.

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How to Use This

Read each point once. If you can't recall the formula or concept immediately → go to the relevant page. This page is for reinforcement, not learning.

🌊 Wave Basics

v = fλ = ω/k. Wave speed = medium property, not amplitude or frequency.

y = A sin(kx−ωt): +kx−ωt → +x direction. Speed = ω/k.

Particle velocity = −v_wave × (∂y/∂x). At crest: v_particle = 0.

Sound = longitudinal. Light/string = transverse. Cannot be mixed up.

🔊 Speed of Sound

Newton: v=√(P/ρ) = 280 m/s ❌ Wrong. Laplace: v=√(γP/ρ) = 332 m/s ✓

v ∝ √T (K), v ∝ 1/√M, v independent of pressure at const T.

Humid air: higher v (water vapour M=18 < air M≈29).

v_solid > v_liquid > v_gas. Steel ≈ 5000, Water ≈ 1500, Air ≈ 340 m/s.

🎸 Standing Waves

y = 2A sin(kx) cos(ωt). Nodes: sin(kx)=0. Antinodes: sin(kx)=±1.

Node-to-node = λ/2. Antinode-to-antinode = λ/2. Node-to-antinode = λ/4.

Open pipe: f = nv/2L (all harmonics). Closed pipe: f = (2n−1)v/4L (odd only).

n loops → n+1 nodes, n antinodes (string). Overtone = harmonic − 1.

🔀 Superposition

Constructive: Δx = nλ, φ = 2nπ. A_max = A₁+A₂.

Destructive: Δx = (2n−1)λ/2. A_min = |A₁−A₂|.

I_max = (√I₁+√I₂)². I_min = (√I₁−√I₂)². Work with amplitudes!

Energy is redistributed, NOT destroyed. Total energy conserved.

🎵 Beats & Doppler

Beats = |f₁−f₂|. Wax → f ↓. Loading → f ↓. Filing → f ↑.

Doppler: f' = f₀(v±v_o)/(v∓v_s). Toward → upper signs → f increases.

Doppler NOT symmetric: v_s=v_o does NOT give same f'. Source moving ≠ observer moving.

Wall reflection: two Doppler shifts. f_final = f₀(v+v_s)/(v−v_s) if source moves toward wall.

📊 Intensity & Sound

I = 2π²f²A²ρv. I ∝ A², I ∝ f², I ∝ 1/r² (point source).

β = 10 log(I/I₀). I₀=10⁻¹² W/m². +10 dB → 10× I. +3 dB → 2× I.

Displacement node = Pressure antinode (sound standing wave).

Resonance tube: v = 2f(L₂−L₁). End correction = e ≈ 0.6r.

⚡ Open Pipe vs Closed Pipe — Side by Side

Property	Open Pipe	Closed Pipe
Boundaries	Both antinodes	Node (closed) + Antinode (open)
Fundamental f₁	v/2L	v/4L (half of open!)
Harmonics present	All: 1,2,3,4...	Odd only: 1,3,5,7...
f ratio	1:2:3:4...	1:3:5:7...
Sound quality	Richer (all harmonics)	Hollow (fewer harmonics)
1st overtone	2nd harmonic = 2f₁	3rd harmonic = 3f₁
nth overtone	(n+1)th harmonic	(2n+1)th harmonic

Wave Speed

v = fλ = ω/k = λ/T

Medium property — independent of amplitude

Wave Equation

y = A sin(kx − ωt)

+kx−ωt: +x direction | −kx−ωt: −x direction

Wave Number

k = 2π/λ [rad/m]

Spatial frequency analogue of ω

Sound in Gas (Laplace)

v = √(γP/ρ) = √(γRT/M)

v ∝ √T(K), ∝ 1/√M, independent of P

Sound in String

v = √(T/μ)

T=tension(N), μ=mass/length(kg/m)

Standing Wave

y = 2A sin(kx) cos(ωt)

Amplitude = 2A sin(kx), varies with position

Open Pipe

fₙ = nv / 2L

All harmonics: n = 1,2,3... Ratio 1:2:3:4

Closed Pipe

fₙ = (2n−1)v / 4L

Odd harmonics only. Ratio 1:3:5:7

String Harmonics

fₙ = (n/2L)√(T/μ)

All harmonics. Fundamental = v/2L

Beat Frequency

f_beat = |f₁ − f₂|

Valid for |f₁−f₂| ≤ 10 Hz perceptible

Doppler Formula

f' = f₀(v±v_o)/(v∓v_s)

Toward → use +/− to get f' > f₀

Resultant Amplitude

A = √(A₁²+A₂²+2A₁A₂cosφ)

φ=0: A=A₁+A₂. φ=π: A=|A₁−A₂|

Intensity (Max/Min)

I_max = (√I₁+√I₂)²

I_min = (√I₁−√I₂)². Work with amplitudes!

Sound Level

β = 10 log(I/I₀) dB

I₀ = 10⁻¹² W/m². +10dB → 10× intensity

Resonance Tube

v = 2f(L₂ − L₁)

Eliminates end correction. L₂ > L₁.

End Correction

e ≈ 0.6r

r = pipe radius. L_eff = L + e (closed end)

Path→Phase

φ = (2π/λ) × Δx

Path difference to phase difference conversion

Intensity & Wave

I = 2π²f²A²ρv

I ∝ A² ∝ f². Also I = P/4πr² for sphere

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Thinking Step — How to Use Flashcards

Click card to flip. If you get it right: press Next. If wrong: review the concept, then try again. Spend 15 minutes on these the night before the exam for maximum retention boost.

Click to flip

What is the speed of sound in terms of temperature?
v ∝ ?

v ∝ √T (Kelvin)
v = √(γRT/M)
Double T → v increases by √2

Click to flip

For a closed organ pipe of length L, what are the allowed frequencies?

fₙ = (2n−1)v/4L
n = 1,2,3...
Odd harmonics only: f, 3f, 5f...

Click to flip

Two coherent waves with intensities I₁ and I₂.
I_max = ?

I_max = (√I₁ + √I₂)²
I_min = (√I₁ − √I₂)²
Always work with AMPLITUDES first!

Click to flip

What is the Doppler formula?
When does f' > f₀?

f' = f₀(v ± v_o)/(v ∓ v_s)
f' > f₀ when source & observer approach each other
"Toward = Top+ Bottom−"

Click to flip

In a sound standing wave:
Displacement NODE corresponds to...?

Pressure ANTINODE
Displacement ANTINODE = Pressure NODE
They are 90° (π/2) out of phase spatially

Click to flip

What is the standing wave equation?
Where are the nodes?

y = 2A sin(kx) cos(ωt)
Nodes: sin(kx) = 0 → x = nλ/2
Antinodes: sin(kx) = ±1 → x = (2n+1)λ/4

Click to flip

Why did Newton's formula give wrong speed of sound?

Newton assumed isothermal (B=P)
Actually: compression is ADIABATIC (too fast for heat exchange)
Laplace: B = γP → v = √(γP/ρ) ✓

Click to flip

A fork gives 5 beats with 440 Hz. After loading with wax, beats = 3. Original frequency of fork?

f = 440 ± 5 = 445 or 435 Hz
Wax ↓ f. Beats decreased → fork was above 440 Hz
f = 445 Hz ✓
(After wax: 442 Hz → 3 beats with 439? Verify!)

Click to flip

What is the 3rd overtone of an open pipe with fundamental f₁?

3rd overtone = 4th harmonic = 4f₁
Pattern: nth overtone = (n+1)th harmonic
For open pipe only. Closed pipe is different!

Click to flip

How does decibel level change if intensity doubles?

Δβ = 10 log(2) ≈ 3 dB
Doubling intensity → +3 dB
10× intensity → +10 dB
100× intensity → +20 dB

1 / 10

Click card to reveal answer

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Closed Pipe — Only ODD Harmonics

Think: closed pipe is like a person with one ear covered — they can only hear odd beats in a conversation.

CLOSED → ODD → 1,3,5,7...

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Doppler — Sign Convention

"TITO" — Toward = Increases (Top + Opposite).
Observer TOward → Top sign + (numerator goes up, f' up). Source TOward → bottom − (denominator goes down, f' up).

T + T-: f'↑ when TOWARD

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Overtone vs Harmonic

Overtone = Harmonic − 1. Always. Like floors in a building: Ground floor = 1st floor = fundamental. 1st overtone = 2nd harmonic.

Overtone_n = Harmonic_(n+1)

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Beats — Wax Logic

WAX = Weight Added → X down (frequency decreases). If beats DECREASE after wax → unknown was ABOVE reference. If beats INCREASE after wax → unknown was BELOW.

Beat↓ after wax → unknown was ABOVE

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Speed of Sound — Temperature

v ∝ √T. If T is in Celsius, convert to Kelvin first! At 0°C = 332 m/s. Every degree Celsius adds ~0.6 m/s. At room temp (27°C=300K): v ≈ 347 m/s.

v_t ≈ 332 + 0.6t m/s (t in °C)

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Node vs Antinode in Sound

At the WALL (displacement node): medium is maximally squeezed/stretched → maximum pressure → pressure ANTINODE. At open end (displacement antinode): medium expands freely → zero pressure change → pressure NODE.

D-Node = P-Antinode | D-Antinode = P-Node

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Intensity Ratio — The 4:1 Pattern

Equal intensities → I_max/I_min = ∞ (if equal) or (A₁+A₂)²/(A₁−A₂)². For intensities I and 4I: A₁=1, A₂=2. Ratio = (3)²/(1)² = 9:1. For I and 9I: ratio = (1+3)²/(3-1)² = 16/4 = 4:1.

Always: work with √I = amplitude, then square ratio

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Newton vs Laplace — Quick Memory

NEWTON was Isothermal (heat escapes — like breathing slowly). LAPLACE was Adiabatic (heat stays — like breathing fast). Sound vibrations are FAST → heat doesn't escape → Adiabatic → Laplace is correct.

Fast = Adiabatic = γP → Laplace ✓

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