📖 Chapter PN04 · NDA Class 11–12 Level🎯 NDA Level : High Priority
Sound is a mechanical longitudinal wave that requires a material medium to travel. This chapter covers everything from the basic nature of sound and its wave characteristics, to the fascinating phenomena of interference, beats, resonance, and the Doppler effect. NDA tests this chapter both conceptually and numerically — with strong emphasis on real-life applications familiar to soldiers (SONAR, echoes, Doppler in defence systems).
📌 What to expect in NDA (based on 2022–2025 pattern): (1) Nature of sound — longitudinal wave, medium requirement, compressions and rarefactions; (2) Wave characteristics — λ, f, T, v, amplitude and their relationships; (3) Speed of sound — in different media, effect of temperature and humidity; (4) Echo and reverberation — minimum distance for echo; (5) Beats — beat frequency, musical applications; (6) Resonance — conditions, column of air, applications; (7) Doppler effect — qualitative shift in frequency for moving source/observer.
Particles vibrate parallel to wave travel — compressions and rarefactions
Sound is produced by vibrating bodies and travels as a longitudinal mechanical wave — meaning particles of the medium vibrate parallel to the direction of wave propagation. It requires a material medium (solid, liquid, or gas) and cannot travel through vacuum.
📢 What Sound Needs
Vibrating source to produce it
Material medium to carry it
Receiver (ear/detector) to sense it
Cannot travel through vacuum (unlike light)
Fastest in solids, slowest in gases
🔌 Types of Sound Waves
Infrasound: f < 20 Hz (earthquake, elephants)
Audible: 20 Hz – 20,000 Hz (human range)
Ultrasound: f > 20,000 Hz (bats, SONAR, medical)
Pitch ↔ Frequency; Loudness ↔ Amplitude
Quality (timbre) ↔ waveform shape
🔸 Compressions & Rarefactions
Compression: region of high pressure & density
Rarefaction: region of low pressure & density
Wavelength = distance between successive compressions
One complete vibration = one compression + one rarefaction
Amplitude = max displacement from equilibrium
Fig. 1 — Longitudinal sound wave. Particles vibrate back and forth (parallel to wave direction). C = Compression (high density); R = Rarefaction (low density). One wavelength spans C–R–C.
⚠ NDA Exam Trap — Sound cannot travel through vacuum. Light is a transverse electromagnetic wave and travels through vacuum. Sound is a longitudinal mechanical wave and requires a medium. In space (vacuum), astronauts cannot hear each other directly — they must use radio waves (EM waves). Explosions in space are silent.
2. Wave Characteristics & Speed of Sound
2.1
Wavelength, Frequency, Time Period, Amplitude & Speed
The complete language of wave description — master these and all problems open up
Every wave — including sound — is completely described by four quantities: amplitude (intensity/loudness), frequency (pitch), wavelength (spatial period), and speed (determined by the medium, not the source). These are related by the fundamental wave equation.
⚡ Wave Characteristics — Core Formulae
Wavelength (λ): Distance between successive compressions (or rarefactions)
Unit: metre (m)
Frequency (f): Number of complete vibrations per second
Unit: Hertz (Hz) = cycles per second
Time Period (T): Time for one complete vibration
T = 1/f Unit: second (s)
Amplitude (A): Maximum displacement of a particle from equilibrium
Determines LOUDNESS (intensity ∝ A²)
Fundamental Wave Equation:
v = f × λ (Speed = Frequency × Wavelength)
Angular frequency: ω = 2πf = 2π/T (rad/s)
Wave number: k = 2π/λ (rad/m)
Key insight: The speed of sound is determined by the medium (its elasticity and density), not by the source's vibration. When a sound wave moves from one medium to another, its frequency stays constant but its wavelength changes (since v changes). This is a common NDA exam concept.
⚡ Speed of Sound — Values & Factors
Speed of sound in air (at 0°C): v₀ = 332 m/s
Speed of sound in air (at 20°C): v ≈ 343 m/s (≈ 344 m/s used in problems)
General rule — speed in different media:
Solids > Liquids > Gases (v_solid > v_liquid > v_gas)
Speed of sound in:
Steel: ≈ 5100 m/s
Water: ≈ 1500 m/s
Air: ≈ 343 m/s (at room temperature)
Effect of temperature on speed in air:
v_T = v₀ + 0.61T (T in °C) approximately
Or: v ∝ √T (T in Kelvin, exact)
Speed INCREASES with temperature
Effect of humidity:
Humid air → speed slightly INCREASES (water vapour lighter than dry air)
Newton's formula: v = √(P/ρ) (isothermal — gives 280 m/s, incorrect)
Laplace correction: v = √(γP/ρ) (adiabatic — gives 332 m/s, correct)
γ = 1.4 for air (ratio of specific heats)
Speed does NOT depend on: frequency, wavelength, amplitude, or loudness of the sound. Two sounds of different frequency from the same source travel at the same speed through the same medium.
📢 Echo & Reverberation
Echo: reflected sound heard distinctly after original ceases
Minimum distance for echo = v × t_persistence / 2
Persistence of hearing = 1/10th second (0.1 s)
Min. distance for echo in air = 343 × 0.1 / 2 ≈ 17.15 m
Reverberation: persistence of sound due to multiple reflections
Reverberation: desirable in concert halls (controlled); harmful in classrooms
📢 SONAR & Ultrasound
SONAR: Sound Navigation And Ranging (uses ultrasound)
Detects submarines, ocean depth, fish shoals
d = v × t / 2 (round-trip time)
Medical ultrasound: tissue imaging, foetal scanning
Bats: echolocation using ultrasound (f > 20 kHz)
Industrial: flaw detection in metals
Worked Example — Echo Distance
A man claps and hears his echo after 0.6 s. How far is the reflecting wall? (Speed of sound = 340 m/s)
Sound travels to wall and back: total distance = v × t = 340 × 0.6 = 204 m.
Distance to wall = 204 / 2 = 102 m.
📝 TOPIC-WISE PYQ
Wave Characteristics & Speed of Sound — NDA Pattern Questions
Q1. The speed of sound is maximum in which medium?
(a) Air at 0°C (b) Water (c) Steel (d) Vacuum
Answer: (c) Steel
Speed of sound: Solids > Liquids > Gases. Steel ≈ 5100 m/s, Water ≈ 1500 m/s, Air ≈ 343 m/s. Sound cannot travel through vacuum at all. Solids have highest elasticity-to-density ratio, so highest speed.
Q2. The frequency of a sound wave is 500 Hz and its speed is 340 m/s. The wavelength is:
(a) 0.68 m (b) 1.47 m (c) 68 m (d) 0.34 m
Answer: (a) 0.68 m
λ = v/f = 340/500 = 0.68 m. Using the fundamental wave equation v = fλ.
Q3. An echo is heard after 2 seconds. If the speed of sound is 340 m/s, the distance of the reflecting surface is:
(a) 170 m (b) 340 m (c) 680 m (d) 85 m
Answer: (b) 340 m
Total distance = v × t = 340 × 2 = 680 m. Distance to wall = 680/2 = 340 m. Sound must travel to the wall and return — hence divide by 2.
Q4. When a sound wave passes from air into water, which property remains unchanged?
(a) Wavelength (b) Speed (c) Frequency (d) Amplitude
Answer: (c) Frequency
When a wave moves from one medium to another, its frequency stays constant (set by the source). Speed changes (water faster than air), and since v = fλ, the wavelength also changes. Frequency is the only invariant.
🤔 TRICKY QUESTIONS
Sound Propagation — Watch Out!
T1. On a hot summer day, does sound travel faster or slower than on a cold winter day? Why?
Faster on a hot summer day.
v ∝ √T (where T is in Kelvin). Higher temperature → molecules move faster → they transmit the disturbance more quickly. Practical rule: v increases by 0.61 m/s for every 1°C rise. At 40°C: v ≈ 332 + 0.61×40 ≈ 356 m/s vs 332 m/s at 0°C. Same reason thunder is heard sooner on hot days.
T2. A soldier puts his ear to a steel rail and hears the sound of an approaching train through the rail before he hears it through air. Why?
Sound travels much faster in steel than in air.
v_steel ≈ 5100 m/s vs v_air ≈ 343 m/s — about 15× faster. So the sound through the rail arrives well before the airborne sound. This is why Native Americans and frontier scouts pressed their ear to the ground/rail to detect approaching horses or trains from great distances. SONAR uses the same principle — sound travels further and faster in water than air.
3. Superposition: Interference, Beats & Resonance
3.1
Principle of Superposition & Interference
When two waves meet — they add algebraically at every point
The principle of superposition states that when two or more waves overlap, the resultant displacement at any point is the algebraic sum of the individual displacements. This leads to constructive interference (waves reinforce) and destructive interference (waves cancel).
🔈 Constructive Interference
Two waves meet in phase (crest meets crest)
Path difference = nλ (n = 0, 1, 2, ...)
Resultant amplitude = A₁ + A₂ (maximum)
Produces loud sound (antinodes in standing waves)
Used in noise-cancelling headphones (destructive)
🔇 Destructive Interference
Two waves meet out of phase (crest meets trough)
Path difference = (n + ½)λ (n = 0, 1, 2, ...)
Resultant amplitude = |A₁ − A₂| (minimum)
If A₁ = A₂: complete cancellation (silence)
Nodes in standing waves — zero displacement
3.2
Beats
Periodic variation in loudness when two slightly different frequencies interfere
Beats are produced when two sound waves of slightly different frequencies travel in the same direction and superpose. The resulting sound alternately becomes loud and soft — the number of these variations per second is the beat frequency.
⚡ Beat Frequency Formula
Beat frequency: f_beat = |f₁ − f₂|
One beat per second = one loud-soft cycle per second
Beats are only audible when |f₁ − f₂| ≤ ~10 Hz (human ear limit)
If f₁ = 256 Hz and f₂ = 260 Hz:
f_beat = |256 − 260| = 4 Hz → 4 beats per second
To identify which frequency is higher:
If loading one fork (increases mass → decreases frequency):
Beats increase → that fork was the higher frequency one
Beats decrease → that fork was the lower frequency one
If loading reduces beats: that fork is approaching the other's freq.
Beats are used for tuning musical instruments — a musician tunes strings until beats disappear (zero beat frequency = same frequency). Piano tuners use this method.
Fig. 2 — Beat pattern. Blue and green: two sound waves of slightly different frequency. Orange envelope: the resulting amplitude modulation. Loud sound occurs where waves reinforce; quiet where they cancel.
💡 Beat frequency application — Tuning an instrument: A musician plays a note alongside a tuning fork (e.g. 440 Hz). If they hear 3 beats/second, the instrument is at either 437 Hz or 443 Hz. Tightening the string raises frequency — if beats reduce to 0, the original was 437 Hz; if beats increase, it was 443 Hz.
3.3
Resonance
When a system vibrates at its natural frequency — maximum amplitude transfer
Resonance occurs when the frequency of an external driving force matches the natural frequency of a system, causing it to vibrate with maximum amplitude. It is a special case of forced vibrations where energy transfer is most efficient.
📌 Conditions for Resonance
Driving frequency = natural frequency of system
Amplitude becomes maximum at resonance
Energy absorbed from driver is maximum
Phase difference between driver and system = 90°
Sharpness of resonance decreases with damping
📢 Examples of Resonance
Tuning a radio: LC circuit resonates at signal frequency
Soldiers break step on bridges: avoid resonance collapse
Glass breaks when opera singer hits natural frequency
Microwave oven: 2.45 GHz resonates water molecules
⚡ Standing Waves in Air Columns
Open pipe (open at both ends):
Harmonics: f_n = nv/(2L) n = 1, 2, 3, ...
Fundamental (1st harmonic): f₁ = v/2L
All harmonics present (1st, 2nd, 3rd ...)
Closed pipe (closed at one end):
Harmonics: f_n = nv/(4L) n = 1, 3, 5, ... (ODD only)
Fundamental: f₁ = v/4L
Only ODD harmonics (1st, 3rd, 5th ...)
f_closed = (1/2) × f_open (same length, closed pipe is an octave lower)
String (fixed at both ends):
f_n = nv/(2L) n = 1, 2, 3, ...
v_string = √(T/μ) T = tension, μ = mass per unit length
NDA key fact: a closed pipe produces only odd harmonics — 1st, 3rd, 5th... A flute (open pipe) produces all harmonics. Closed organ pipe sounds an octave lower than an open pipe of the same length.
📝 TOPIC-WISE PYQ
Beats & Resonance — NDA Pattern Questions
Q1. Two tuning forks have frequencies 256 Hz and 260 Hz. The number of beats heard per second is:
(a) 516 (b) 2 (c) 4 (d) 8
Answer: (c) 4
Beat frequency = |f₁ − f₂| = |256 − 260| = 4 beats per second. The sum (516) is irrelevant — beats depend on the difference, not the sum.
Q2. A closed organ pipe and an open organ pipe have the same length. The ratio of their fundamental frequencies is:
(a) 1 : 2 (b) 2 : 1 (c) 1 : 1 (d) 1 : 4
Answer: (a) 1 : 2
f_closed = v/4L; f_open = v/2L. Ratio = (v/4L)/(v/2L) = 2L/4L = 1:2. The closed pipe's fundamental is half the open pipe's — it sounds an octave lower.
Q3. Soldiers are ordered to break step while marching on a suspension bridge. This is to avoid:
Answer: (c) Resonance
If the marching frequency matches the bridge's natural frequency, resonance occurs — amplitude builds up to dangerous levels and the bridge could collapse. Breaking step prevents this. The Tacoma Narrows Bridge (1940) is a famous resonance-related collapse.
🤔 TRICKY QUESTIONS
Beats & Resonance — Deep Concepts
T1. Two tuning forks A (256 Hz) and B (unknown) produce 5 beats/s. When B is loaded with wax, the beats reduce to 3 per second. What is the frequency of B?
B = 261 Hz.
f_B = 256 ± 5, so f_B = 261 Hz or 251 Hz. Loading B with wax adds mass → reduces B's frequency. If beats reduce (from 5 to 3), B is getting closer to A's frequency — so B was above A. Therefore f_B = 261 Hz. (If B were 251 Hz, loading would take it further from 256, increasing beats.)
T2. Why does resonance not build up infinitely? What limits the amplitude in a real resonating system?
Damping — energy loss to friction and air resistance.
In a real system, as amplitude increases, the energy dissipated per cycle also increases (friction, air resistance, internal material damping). Equilibrium is reached when energy absorbed from the driver per cycle equals energy dissipated per cycle. This sets the maximum resonant amplitude. In an ideal (undamped) system, amplitude would grow without limit — but no real system is perfectly undamped.
4. Doppler Effect
4.1
Doppler Effect — Apparent Change in Frequency
Relative motion between source and observer changes the perceived pitch
The Doppler effect (named after Christian Doppler, 1842) is the apparent change in the frequency (pitch) of a wave as the source and observer move relative to each other. It applies to both sound and light. When source and observer approach each other, frequency appears to increase; when they move apart, it appears to decrease.
⚡ Doppler Effect — Formula & Sign Convention
General Doppler Formula:
f' = f₀ × (v + v_observer) / (v + v_source)
Sign convention (always apply):
v = speed of sound in medium
v_observer: POSITIVE (+) if observer moves TOWARD source
NEGATIVE (−) if observer moves AWAY from source
v_source: NEGATIVE (−) if source moves TOWARD observer
POSITIVE (+) if source moves AWAY from observer
Special cases:
Source approaching, observer stationary:
f' = f₀ × v / (v − v_s) → f' > f₀ (higher pitch)
Source receding, observer stationary:
f' = f₀ × v / (v + v_s) → f' < f₀ (lower pitch)
Observer approaching, source stationary:
f' = f₀ × (v + v_o) / v → f' > f₀ (higher pitch)
Observer receding, source stationary:
f' = f₀ × (v − v_o) / v → f' < f₀ (lower pitch)
Simple rule to remember: Approaching = frequency increases (higher pitch). Receding = frequency decreases (lower pitch). Source motion and observer motion are NOT symmetric — source moving changes wavelength; observer moving does not.
Fig. 3 — Doppler effect for a moving source. Wavefronts bunch up ahead (shorter wavelength, higher frequency) and spread out behind (longer wavelength, lower frequency). The pitch heard by the observer ahead is higher than that heard by the observer behind.
Air traffic control: Doppler identifies aircraft approach/departure
Echolocation (bats): Doppler shift gives prey speed info
⚠ NDA Exam Trap — Doppler Effect is asymmetric. The effect is NOT the same whether the source moves or the observer moves (even at the same relative speed). Source motion changes the wavelength of waves in the medium; observer motion does not. The formulas give different values for equal relative speeds. For NDA, focus on the qualitative rule: approaching → higher pitch; receding → lower pitch.
📝 TOPIC-WISE PYQ
Doppler Effect — NDA Pattern Questions
Q1. An ambulance with a siren of frequency 680 Hz approaches a stationary observer. The speed of ambulance is 20 m/s and speed of sound is 340 m/s. The frequency heard by the observer is:
Q3. The Doppler effect is observed in which type of waves?
(a) Sound waves only (b) Light waves only (c) Both sound and light (d) Neither
Answer: (c) Both sound and light
The Doppler effect applies to all waves — mechanical (sound) and electromagnetic (light, radio waves). In astronomy, the red shift of distant galaxy light confirms galaxies are receding (light Doppler). Radar speed guns use radio wave Doppler.
🤔 TRICKY QUESTIONS
Doppler Effect — Careful Reasoning
T1. Source and observer are moving toward each other, each at 20 m/s. Speed of sound = 340 m/s, source frequency = 500 Hz. What frequency is heard?
f′ = 562.5 Hz (both moving toward each other).
Observer moving toward source: +v_o in numerator. Source moving toward observer: −v_s in denominator.
f′ = 500 × (340 + 20)/(340 − 20) = 500 × 360/320 = 500 × 1.125 = 562.5 Hz.
Compare to only source moving at 40 m/s: f′ = 500 × 340/300 = 566.7 Hz. Not the same — source and observer motion are not interchangeable.
T2. A bat flying toward a wall emits a 40,000 Hz ultrasound. Why does the bat hear a higher frequency echo than what it emitted?
Double Doppler effect — bat is both moving source (emission) and moving observer (reception of echo).
When bat emits sound moving toward wall: wavefronts compress → wall receives higher frequency. Wall reflects this higher frequency. Now bat (moving toward wall) receives the reflection: it acts as a moving observer approaching the reflected source → frequency increases again. The result is a doubly blue-shifted echo, significantly higher than 40,000 Hz. Bats use this shift to detect prey speed and direction.
⚡ High-Yield Formula Sheet — PN04 Sound
📢 Wave Fundamentals
v = fλ (wave equation)
T = 1/f (period vs frequency)
ω = 2πf = 2π/T
Loudness ∝ Amplitude²
Pitch ↔ Frequency; Quality ↔ waveform
📈 Speed of Sound
In air (0°C): 332 m/s; (20°C): 343 m/s
Order: v_solid > v_liquid > v_gas
v_T ≈ v₀ + 0.61T (°C) or v ∝ √T (K)
Laplace: v = √(γP/ρ)
Humid air: v slightly higher than dry air
🔌 Echo & SONAR
Echo: min distance = v × 0.1/2 ≈ 17.15 m
SONAR: d = v × t/2
Persistence of hearing = 0.1 s
Reverberation = multiple reflections
♫ Beats
f_beat = |f₁ − f₂|
Audible beats: f_beat ≤ ~10 Hz
Loading fork (wax): reduces frequency
Beats = 0 when f₁ = f₂ (tuned)
🎤 Resonance / Air Columns
Open pipe: f_n = nv/2L (all harmonics)
Closed pipe: f_n = nv/4L (odd harmonics only)
f_closed = f_open/2 (same length)
String: v = √(T/μ)
🚨 Doppler Effect
f′ = f₀ × (v + v_obs)/(v + v_src)
Approaching: f′ > f₀ (higher pitch)
Receding: f′ < f₀ (lower pitch)
v_obs: + toward source; − away
v_src: − toward observer; + away
📌 Key Numbers: Speed of sound in air ≈ 340 m/s; Min. echo distance ≈ 17 m; Human audible range: 20 Hz – 20 kHz; Ultrasound > 20 kHz; Infrasound < 20 Hz; Beat freq = |f₁ − f₂|.
⚡ Quick Revision Booster — PN04 Sound
📢 Nature of Sound
Longitudinal mechanical wave (NOT transverse)
Particles vibrate parallel to propagation
Cannot travel through vacuum (light can)
C = compression (high P); R = rarefaction (low P)
One λ = distance between two successive C or R
📈 Speed of Sound
Solid > Liquid > Gas (elasticity vs density)
v_air ≈ 340 m/s at room temp
Temperature up → v up (v ∝ √T in K)
Humidity up → v slightly up
Frequency/amplitude do NOT affect speed
🔌 Echo Facts
Min echo distance ≈ 17 m
Echo distance = v×t/2
Reverberation = echo overlapping with original
SONAR: ultrasound echo for depth/submarine
Bats: echolocation using 40–100 kHz
♫ Beats
f_beat = |f₁ − f₂| (difference, not sum)
Loading fork with wax → frequency drops
If beats reduce after loading → that fork was higher
Beats = 0 → forks are in tune (same frequency)
Audible beats: difference ≤ ~10 Hz
🎤 Resonance
Open pipe: all harmonics; Closed: odd harmonics only
This material is for personal NDA exam preparation only.
Unauthorised reproduction or distribution is prohibited.
All rights reserved. · contact@olivedefence.com · olivedefence.com