Oscillations & Waves – NDA Physics Notes

Look around you. The world is full of things that oscillate. A pendulum swings back and forth. A guitar string vibrates. Your eardrums move in and out. All of these are oscillations, meaning repeated back-and-forth motion about a fixed position.

Now notice what happens when you pluck a guitar string. The string oscillates, and the air around it begins to oscillate too. Then the air farther away oscillates, and so on, outward in all directions. By the time that oscillation reaches your ear, it has become a wave, a travelling oscillation that carries energy across space without transporting matter.

1. Why Study Oscillations?

Oscillatory motion is everywhere. The motion of the Moon around Earth is periodic. The ticking of a clock is periodic. The vibration of a bridge in the wind is oscillatory. Heart valves open and close rhythmically. Electrons in an antenna oscillate to produce radio waves.

Understanding oscillations is the entry point to understanding waves, acoustics, optics, and quantum mechanics. Every wave (sound, light, X-rays, radio) is ultimately an oscillation propagating through a medium or through space.

2. Periodic Motion and Oscillatory Motion

Periodic motion is any motion that repeats itself at regular intervals of time. Earth revolving around the Sun: periodic. Clock ticking: periodic. A bouncing ball: periodic.

Oscillatory motion is periodic motion that is back-and-forth about a fixed equilibrium position. A pendulum swings to the left, then to the right, repeatedly. A spring stretches, then compresses, repeatedly. All oscillatory motion is periodic, but not all periodic motion is oscillatory.

Property	Periodic Motion	Oscillatory Motion
Definition	Repeats at regular time intervals	Back-and-forth about a fixed equilibrium position
Includes oscillation?	Not necessarily	Yes: oscillatory motion is always periodic
Requires equilibrium?	No	Yes: always about an equilibrium point
Example	Earth orbiting the Sun, clock ticking	Pendulum, spring-mass, vibrating string
Has restoring force?	Not necessarily	Yes : always restoring toward equilibrium

3. Equilibrium and Restoring Force

Every oscillating system has a mean position (equilibrium position), where the net force on the object is zero.

When the object is displaced from this equilibrium, a special force acts on it: the restoring force. This force always points back toward the mean position. It tries to restore the object to equilibrium. Without a restoring force, there is no oscillation. The restoring force is what makes the object turn around and come back. In a pendulum, the restoring force is provided by gravity, specifically the component of gravitational force pointing along the arc toward the mean position. [NDA 2018-I]

4. Simple Harmonic Motion: The Three Conditions

Not every oscillation is simple harmonic motion (SHM). SHM is a special type of oscillation that satisfies three specific conditions. Learn these conditions before any equation is introduced.

Condition 1: The object has a stable equilibrium position. If disturbed, it always returns rather than moving away indefinitely.

Condition 2: A restoring force always acts toward the equilibrium position. The force always opposes displacement.

Condition 3: The restoring force is directly proportional to the displacement from equilibrium. Double the displacement and you double the restoring force.

These three conditions together define SHM. Mathematically:

F = −kx

a = −ω²x

F = restoring force (N). k = force constant (N m⁻¹). x = displacement from equilibrium (m). a = acceleration (m s⁻²). ω = angular frequency (rad s⁻¹). The negative sign is essential. It means the force (and acceleration) are directed opposite to the displacement.

Which of these represents SHM?

(a) a = +3x : positive sign → force away from equilibrium → NOT SHM

(b) a = +3x² : positive sign AND squared displacement → NOT SHM

(c) a = −3x² : squared displacement → force non-linear → NOT SHM

(d) a = −3x : negative sign, linear in x → SHM [NDA 2016-II | NDA 2021-I]

The motion x = cos²ωt is periodic but NOT SHM. Rewriting: cos²ωt = (1 + cos 2ωt)/2. This contains a constant offset. The particle oscillates about a shifted mean, and the restoring force is not proportional to displacement in SHM form. [NDA 2013-I]

5. Displacement, Velocity, and Acceleration in SHM

In SHM, displacement varies sinusoidally with time. If we take x = 0 at the mean position:

x = A sin(ωt + φ)

v = Aω cos(ωt + φ)

a = −Aω² sin(ωt + φ) = −ω²x

A = amplitude, the maximum displacement from equilibrium (m). ω = angular frequency = 2πf (rad s⁻¹). φ = initial phase (rad). t = time (s).

At the mean position (x = 0): velocity is maximum (v = Aω), acceleration is zero.

At the extreme positions (x = ±A): velocity is zero, acceleration is maximum in magnitude (a = ±Aω²).

A particle executing SHM: acceleration is minimum (zero) when speed is maximum, at the mean position. [NDA 2016-II] These two facts are among the most tested in NDA.

Velocity at any position x: from energy conservation, v = ω√(A² − x²). This gives v_max = Aω (when x = 0) and v = 0 (when x = ±A).

6. Phase and Phase Relationships in SHM

The phase of an oscillating particle at time t describes where in the cycle the particle is. Two particles are in phase if they are at the same position moving in the same direction at the same time.

For a single SHM oscillator, velocity = Aω cos(ωt) and acceleration = −Aω² sin(ωt). The phase difference between velocity and acceleration is 90° (π/2 radians). [NDA 2010-I]

Two particles in SHM have the same phase if they are separated by an integer multiple of the time period T. For a displacement-time graph with period T = 4 s: a particle at t = 3 s is in phase with the particle at t = 3 + 4 = 7 s (same phase, one full cycle apart). [NDA 2017-I]

The correct relation between angular frequency and frequency: ω = 2πf. Equivalently, f = ω/(2π). The expressions f = πω and f = 2πω are both wrong. [NDA 2017-I]

Two oscillators executing SHM cannot maintain a constant phase relationship if they have different time periods. Differing periods means differing frequencies. The two oscillators continuously drift relative to each other. [NDA 2011-II]

7. Energy in SHM

Total mechanical energy in SHM is constant. It simply converts between kinetic and potential form as the particle moves.

KE = ½mω²(A² − x²)

PE = ½mω²x² = ½kx²

Total E = KE + PE = ½mω²A²

m = mass of oscillating body (kg). ω = angular frequency (rad s⁻¹). A = amplitude (m). x = current displacement (m). k = force constant = mω² (N m⁻¹). Total energy is proportional to A². Doubling the amplitude quadruples the energy.

Quantity	Mean Position (x = 0)	Extreme Position (x = ±A)
Displacement	Zero	Maximum (A)
Velocity	Maximum (Aω)	Zero
Acceleration	Zero	Maximum (Aω²)
Kinetic Energy	Maximum (½mω²A²)	Zero
Potential Energy	Zero	Maximum (½mω²A² = ½kA²)

In SHM, as the particle moves away from the mean position, potential energy increases and kinetic energy decreases. Total mechanical energy remains constant. [NDA 2010-I | NDA 2012-II]

8. SHM Graphs

Graphs are among the most tested elements of SHM in NDA. Three graphs describe the motion completely.

Displacement-time graph: Sinusoidal (sine or cosine). Amplitude = peak value. Period T = one complete cycle.

Velocity-time graph: Cosine curve. Leads displacement by 90°. Maximum at mean position, zero at extremes.

Acceleration-time graph: Negative sine curve. Exactly opposite to displacement (180° out of phase). Maximum at extremes, zero at mean position.

A displacement-time graph showing decreasing amplitude over successive oscillations represents damped oscillation. Amplitude decays due to energy loss. A simple pendulum immersed in water shows this strongly damped pattern. [NDA 2014-I]

9. The Simple Pendulum

A simple pendulum consists of a bob (a mass) attached to a light inextensible string of length L, swinging freely under gravity. This is the most asked mechanical system in NDA Physics.

The Period Formula

T = 2π√(L/g)

T = time period (s). L = length of the pendulum (m). g = local acceleration due to gravity (m s⁻²). The formula reveals three critical facts about the pendulum period:

T ∝ √L: Length controls the period. Double the length → T multiplies by √2. Quadruple the length → T doubles.

T ∝ 1/√g: Gravity controls the period. Halve g → T multiplies by √2.

T is completely independent of both the mass of the bob and the amplitude (for small angles).

A 1 m pendulum has T = 2π√(1/9.8) ≈ 2 s. This is called the “seconds pendulum.” [NDA 2022-I]

The T² vs L Graph

From T² = 4π²L/g: T² is directly proportional to L. A graph of T² against L is a straight line through the origin. [NDA 2012-II] Slope = 4π²/g, from which g can be measured experimentally.

Period Independence from Mass and Amplitude

Mass has zero effect on the period. Doubling or halving the bob’s mass leaves T unchanged, as mass cancels out in the derivation. Amplitude has zero effect, but only for small angles (typically below about 15°). The small-angle approximation sin θ ≈ θ makes the restoring force proportional to displacement. [NDA 2024-I]

Large-Angle Pendulum

At large amplitudes (such as θ = 60°), the true restoring force is proportional to sin θ, not θ itself. The small-angle approximation underestimates the restoring force at large angles, effectively elongating the period. Therefore, for large-angle oscillations, T is slightly greater than T₀ = 2π√(L/g). [NDA 2026-I] The mass of the bob still has no effect on the period.

Six Problem Patterns: All asked in NDA

Pattern	What Changes	Formula Applied	Result
1. Basic period	Nothing: just calculate T	T = 2π√(L/g)	Standard calculation
2. Length quadrupled (mass unchanged)	L → 4L	T_new = 2π√(4L/g) = 2T	Period doubles
3. Length halved (mass doubled)	L → L/2, m → 2m	T_new = 2π√(L/2 / g) = T/√2	Mass irrelevant; period = T/√2
4. Length 4×, mass 2×	L → 4L, m → 2m	T_new = 2π√(4L/g) = 2T	Mass irrelevant; period doubles
5. Different gravity location	g → g/2 (same L)	T_new = 2π√(L/(g/2)) = √2 × T	Period multiplied by √2
6. Pendulum beats faster	T_current < T_standard	Increase L (mass useless)	Longer pendulum, longer period

[NDA 2018-II]: length 4L, mass 2m → T_new = 2T.  

[NDA 2022-II]: length L/2, mass 2m → T_new = T/√2.  

[NDA 2025-I]: length 4L, mass 2m → new:old = 2:1.  

[NDA 2019-II]: same length, g halved → T_new = √2 T.

A pendulum beating faster than standard has a shorter period. To slow it to standard, increase the length. Mass adjustment has no effect. [NDA 2010-II]

10. Spring–Mass System

A mass m attached to a spring of spring constant k oscillates horizontally (or vertically) with period:

T = 2π√(m/k)

T = period (s). m = mass (kg). k = spring constant (N m⁻¹). Unlike the pendulum, mass matters here. Larger mass means longer period. Spring constant k means larger k → shorter period (stiffer spring, faster oscillation). The spring-mass system does not depend on gravity : it oscillates the same way in orbit.

11. Damped and Forced Oscillations

Damped Oscillations

In real oscillating systems, energy is gradually lost, to air resistance, friction, or viscous drag. The amplitude decreases with each cycle. This is damped oscillation. A displacement-time graph showing a sinusoidal curve with decreasing amplitude is characteristic of a damped oscillator. A pendulum in water (high viscosity) damps rapidly. A pendulum in vacuum maintains constant amplitude indefinitely. [NDA 2014-I]

Forced Oscillations

When an external periodic force is applied to an oscillating system at a frequency different from the system’s natural frequency, the system oscillates at the forcing frequency with an amplitude determined by how close the forcing frequency is to the natural frequency.

12. From One Oscillating Particle to a Travelling Wave

Concept Builder: Read This Carefully You have studied oscillating particles: pendulums, springs, strings vibrating back and forth about their equilibrium positions. Now ask: what happens to the particles around an oscillating particle? Imagine a row of identical pendulums connected by springs. Displace the first pendulum and it begins to oscillate. Because it is connected to the second, the second begins to oscillate with a slight time delay. The third follows the second. And so on down the row. Notice: the disturbance moves along the row. Each individual pendulum oscillates in place. It does not travel anywhere. Only the disturbance moves forward.

An oscillation is motion at one place. A wave is that oscillation travelling through a medium, carrying energy without carrying matter.

A guitar string vibrates. Each point oscillates up and down. The vibration propagates outward through the air as a sound wave. Each air molecule oscillates about its rest position, but the wave carries energy to your ear.

13. What Is a Wave?

A wave is a periodic disturbance that propagates through a medium (or through space), transferring energy and momentum without transferring matter. The medium particles oscillate about their equilibrium positions. They do not travel with the wave. A sound wave passes through air: each air molecule oscillates back and forth, but the molecules do not move from source to your ear. The energy does. [NDA 2016-II]

14. Mechanical vs Electromagnetic Waves

All waves fall into two fundamental categories based on whether they need a medium to propagate.

Property	Mechanical Waves	Electromagnetic Waves
Medium required?	Yes: must have a medium	No: can travel in vacuum
Examples	Sound, water waves, waves in strings	Light, radio, X-rays, microwaves, gamma rays
Speed in vacuum	Cannot travel in vacuum	Speed of light (c ≈ 3 × 10⁸ m s⁻¹)
Type (transverse or longitudinal)	Either (depends on medium and wave type)	Always transverse
Travels through Earth’s core?	Yes (seismic waves do)	Electromagnetic waves can travel through space
NDA tested example	Sound is mechanical   [NDA 2020-I&II]	Light, X-rays, radio, microwaves are electromagnetic

Sound waves, water waves, and seismic waves are all mechanical. Radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays are all electromagnetic. Sound does not belong in the same category as X-rays, microwaves, or radio waves. [NDA 2017-II]

All waves (electromagnetic, sound, and water) can reflect and carry energy. However, only electromagnetic waves can travel in vacuum. [NDA 2018-I]

15. Why Can’t Sound Travel in Vacuum?

Concept Builder Sound is a mechanical wave. It propagates through the collisions between adjacent particles in a medium. When a tuning fork vibrates, it pushes the air molecules next to it. Those molecules push the ones next to them. This chain of pushes and pulls (compressions and rarefactions) travels outward as a sound wave. Vacuum contains no particles. There is nothing to push or pull. Without particles to carry the disturbance, no sound wave can form or travel. This is why the Moon is completely silent. There is no air on the Moon. Two astronauts on the lunar surface cannot hear each other speak without radio communication, even standing side by side.

★  IMPORTANT Sound cannot travel through vacuum. This is the most tested single fact about sound in NDA, tested in at least 4 separate papers. [NDA 2020-I & II | NDA 2021-II | NDA 2023-II | NDA 2024-I]

16. Transverse vs Longitudinal Waves

The two major types of mechanical waves are classified by how the particles move relative to the direction the wave travels.

Property	Transverse Wave	Longitudinal Wave
Particle motion	Perpendicular to wave direction	Parallel to wave direction
Examples	Light (EM), waves in strings, water surface waves	Sound in air, sound in water, sound in solids
Features	Crest and trough	Compression and rarefaction
Can be polarised?	Yes	No
Sound in fluids?	No : sound in fluids is always longitudinal	Yes : sound in air, water is longitudinal
NDA Confusion	Sound is NOT transverse in fluids	Only transverse waves can be polarised

Sound in air is longitudinal. Air molecules oscillate parallel to the direction sound travels, creating compressions and rarefactions. Sound is never transverse in fluids.

Only transverse waves can be polarised (oscillations restricted to one plane). Longitudinal sound cannot be polarised because there is no perpendicular oscillation to restrict. [NDA 2011-I | NDA 2012-I]

Sound waves are similar to waves generated in a pipe filled with air by moving a piston, as both are longitudinal pressure waves. Not similar to waves in a stretched wire (transverse), not similar to laser light (electromagnetic), not similar to mobile tower waves (electromagnetic). [NDA 2014-I]

17. Wave Terminology

Amplitude (A): Maximum displacement of a particle from its equilibrium position. Measured in metres. [NDA 2022-II] Amplitude determines the energy the wave carries. Larger amplitude means more energy.

Wavelength (λ): Distance between two successive points in the same phase (e.g., two successive crests or two successive compressions). SI unit: metres (m).

Frequency (f): Number of complete oscillations per second. SI unit: Hertz (Hz = s⁻¹).

Time period (T): Time for one complete oscillation. T = 1/f. SI unit: seconds (s).

Wave speed (v): Speed at which the wave front propagates. SI unit: m s⁻¹.

18. The Wave Equation and Wave Speed

The most fundamental relationship connecting wave speed, frequency, and wavelength:

v = fλ

v = wave speed (m s⁻¹). f = frequency (Hz). λ = wavelength (m). The correct form is v = fλ, meaning the speed equals frequency times wavelength. [NDA 2025-I]

Common wrong forms: f = vλ (wrong units), λ = vf (wrong), f = λ/v (wrong).

Numerical Applications

A sound wave has f = 2 kHz = 2000 Hz and λ = 35 cm = 0.35 m. Speed v = fλ = 2000 × 0.35 = 700 m/s. Time to travel 1.4 km: t = d/v = 1400/700 = 2 s. [NDA 2014-I]

A sound wave has f = 1 kHz = 1000 Hz and λ = 50 cm = 0.5 m. Speed v = 1000 × 0.5 = 500 m/s. Time to travel 1 km: t = 1000/500 = 2 s. [NDA 2022-I]

The wave equation in mathematical form: y = A sin(kx − ωt), where k = 2π/λ is the wave number (rad m⁻¹).

19. Speed of Sound in Different Media

The speed of sound in any medium depends on two properties of the medium: how elastic it is and how dense it is.

v = √(B/ρ)

B = bulk modulus (elastic modulus, Pa). ρ = density of medium (kg m⁻³). Speed depends on both elastic and inertia properties, not just one of them. [NDA 2016-II]

Medium	Speed of Sound	Notes
Steel (solid)	~5000 m/s	Fastest : highest bulk modulus
Water (liquid)	~1500 m/s at 20°C	Intermediate
Air (gas)	~340 m/s at room temperature	Slowest : lowest bulk modulus

Speed of sound in water ≈ 1500 m/s. [NDA 2019-I]

The order is always: solids > liquids > gases.

Effect of Temperature

Speed of sound in air increases with increasing temperature. At higher temperatures, air molecules move faster and collide more rapidly, transmitting the pressure disturbance more quickly. The incorrect statement, that speed decreases with temperature, is a direct NDA confusion. [NDA 2022-II]

Effect of Humidity

Speed of sound in moist air is greater than in dry air. Water vapour (molar mass 18 g/mol) is lighter than dry air (average molar mass ~29 g/mol). Lower density → higher speed. This is why sound travels slightly faster in humid conditions.

Independence from Frequency and Pressure

Speed of sound in a given medium is the same for all frequencies. Ultrasonic sound and audible sound travel at identical speeds in the same medium. [NDA 2023-I | NDA 2019-II]

At constant temperature, doubling the pressure of an ideal gas does not change the speed of sound. Doubling pressure doubles both the elastic modulus and the density. The ratio B/ρ stays constant, so v stays constant. [NDA 2026-I]

20. Sound Characteristics: Loudness, Pitch, and Timbre

Three properties describe a sound as we hear it: how loud it is, how high-pitched it is, and what instrument or voice produced it. Each depends on a different physical property of the sound wave.

Characteristic	Physical Property	SI Unit of Physical Property	Common NDA Confusion
Loudness	Amplitude (and intensity)	Amplitude in metres (m)	Confused with frequency or pitch
Pitch	Frequency	Hertz (Hz)	Confused with loudness or amplitude
Timbre (Quality)	Waveform (harmonic content)	—	Often ignored; it distinguishes instruments

Loudness is determined by the amplitude (or intensity) of the sound wave. Larger amplitude = louder sound. [NDA 2015-II | NDA 2019-II | NDA 2022-II]

Amplitude of sound waves is measured in metres, the maximum displacement of air particles from equilibrium. [NDA 2022-II]

Pitch is determined by frequency. Higher frequency = higher pitch. A female voice has higher pitch than a male voice because of higher fundamental frequency. [NDA 2016-II | NDA 2022-II]

Timbre is the quality that lets us distinguish a piano from a guitar playing the same note at the same loudness. It is determined by the waveform, the mixture of harmonics (overtones) present alongside the fundamental frequency. [NDA 2016-II]

The decibel (dB) measures sound intensity level on a logarithmic scale. It is not a unit of frequency. Valid frequency units: Hz, s⁻¹, min⁻¹ (after conversion). dB cannot express frequency. [NDA 2021-I]

21. Frequency Ranges : Audible, Infrasonic, and Ultrasonic

Not all sound waves are audible to humans. The human ear responds to a specific range of frequencies.

Range	Frequency	Examples
Infrasonic	< 20 Hz	Earthquakes, elephants communicating, whales
Audible (human)	20 Hz – 20,000 Hz (20 kHz)	Human speech, music, environmental sounds
Ultrasonic	> 20 kHz (20,000 Hz)	Bat echolocation, SONAR, medical imaging, defect detection

Ultrasonic frequency is greater than 20 kHz. [NDA 2011-II | NDA 2018-II] Not between 20 Hz and 1000 Hz (that is audible). Not between 1000 Hz and 20000 Hz. Greater than 20,000 Hz.

Ultrasonic waves travel at the same speed as audible sound in the same medium, not faster. They have higher frequency and shorter wavelength, not higher speed. [NDA 2019-II]

Ultrasonic waves can be reflected, refracted, diffracted, and absorbed. The incorrect statement is that they cannot get reflected, refracted, or absorbed. [NDA 2017-II]

22. Reflection, Echo, and Reverberation

Echo

An echo is a reflected sound that reaches the listener after a perceptible time delay after the original sound. For a distinct echo, the reflecting surface must be at least about 17 m away (sound travels at ~340 m/s; minimum 0.1 s delay required by human hearing persistence). [NDA 2025-I] An echo is caused by reflection of sound from a distant surface.

Reverberation

Reverberation is the persistence of sound in an enclosed space due to multiple successive reflections from walls, ceiling, and floor. After the sound source stops, the sound appears to linger because waves keep reflecting from all surfaces. [NDA 2021-I | NDA 2021-II] Reverberation is not refraction and not diffraction. It is multiple reflections inside an enclosed space.

Property	Echo	Reverberation
Definition	Distinct reflected sound heard after a delay	Persistence of sound due to repeated reflections
Cause	Single reflection from a distant surface	Multiple reflections from all surfaces in enclosed space
Number of reflections	Usually one clear reflection	Many successive overlapping reflections
Required distance / time	Reflecting surface ~17 m away; delay ≥ 0.1 s	Enclosed space : no minimum distance
Duration of effect	Brief: one clear repeat heard	Continues for seconds after source stops
Examples	Shouting in mountains; cliff echo	Large hall, cave, empty auditorium
Applications	Measuring distances (SONAR, bats)	Acoustic design of auditoriums and concert halls
NDA Confusion	Echo is not reverberation: needs minimum distance	Reverberation is not refraction or diffraction

23. Beats

When two sound waves of nearly equal (but not identical) frequencies travel in the same direction and superpose, the combined wave shows periodic amplitude fluctuations. These alternating loud and soft sounds are called beats.

Beat frequency = |f₁ − f₂|

The listener hears one beat per second for every 1 Hz difference between the two frequencies.

Beats occur only when frequencies are nearly equal, not equal (no fluctuation), not far apart (too rapid to perceive as beats), and not simple multiples of each other. [NDA 2021-I]

Application: A tuner hears beats between a guitar string and a reference frequency. As the string is tightened or loosened until beats disappear, the string is in tune (frequencies are equal).

24. Standing Waves and Resonance

Standing Waves

When two waves of the same frequency and amplitude travel in opposite directions and superpose, they create a standing wave (stationary wave). The wave does not appear to travel. Certain points remain stationary.

Nodes: Points of zero displacement, where waves from both directions cancel exactly at all times.

Antinodes: Points of maximum displacement, where waves from both directions reinforce each other maximally.

Progressive vs Standing Waves

Property	Progressive (Travelling) Wave	Standing (Stationary) Wave
Energy transfer	Transports energy through medium	Does not transport energy : energy is localised
Amplitude	Same for all particles	Varies from zero (node) to maximum (antinode)
Phase	Varies continuously with position	All particles between two nodes are in phase
Produced by	Single source	Reflection causing two waves to superpose
Example	Sound wave from a speaker	Vibrating guitar string, organ pipe

In progressive (travelling) waves, the amplitude of all particles is the same (constant), but neighbouring particles are out of phase. Each particle has a different phase depending on its position along the wave. [NDA 2011-I]

Resonance

Every physical system has a natural frequency at which it vibrates most readily. When an external driving force is applied at this natural frequency, the system resonates. It absorbs energy most efficiently and vibrates with maximum amplitude.

Child on a swing: Push at the swing’s natural frequency → amplitude builds rapidly. Push at the wrong frequency → little or no effect.

Tuning forks: Strike one tuning fork of 440 Hz and a second 440 Hz fork nearby begins to ring sympathetically. The sound wave from the first drives the second at its natural frequency.

Soldiers crossing a bridge: If soldiers march in step at a frequency close to the bridge’s natural frequency, they continuously add energy to the bridge at resonance. This can cause dangerous amplitude buildup. That is why soldiers break step (march out of sync) when crossing bridges, to avoid resonance.

25. The Doppler Effect

The Doppler effect is the change in observed frequency of a wave when the source and observer are moving relative to each other.

Source approaches: wavefronts compressed → shorter wavelength → higher observed frequency → higher pitch.

Source moves away: wavefronts stretched → longer wavelength → lower observed frequency → lower pitch.

Classic example: An ambulance siren sounds higher-pitched as it approaches and drops in pitch as it passes and moves away. The siren’s own frequency has not changed. Only the observer’s perception has.

The Doppler Formula (NDA level)

For a moving source and stationary observer:

f’ = f × v / (v ± v_s)

f’ = observed frequency (Hz). f = source frequency (Hz). v = speed of sound in medium (m s⁻¹). v_s = speed of source (m s⁻¹). Use (v − v_s) when source approaches → f’ > f (higher observed frequency). Use (v + v_s) when source moves away → f’ < f (lower observed frequency).

26. Everyday Applications

SONAR : Sound Navigation and Ranging

SONAR stands for Sound Navigation and Ranging. [NDA 2025-II] It uses ultrasonic waves (not audible sound, not infrasound) to measure distances to underwater objects. A SONAR device sends an ultrasonic pulse downward from a ship. The pulse reflects off the seabed or a submarine and returns to the ship.

Time of travel gives distance: d = v × t/2 (divide by 2 because the pulse travels down and back). [NDA 2012-II | NDA 2022-II | NDA 2025-II]

SONAR is primarily used by navigators : ships and submarines. [NDA 2012-II]

Bat Echolocation

Bats navigate and hunt in complete darkness using echolocation. A bat emits ultrasonic waves from its own body. These waves reflect off obstacles or prey and return to the bat’s ears. The time delay between emission and detection gives distance; the direction of the returned echo gives location. [NDA 2011-I | NDA 2017-II] The reflected waves come from distant objects, not from the bat itself. Ultrasonic, not supersonic: ultrasonic means high frequency; supersonic means faster than sound.

Microphone

A microphone converts sound waves into electrical signals. It is a transducer operating in the sound → electricity direction. A speaker does the reverse (electrical → sound). [NDA 2022-II]

Flute

In a flute, sound comes from a vibrating column of air inside the flute. The pitch is controlled by which holes are open or closed, changing the effective length of the air column and thus the resonant frequency. [NDA 2023-I]

Ultrasound in Medicine and Industry

Medical imaging: Ultrasound images (sonography) use reflection of high-frequency sound to produce images of internal organs without radiation.

Industrial defect detection: Cracks and porosity inside metal components can be detected without damaging the material. Ultrasound pulses reflect from internal defects. [NDA 2017-II]

Ultrasonic drilling: Ultrasonic vibrations can drill holes in hard materials like diamond, where conventional drills struggle. [NDA 2017-II]

Important Distinctions

Periodic Motion vs Oscillatory Motion vs SHM

Periodic motion: repeats at regular intervals. Oscillatory motion: back-and-forth about equilibrium, a specific type of periodic motion. SHM: oscillatory motion where restoring force is proportional to displacement, a specific type of oscillatory motion. SHM ⊂ Oscillatory ⊂ Periodic.

Mechanical vs Electromagnetic Waves

Mechanical waves require a physical medium (sound, water waves). Electromagnetic waves need no medium. They travel through vacuum at the speed of light (radio, light, X-rays). Sound and light are completely different types of waves.

Transverse vs Longitudinal

Transverse: particles move perpendicular to wave direction : can be polarised (light). Longitudinal: particles move parallel to wave direction : cannot be polarised (sound in air). Sound in all fluids is always longitudinal.

Amplitude vs Wavelength vs Frequency

Amplitude: maximum particle displacement from equilibrium : determines loudness (metres). Wavelength: distance between successive identical points : determines spatial extent (metres). Frequency: oscillations per second : determines pitch (Hz). These are three independent properties of a wave.

Echo vs Reverberation Echo: one distinct reflected sound from a distant surface (single reflection). Reverberation: persistence of sound from many successive reflections in enclosed spaces (multiple reflections). An echo is heard distinctly; reverberation is heard as a prolonged decay.

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