1. Introduction: The Universal Language of Energy

Welcome, students, to the grand finale of our mechanics syllabus. We have spent months studying particles—balls colliding, cars accelerating, planets orbiting. In all those cases, matter itself moved from Point A to Point B. But today, we unlock a new concept. We are going to talk about how Energy travels from Point A to Point B without the matter itself travelling.

Think about a line of students standing in a queue. If the last student pushes the one in front, and that student pushes the next, a “disturbance” travels all the way to the front of the line. But did the last student run to the front? No. The students stayed in their spots (roughly), but the push traveled. This is the essence of a Wave.

Waves are everywhere. The light allowing you to read this, the sound of my voice, the Wi-Fi signal on your phone, and the heat from the sun are all waves. Even the electrons inside atoms behave like waves. Understanding waves is not just about passing an exam; it is about understanding the operating system of the universe.

2. Classification of Waves

Before we start deriving equations, we need to categorize the different types of waves we encounter in nature. We can classify them based on two main criteria: what they need to travel, and how they wiggle.

2.1 Classification by Medium Requirement

Based on whether a wave needs a material medium (like air, water, or string) to propagate, we have:

• Mechanical Waves: These waves act like “gossip.” They need a neighbor to pass the message to. They require a material medium possessing elasticity and inertia. Examples include sound waves, water waves, seismic waves, and waves on a string. Without a medium (vacuum), these waves cannot exist. This is why space is silent.
• Electromagnetic Waves: These are the “independent travelers.” They do not need a medium. They consist of oscillating electric and magnetic fields that sustain each other. Examples include Light, Radio waves, X-rays, and Microwaves. They can travel through the vacuum of space.
• Matter Waves: These are associated with microscopic particles like electrons, protons, and neutrons. This is the realm of Quantum Mechanics (which you will study in Class 12).

2.2 Classification by Mode of Vibration

This is the most important classification for mechanics. It depends on the direction of particle oscillation relative to the wave’s motion.

3. Mathematical Description of a Wave

Now, let’s get rigorous. How do we describe a wave mathematically? A wave is a function of both space (x) and time (t). We need a function y(x,t) that tells us the displacement of any particle at position ‘x’ at any instant ‘t’.

3.1 The Harmonic Wave Function

The simplest type of wave is the sinusoidal (harmonic) wave. We describe it using the Sine or Cosine function:

y(x, t) = A sin(kx – ωt + φ)

Let’s dissect this equation carefully. Every symbol has a deep physical meaning:

• y(x, t): Displacement of the particle at position x at time t.
• A (Amplitude): The maximum displacement from the mean position. It represents the “loudness” or “brightness” of the wave.
• (kx – ωt + φ): This entire term inside the bracket is called the Phase. It tells us the current state of vibration.
• φ (Initial Phase Constant): It accounts for the initial conditions (where the wave started at t=0, x=0).

3.2 Wave Parameters

We need to define two crucial new constants that appear in the equation:

1. Angular Wave Number (k):

Just as frequency tells us how many cycles occur in unit time, ‘k’ tells us how many radians of phase change occur in unit distance.

Formula: k = 2π / λ

Unit: Radians per meter (rad/m).

2. Angular Frequency (ω):

This tells us how fast the particle oscillates in time.

Formula: ω = 2π / T = 2πf

Unit: Radians per second (rad/s).

3.3 Wave Speed (v) vs. Particle Speed (v_p)

This is the most common point of confusion.

Wave Speed (v): The speed at which the shape or energy moves. It is constant for a given medium.

Formula: v = ω / k = λ / T = fλ

Particle Speed (v_p): The speed at which the matter oscillates up and down. It varies with time (Simple Harmonic Motion).

Formula: v_p = dy/dt = -Aω cos(kx - ωt)

Note that the particle speed is maximum at the mean position and zero at the extreme positions, while the wave speed is constant throughout.

4. The Speed of Traveling Waves

The speed of a mechanical wave is not arbitrary. You cannot just “choose” the speed. It is strictly determined by the properties of the medium. Specifically, it depends on two competing factors:

1. Elasticity (E): The ability of the medium to return to its original shape. Stiffer media pull back harder, transmitting the wave faster.
2. Inertia (Density, ρ): The resistance to motion. Heavier media are harder to move, slowing the wave down.

v = √(Elastic Property / Inertial Property)

4.1 Speed of a Transverse Wave on a Stretched String

For a string (like a guitar string), the elasticity is provided by the Tension (T) and the inertia is provided by the Linear Mass Density (μ = mass/length).

v = √(T / μ)

This explains why tuning a guitar works:

• Tightening the string (increasing T) increases speed (and frequency/pitch).

• Using a thicker string (increasing μ) decreases speed (and frequency/pitch).

4.2 Speed of a Longitudinal Wave (Sound)

For a fluid (liquid or gas), the elasticity is provided by the Bulk Modulus (B) and inertia by Volume Density (ρ).

v = √(B / ρ)

5. The Principle of Superposition

What happens when two waves collide? Do they bounce off like billiard balls? No. They pass right through each other like ghosts! While they overlap, they combine mathematically.

The Principle: The resultant displacement of a particle acted upon by two or more waves is the vector sum (algebraic sum) of the individual displacements.

y_net = y₁ + y₂

5.1 Reflection of Waves

When a wave hits a boundary, it reflects. But how it reflects depends on the boundary.

• Rigid Boundary (Fixed End): Imagine a rope tied to a wall. The wall cannot move. By Newton’s 3rd Law, the wall exerts a downward force on the rope.
  
  Result: The wave flips upside down. There is a phase change of π radians (180°). A crest reflects as a trough.
• Free Boundary (Open End): Imagine a ring sliding on a frictionless pole. The end is free to move.
  
  Result: The wave does not flip. There is zero phase change. A crest reflects as a crest.

6. Standing Waves (Stationary Waves)

This is arguably the most beautiful phenomenon in wave physics. It is the basis of all music.

When two identical waves (same amplitude, same frequency) travel in opposite directions along the same line, they interfere to produce a pattern that does not appear to move. Energy is not transported; it is trapped in loops. These are called Standing Waves.

6.1 Nodes and Antinodes

• Nodes (N): Points of destructive interference where the displacement is always zero. These points never move. (Think of the fixed ends of a guitar string).
• Antinodes (A): Points of constructive interference where the displacement is maximum. These points swing wildly up and down.

Key Distances:

• Distance between two consecutive Nodes = λ/2

• Distance between two consecutive Antinodes = λ/2

• Distance between a Node and the next Antinode = λ/4

6.2 Harmonics on a Stretched String

A string fixed at both ends can only vibrate at specific frequencies. These are called Harmonics.

• Fundamental Mode (1st Harmonic):
  
  The string vibrates in one single loop.
  
  Length L = λ/2 → λ = 2L.
  
  Frequency f₁ = v / λ = v / 2L.
• Second Harmonic (1st Overtone):
  
  The string vibrates in two loops.
  
  Length L = λ.
  
  Frequency f₂ = v / L = 2(v / 2L) = 2f₁.
• General Formula:
  
  For the n-th harmonic: f_n = n (v / 2L)
  
  Strings produce all harmonics (odd and even), making them sound rich and musical.

6.3 Organ Pipes (Air Columns)

Air columns also form standing waves. The rules are:

• Closed Organ Pipe (One end closed):
  
  Closed end is a Node (air can’t move). Open end is an Antinode.
  
  Produces only Odd Harmonics (f, 3f, 5f…).
  
  Fundamental: f₁ = v / 4L.
• Open Organ Pipe (Both ends open):
  
  Both ends are Antinodes.
  
  Produces All Harmonics (f, 2f, 3f…).
  
  Fundamental: f₁ = v / 2L.
  
  Note: Open pipes sound richer because they have more overtones.

7. Beats

What happens if two sound waves are almost the same frequency, but not quite? Say, 440 Hz and 442 Hz?

They interfere. Sometimes they are in phase (Loud), and a moment later they are out of phase (Silence). This periodic rise and fall in intensity is called Beats.

Beat Frequency = |f₁ – f₂|

If you play 440 Hz and 442 Hz together, you will hear a “Wah-Wah-Wah” pulsing sound 2 times every second. Musicians use this to tune instruments. If they hear beats, they are out of tune. They adjust until the beats disappear (beat freq = 0), meaning frequencies are identical.

8. The Doppler Effect

Note: This topic is for JEE/NEET aspirants. It is deleted from CBSE Board Syllabus.

Have you ever noticed how the siren of an ambulance sounds high-pitched as it approaches you, and suddenly drops to a low pitch as it passes you? This phenomenon is the Doppler Effect.

It occurs whenever there is relative motion between the Source (S) of the sound and the Observer (O).

9. Comprehensive Practice Set

Physics is not a spectator sport. Let’s solve some high-quality problems to cement these concepts.

Part A: Conceptual Questions

1. Q: Can two astronauts talk to each other on the Moon?
   
   A: No. Sound is a mechanical wave and requires a medium (air). The Moon is a vacuum. They must use radio waves (electromagnetic) which can travel through a vacuum.
2. Q: Why does sound travel faster in steel than in air?
   
   A: Speed v = √(E/ρ). Although steel is denser than air (which should slow it down), its Elasticity (E) is thousands of times greater than air. The elasticity factor wins, making sound travel ~15 times faster in steel.
3. Q: What happens to the wavelength if the frequency is doubled?
   
   A: Since v = fλ, and wave speed ‘v’ is constant for a given medium, if ‘f’ doubles, ‘λ’ must be halved.

Part B: Numerical Problems (Detailed Solutions)