Waves – Chapter 14 | AAI ATC Physics

✈️ AAI ATC Relevance: Communication & radar signals use wave principles

🌊

14.1 Introduction to Waves

A wave is a disturbance that transfers energy from one point to another without the physical transfer of matter. When a stone is dropped in water, the ripples move outward — but the water itself does not travel with the ripples.

Waves are classified into three main types:

Mechanical Waves — require a material medium (sound waves, waves on a string, water waves, seismic waves).
Electromagnetic Waves — do not require a medium; travel through vacuum (light, radio, X-ray). Speed in vacuum: c = 3 × 10⁸ m/s.
Matter Waves — associated with sub-atomic particles (electrons, protons); quantum mechanical in nature.

🖼️

[ Add Image: Types of Waves Diagram ]

Energy TransferWaves transport energy & information, not matter.

MediumMechanical waves need elastic medium; EM waves don't.

Speed of Lightc = 299,792,458 m/s ≈ 3 × 10⁸ m/s

AAI ATC LinkRadar & communication rely on EM wave propagation.

🔢 Practice MCQs — Introduction

Q1. A sound wave travels from air into water. Which property of the wave remains unchanged?

Q2. Electromagnetic waves travel through vacuum with speed c = 3 × 10⁸ m/s. What is the time taken by a radar signal to travel a distance of 150 km and return?

Q3. A mechanical wave requires a medium for propagation because it depends on:

✈️ AAI ATC Relevance: Sound waves in ATC communication are longitudinal

↕️

14.2 Transverse and Longitudinal Waves

Waves are classified by the direction of particle oscillation relative to wave propagation:

🔀 Transverse Waves

Particle oscillation is perpendicular to the direction of propagation.

Examples: Waves on a string, light (EM waves), water surface waves

Medium: Only in solids (shear stress needed)

↔️ Longitudinal Waves

Particle oscillation is parallel to the direction of propagation.

Examples: Sound waves, waves in spring (slinky)

Medium: Solids, liquids and gases (bulk modulus)

🖼️

[ Add Image: Transverse vs Longitudinal Wave Diagram ]

Water waves are a combination of both transverse and longitudinal components. Ocean waves involve particles moving in elliptical paths.

🔢 Practice MCQs — Transverse & Longitudinal Waves

Q4. Ultrasonic sound waves used in ATC equipment are what type of waves?

Q5. Transverse waves cannot propagate in fluids because fluids:

Q6. A kink is created in a longitudinal spring by displacing one end sideways. The wave produced is:

✈️ AAI ATC Relevance: Signal modulation and wave parameters appear in technical papers

📐

14.3 Displacement Relation in a Progressive Wave

A sinusoidal travelling wave moving along the positive x-axis is described by:

y(x, t) = a sin(kx − ωt + φ)

a — AmplitudeMaximum displacement of particles from equilibrium. Unit: metre (m).

k — Angular Wave Numberk = 2π/λ. Unit: rad m⁻¹.

ω — Angular Frequencyω = 2π/T = 2πν. Unit: rad s⁻¹.

φ — Initial Phase AnglePhase at x = 0, t = 0. Unit: radian.

🖼️

[ Add Image: Sinusoidal Wave with Labels (a, λ, T) ]

Key relations:

λ = 2π/k | T = 2π/ω | ν = 1/T = ω/2π

A wave moving in the negative x-direction: y(x,t) = a sin(kx + ωt + φ)

🔢 Practice MCQs — Displacement Relation

Q7. A wave is described by y(x, t) = 0.005 sin(80x − 3t). What is the wavelength of the wave?

Q8. For the wave y = 0.005 sin(80x − 3t), what is the time period of oscillation?

Q9. The equation y = a sin(kx + ωt) represents a wave travelling in which direction?

✈️ AAI ATC Relevance: Speed of sound is critical in voice communication clarity

🚀

14.4 Speed of a Travelling Wave

The speed of a progressive wave is given by the general relation:

v = ω/k = λ/T = λν

Speed of Transverse Wave on a Stretched String:

v = √(T/μ)

where T = tension in the string (N), μ = linear mass density (kg/m)

Speed of Longitudinal Wave (Sound):

v = √(B/ρ) [Fluids] | v = √(Y/ρ) [Solid bar]

where B = Bulk modulus, Y = Young's modulus, ρ = density

Newton's Formula for speed of sound in gas: v = √(P/ρ) — gave ~280 m/s (wrong)

Laplace Correction (adiabatic):

v = √(γP/ρ) → 331 m/s at 0°C for air (γ = 1.4)

🖼️

[ Add Image: Speed of Sound in Different Media – Table / Chart ]

🔢 Practice MCQs — Speed of Wave

Q10. A steel wire of length 0.72 m has mass 5 × 10⁻³ kg and is under tension 60 N. What is the speed of transverse waves on the wire?

Q11. The speed of sound in air at 0°C using Laplace's correction is approximately:

Q12. If the tension in a stretched string is increased to 4 times its original value, the speed of the transverse wave becomes:

✈️ AAI ATC Relevance: Signal interference in VHF radio communication

➕

14.5 Principle of Superposition of Waves

When two or more waves simultaneously traverse the same medium, the net displacement of any particle is the algebraic sum of the displacements due to each individual wave.

y(x,t) = y₁(x,t) + y₂(x,t)

For two waves with equal amplitude a, same frequency, but phase difference φ:

y(x,t) = 2a cos(φ/2) · sin(kx − ωt + φ/2)

✅ Constructive Interference

φ = 0 → Amplitude = 2a (maximum)

Waves are in phase

❌ Destructive Interference

φ = π → Amplitude = 0

Waves are out of phase

🖼️

[ Add Image: Constructive and Destructive Interference Diagrams ]

🔢 Practice MCQs — Superposition

Q13. Two waves each of amplitude 4 cm are superimposed with a phase difference of 180°. What is the resultant amplitude?

Q14. Two waves of equal amplitude 3 cm are in phase (φ = 0). The resultant amplitude is:

Q15. The principle of superposition is the basis for which of the following phenomena?

✈️ AAI ATC Relevance: Resonance in microphone/speaker cavities; organ pipe analogy

🏗️

14.6 Reflection of Waves & Standing Waves

Reflection at rigid boundary: Wave suffers a phase change of π (180°). Reflected wave: y_r = −a sin(kx + ωt)

Reflection at open boundary: No phase change. Reflected wave: y_r = a sin(kx + ωt)

Two waves of equal amplitude travelling in opposite directions produce Standing Waves:

y(x,t) = 2a sin(kx) cos(ωt)

Nodes (zero displacement): x = nλ/2 | Antinodes (max displacement): x = (n+½)λ/2

String fixed at both ends (Normal modes):

ν_n = nv/2L ; n = 1, 2, 3, …

Pipe closed at one end: Only odd harmonics → ν = (2n+1)v/4L

Pipe open at both ends: All harmonics → ν_n = nv/2L

🖼️

[ Add Image: Standing Wave Modes – String Fixed at Both Ends ]

🔢 Practice MCQs — Standing Waves

Q16. A pipe 30 cm long is open at both ends. Speed of sound = 330 m/s. What is the frequency of the second harmonic?

Q17. In a standing wave on a string, the distance between two consecutive nodes is:

Q18. A pipe closed at one end of length 20 cm resonates with a 430 Hz source (v = 340 m/s). Which harmonic is excited?

✈️ AAI ATC Relevance: Beat frequency concept used in radio tuning and signal detection

🥁

14.7 Beats

When two sound waves of nearly equal (but not identical) frequencies are superimposed, the resultant intensity waxes and wanes periodically. This phenomenon is called beats.

Beat Frequency = ν_beat = |ν₁ − ν₂|

The resultant wave can be written as:

s = [2a cos(ω_bt)] cos(ω_at)

where ω_a = (ω₁+ω₂)/2 (average frequency) and ω_b = (ω₁−ω₂)/2 (beat frequency modulation)

🖼️

[ Add Image: Beat Pattern – Two waves (11 Hz & 9 Hz) producing 2 Hz beats ]

💡 Important for AAI ATC Exam

Musicians use beats to tune instruments. If beats increase on changing tension → original frequency was lower than reference. If beats decrease → original was higher.

🔢 Practice MCQs — Beats

Q19. Two sitar strings A and B playing note 'Dha' produce 5 beats/s. Frequency of A = 427 Hz. Tension of B is slightly increased and beat frequency drops to 3 Hz. What is the original frequency of B?

Q20. Two tuning forks of frequencies 256 Hz and 260 Hz are sounded together. The beat frequency is:

Q21. A string of frequency 324 Hz is played with another string producing 6 beats/s. When tension in the first string is reduced, beat frequency drops to 3 Hz. What is the frequency of the second string?

📋

Chapter 14 — Quick Summary Table

Quantity	Symbol	Formula	Unit
Wavelength	λ	2π/k	m
Angular Wave Number	k	2π/λ	rad m⁻¹
Time Period	T	2π/ω	s
Frequency	ν	ω/2π = 1/T	Hz
Wave Speed (general)	v	λν = ω/k	m s⁻¹
Speed on string	v	√(T/μ)	m s⁻¹
Speed of sound (fluid)	v	√(B/ρ)	m s⁻¹
Speed of sound (gas, Laplace)	v	√(γP/ρ)	m s⁻¹
Beat Frequency	ν_beat	\|ν₁ − ν₂\|	Hz
Harmonics (open/both-end string)	ν_n	nv/2L	Hz
Harmonics (closed one end)	ν_n	(2n+1)v/4L	Hz

Chapter 14 — Waves

🚀 Level Up Your AAI ATC Prep!