Waves & Sound — vidhyapath 2027

Concept Core

Wave motion, sound characteristics, beats, and resonance — the complete waves framework.

Wave Equation & Fundamental Parameters

A progressive wave: y = A sin(ωt − kx) where A = amplitude, ω = angular frequency, k = wave number.

v = fλ = ω/k f = 1/T = ω/2π k = 2π/λ ω = 2πf Speed of sound in a medium: v = √(B/ρ) (B=bulk modulus, ρ=density) Speed of sound in air (Newton-Laplace): v = √(γP/ρ) ≈ 343 m/s at 20°C

Speed of Sound — Temperature Dependence

v = v₀ + 0.6t (approximate; t in °C; v₀ = 332 m/s at 0°C) v ∝ √T (T in Kelvin) → v₁/v₂ = √(T₁/T₂)

Sound travels faster in denser solids (not denser gases — in gases, v ∝ 1/√ρ). v in solid > v in liquid > v in gas.

Beats

Beats occur when two sounds of slightly different frequencies interfere. The beat frequency = difference in frequencies:

f_beat = |f₁ − f₂|

Beat frequency = number of loud-soft cycles per second. Maximum (constructive interference) when path difference = nλ. Minimum (destructive) at half-wavelength path difference.

Doppler Effect

Apparent frequency when source/observer move relative to each other:

f' = f × (v + v₀)/(v − vₛ)

v = speed of sound; v₀ = observer speed (+ toward source); vₛ = source speed (+ toward observer). Apply signs carefully: approach → higher frequency; recession → lower frequency.

Resonance in Pipes (Organ Pipes)

Open pipe harmonics: fₙ = nv/(2L), n = 1,2,3,... (all harmonics) Closed pipe harmonics: fₙ = nv/(4L), n = 1,3,5,... (odd harmonics only) Fundamental (open): v/2L Fundamental (closed): v/4L

Open pipe has all harmonics; closed pipe has only odd harmonics.

Intensity & Loudness

Intensity I = Power/Area = P/(4πr²) (for point source) I ∝ A² (intensity ∝ square of amplitude) Loudness (dB) = 10 log₁₀(I/I₀) where I₀ = 10⁻¹² W/m²

Every 10 dB increase corresponds to 10× intensity increase. 20 dB louder → 100× more intense.

Formula Vault

All wave and sound formulas for EAPCET.

Wave Speed

v = fλ = ω/k

f in Hz; λ in m; v in m/s

Speed of Sound in Air

v = 332 + 0.6t m/s

t = temperature in °C

Speed Ratio (temp.)

v₁/v₂ = √(T₁/T₂)

T in Kelvin

Beat Frequency

f_beat = |f₁ − f₂|

Beats per second

Doppler Effect

f' = f·(v±v₀)/(v∓vₛ)

Upper signs: motion toward each other

Open Pipe (fundamental)

f₁ = v/(2L)

All harmonics present

Closed Pipe (fundamental)

f₁ = v/(4L)

Only odd harmonics

Intensity from Point Source

I = P/(4πr²)

Inverse square law

Intensity vs Amplitude

I ∝ A²

Doubling A → 4× intensity

Loudness (dB)

β = 10 log₁₀(I/I₀)

I₀ = 10⁻¹² W/m² (threshold)

Worked Examples

5 problems — wave speed, Doppler, beats, organ pipes, and intensity.

EasyFind the frequency of a wave with v = 340 m/s and λ = 0.5 m▾

A sound wave has speed 340 m/s and wavelength 0.5 m. Find its frequency.

v = fλ → f = v/λ = 340/0.5 = 680 Hz

✓ f = 680 Hz

EasyTwo tuning forks give 5 beats/s. One has f = 256 Hz. Find the other's frequency.▾

Two tuning forks produce 5 beats per second. One has frequency 256 Hz. Find the other.

f_beat = |f₁ − f₂| = 5

f₂ = 256 ± 5 = 261 Hz or 251 Hz

(Additional experiment needed to determine which one)

✓ f₂ = 261 Hz or 251 Hz

MediumFind fundamental frequency of a closed pipe of length 0.5 m (v = 340 m/s)▾

Calculate the fundamental frequency of a closed organ pipe of length 0.5 m. Speed of sound = 340 m/s.

Closed pipe fundamental: f₁ = v/(4L) = 340/(4 × 0.5) = 340/2 = 170 Hz

✓ Fundamental frequency = 170 Hz

EAPCET LevelTrain approaches at 20 m/s sounding 500 Hz horn. Find apparent frequency (v=340 m/s).▾

A train moving at 20 m/s sounds a horn of frequency 500 Hz. Find the apparent frequency heard by a stationary observer as the train approaches. (v_sound = 340 m/s)

Observer is stationary (v₀ = 0), source moves toward observer (vₛ = 20 m/s).

Doppler: f' = f × (v + v₀)/(v − vₛ) = 500 × (340 + 0)/(340 − 20)

= 500 × 340/320 = 500 × 1.0625 = 531.25 Hz

✓ Apparent frequency = 531.25 Hz (higher as source approaches)

Trap QuestionSound travels faster in denser media — True or False?▾

A student states: 'Sound travels faster in air than in water because air is less dense, so particles vibrate more freely.' Evaluate.

The trap: For sound in solids and liquids, v = √(B/ρ). Both bulk modulus B and density ρ affect speed.

While denser materials do resist motion (larger ρ reduces v), denser materials are also usually stiffer (larger B increases v).

The net effect: v in solid (e.g., steel ≈ 5000 m/s) >> v in water (≈ 1500 m/s) >> v in air (≈ 340 m/s).

For gases specifically: v ∝ 1/√ρ (at same temperature). But comparing different states of matter requires considering both B and ρ.

✓ False (in general) — sound is fastest in solids, then liquids, then gases; stiffness (bulk modulus) also matters

Mistake DNA

4 waves errors from EAPCET distractor analysis.

📏

Doppler: Wrong Signs for Observer and Source Motion

Observer moving toward source: +v₀. Observer moving away: −v₀. Source moving toward observer: −vₛ. Source moving away: +vₛ.

❌ Wrong

Source moves toward observer: f' = f(v+vₛ)/(v) ✗ (source motion: wrong sign)

✓ Correct

f' = f(v+v₀)/(v−vₛ) ✓ Source toward: subtract vₛ ✓ (denominator shrinks, f' rises)

Memory device: motion of approach → frequency increases. In formula: observer approach adds to numerator; source approach subtracts from denominator. Both make f' larger than f.

🎵

Closed Pipe Has All Harmonics

Closed pipes (one end closed) support only odd harmonics: 1st, 3rd, 5th, ... Open pipes support all harmonics.

❌ Wrong

Closed pipe harmonics: f₁, 2f₁, 3f₁, 4f₁... ✗ (that's open pipe!)

✓ Correct

Closed: only odd harmonics ✓ f₁, 3f₁, 5f₁ ... ✓ fₙ = nv/(4L), n = 1,3,5...

Closed pipe: node at closed end, antinode at open end. This forces only odd-harmonic standing waves. Open pipe: antinodes at both ends, all harmonics fit.

🔊

Beat Frequency Can Be Negative

Beat frequency is the absolute value of the difference. It is always positive.

❌ Wrong

f_beats = f₁ − f₂ = −5 Hz ✗ (negative frequency makes no sense)

✓ Correct

f_beats = |f₁−f₂| ✓ = 5 Hz always positive ✓ Beats/second is a count

Beat frequency represents the number of amplitude maxima per second — it's a count, always non-negative. If f₁ < f₂, the beat frequency is still |f₁ − f₂|.

🌡️

Speed of Sound Depends Only on Temperature (Not Pressure at Constant T)

At constant temperature, speed of sound in an ideal gas is independent of pressure. Pressure cancels in v = √(γP/ρ).

❌ Wrong

Increasing pressure increases speed of sound ✗ (at constant temperature, pressure doesn't affect v)

✓ Correct

v = √(γRT/M) ✓ Only temperature matters ✓ At constant T, v constant regardless of P

In v = √(γP/ρ): when P increases, ρ also increases (same T, same volume). The ratio P/ρ = RT/M is constant at fixed T. So v depends only on T.

Chapter Intelligence

Waves connects to SHM (wave = propagating SHM), optics (wave optics), and modern physics (wave-particle duality).

EAPCET Weightage (2019–2024)

Doppler effect

Beats and superposition

Organ pipes resonance

Wave speed v=fλ

Sound level in dB

High-Yield PYQ Patterns

Doppler: train/source approachingBeat frequency from two frequenciesFundamental of open vs closed pipeSpeed of sound at different TTwo frequencies giving same beatHarmonics of open/closed pipesIntensity at distance r

Exam Strategy

Doppler formula: f' = f(v+v₀)/(v−vₛ). Signs: + for motion toward, − for motion away. Approaching → higher frequency always.
Beats: f_beat = |f₁ − f₂|. Given one frequency and beat count, two answers are possible (f₂ = f₁ ± f_beat). Extra info (like tuning up/down) resolves ambiguity.
Organ pipes: Open pipe f = nv/2L (all harmonics). Closed pipe f = nv/4L, n = odd only. Closed pipe is half the frequency of equal-length open pipe.
At same temperature: heavier gas (greater M) → lower sound speed. v = √(γRT/M) — larger M in denominator.