Waves and Oscillations: Complete Guide to Wave Motion in Physics | HRK Notes

House of Physics October 05, 2025
Physics HRK Notes: Waves and Oscillations - Complete Guide to Wave Motion
Mastering Mechanical Waves, Traveling Waves, Sinusoidal Waves, Wave Equation, Interference, Standing Waves, and Resonance
Waves and Oscillations Wave Motion Mechanical Waves Wave Equation Interference Standing Waves Reading Time: 20 min

📋 Table of Contents

📜 Historical Background

The study of waves and oscillations has been fundamental to physics for centuries:

  • Ancient Times: Early observations of water waves and vibrating strings
  • 17th Century: Robert Hooke's work on springs and elasticity
  • 18th Century: Daniel Bernoulli's analysis of vibrating strings
  • 19th Century: Development of wave equation by d'Alembert and Fourier analysis
  • 20th Century: Quantum mechanics established wave-particle duality

Wave motion appears in almost every branch of physics, making it one of the unifying themes of the subject.

Introduction to Wave Motion

🔬 What is Wave Motion?

Wave motion appears in almost every branch of physics. Surface waves on bodies of water are commonly observed. Sound waves and light waves are essential to our perception of the environment, because we have evolved receptors (eyes and ears) capable of their detection.

In the past century, we have learned how to produce and use radio waves. We can understand the structure of atoms and subatomic systems based on the wavelike properties of their constituent particles. The similarity of the physical and mathematical description of these different kinds of waves indicates that wave motion is one of the unifying themes of physics.

📝 Key Characteristics of Waves

All waves share certain fundamental characteristics:

  • Transfer of Energy: Waves transfer energy from one point to another without transferring matter
  • Periodic Motion: Waves exhibit periodic oscillations in time and space
  • Wave Parameters: All waves can be described using parameters like amplitude, wavelength, frequency, and speed
  • Superposition: Waves can pass through each other and combine according to the principle of superposition

Mechanical Waves

🌊 What are Mechanical Waves?

"The waves which require a material medium for their propagation are called mechanical waves."

Mechanical waves cannot travel through vacuum - they need a material medium such as air, water, or solids to propagate. Examples include sound waves, water waves, and seismic waves.

Type Definition Examples Particle Motion
Transverse Waves Particles vibrate perpendicular to wave direction Waves on strings, electromagnetic waves Perpendicular to propagation
Longitudinal Waves Particles vibrate parallel to wave direction Sound waves, spring waves Parallel to propagation

Transverse Waves

⚙️ Transverse Wave Demonstration

Transverse Wave
Wave Direction →
Particle Motion ↕

Working Principle: In transverse waves, the particles of the medium vibrate perpendicular to the direction of wave propagation.

Example: When a string is tied to a hook from one end and the free end is displaced up and down, transverse waves are produced. The disturbance moves along the string but the particles vibrate perpendicularly to the direction of propagation of disturbance.

Real-world Example: Water waves are also transverse in nature, though they have a more complex motion.

Longitudinal Waves

⚙️ Longitudinal Wave Demonstration

Longitudinal Wave
Compressions and Rarefactions
Wave Direction ↔
Particle Motion ↔

Working Principle: In longitudinal waves, the particles of the medium vibrate parallel to the direction of wave propagation.

Example: When a spring under tension is made to vibrate back and forth at one end, a longitudinal wave is produced. The coils of the spring vibrate back and forth parallel to the direction of propagation of disturbance.

Real-world Example: Sound waves in a gas are longitudinal waves or compressional waves.

Traveling Waves

🚀 What are Traveling Waves?

"The waves which transfer energy from one point to another moving away from source of disturbance are called traveling waves."

Traveling waves carry energy and momentum with them as they propagate through a medium. They are characterized by their ability to maintain their shape as they move through space.

Wave Function and Phase Velocity

🧮 Mathematical Description of Traveling Waves

Step 1: Wave Function at t = 0

The wave form of a pulse at t = 0 is given by:

\[ y(x, 0) = f(x) \]

where the function 'f' gives the shape of the wave.

Step 2: Wave Function at Time t

If the wave moves without changing its shape with velocity v, then at time t, the wave function becomes:

\[ y(x, t) = f(x - vt) \]

This represents a wave traveling in the positive x-direction.

Step 3: Wave Traveling in Negative Direction

For a wave traveling in the negative x-direction:

\[ y(x, t) = f(x + vt) \]

💡 Key Insight

The quantity (x - vt) is called the phase of the wave. The velocity v is called the phase velocity because it represents the speed at which a particular phase of the wave (like a crest or trough) travels through space.

Sample Problems

Sample Problem 1: Wave Function Analysis

A wave pulse is described by the function:

\[ y(x, t) = \frac{6}{2 + (x - 3t)^2} \]

where x and y are in meters and t is in seconds.

(a) What is the velocity of the pulse?

(b) What is the displacement y at x = 2 m and t = 2 s?

Given wave function:
\[ y(x, t) = \frac{6}{2 + (x - 3t)^2} \]
(a) Comparing with the standard form y(x, t) = f(x - vt):
\[ v = 3 \, \text{m/s} \]
The pulse is moving in the positive x-direction with speed 3 m/s.
(b) At x = 2 m and t = 2 s:
\[ y(2, 2) = \frac{6}{2 + (2 - 3 \times 2)^2} \]
\[ = \frac{6}{2 + (2 - 6)^2} \]
\[ = \frac{6}{2 + (-4)^2} \]
\[ = \frac{6}{2 + 16} \]
\[ = \frac{6}{18} \]
\[ = 0.333 \, \text{m} \]

Sinusoidal Waves

📈 What are Sinusoidal Waves?

Sinusoidal waves are periodic waves with a sine or cosine shape. They are particularly important because:

  • They are mathematically simple to analyze
  • Any periodic wave can be expressed as a sum of sinusoidal waves (Fourier analysis)
  • Many physical systems naturally oscillate with sinusoidal motion

Wave Parameters

🌊 Amplitude (A)

The maximum displacement from equilibrium position. Determines the energy carried by the wave.

📏 Wavelength (λ)

The distance between two successive points that are in phase. Measured in meters.

⏱️ Period (T)

The time for one complete cycle of the wave. Measured in seconds.

📊 Frequency (f)

The number of complete cycles per unit time. f = 1/T. Measured in Hertz (Hz).

🚀 Wave Speed (v)

The speed at which the wave propagates through the medium. v = fλ.

📐 Wave Number (k)

Spatial frequency of the wave. k = 2π/λ. Measured in radians per meter.

Mathematical Representation

🧮 Sinusoidal Wave Equation

Step 1: Wave Traveling in Positive x-Direction

\[ y(x, t) = A \sin(kx - \omega t + \phi) \]

where:

  • A = amplitude
  • k = wave number = 2π/λ
  • ω = angular frequency = 2πf = 2π/T
  • φ = phase constant

Step 2: Wave Traveling in Negative x-Direction

\[ y(x, t) = A \sin(kx + \omega t + \phi) \]

Step 3: Relationship Between Parameters

\[ v = f\lambda = \frac{\omega}{k} \]

📈 Sinusoidal Wave Visualization

[Graph: Sinusoidal wave showing amplitude, wavelength, crest, and trough]

A sinusoidal wave exhibits periodic oscillations with characteristic points: crests (maximum positive displacement), troughs (maximum negative displacement), and nodes (points of zero displacement).

Wave Speed and Wave Equation

⚡ Wave Speed Fundamentals

The speed of a mechanical wave depends on the properties of the medium through which it travels. For different types of waves:

  • Transverse waves on a string: v = √(T/μ), where T is tension and μ is linear mass density
  • Longitudinal waves in a solid: v = √(Y/ρ), where Y is Young's modulus and ρ is density
  • Longitudinal waves in a fluid: v = √(B/ρ), where B is bulk modulus and ρ is density
  • Sound waves in air: v = √(γRT/M), where γ is adiabatic index, R is gas constant, T is temperature, and M is molar mass

Dimensional Analysis

🧮 Dimensional Analysis of Wave Speed

Step 1: Identify Relevant Parameters

For transverse waves on a string, the wave speed depends on:

  • Tension (T) - force that restores the string to equilibrium
  • Linear mass density (μ) - inertial property that resists acceleration

Step 2: Dimensional Analysis

\[ [v] = LT^{-1} \]
\[ [T] = MLT^{-2} \]
\[ [\mu] = ML^{-1} \]
\[ v \propto T^a \mu^b \]
\[ LT^{-1} = (MLT^{-2})^a (ML^{-1})^b \]

Step 3: Solve for Exponents

For M: 0 = a + b
For L: 1 = a - b
For T: -1 = -2a
Solving: a = 1/2, b = -1/2
\[ v \propto \sqrt{\frac{T}{\mu}} \]

Mechanical Analysis

🧮 Mechanical Derivation of Wave Speed

Step 1: Consider a Small Element

Consider a small element of the string of length Δx. The net force on this element comes from the tension at both ends.

Step 2: Apply Newton's Second Law

\[ F_{net} = ma \]
\[ T \sin\theta_2 - T \sin\theta_1 = \mu \Delta x \frac{\partial^2 y}{\partial t^2} \]

Step 3: Small Angle Approximation

For small angles: sinθ ≈ tanθ = ∂y/∂x
\[ T\left(\frac{\partial y}{\partial x}\right)_2 - T\left(\frac{\partial y}{\partial x}\right)_1 = \mu \Delta x \frac{\partial^2 y}{\partial t^2} \]

Step 4: Take Limit as Δx → 0

\[ T \frac{\partial^2 y}{\partial x^2} = \mu \frac{\partial^2 y}{\partial t^2} \]
\[ \frac{\partial^2 y}{\partial x^2} = \frac{\mu}{T} \frac{\partial^2 y}{\partial t^2} \]

Step 5: Identify Wave Speed

Comparing with the standard wave equation:
\[ \frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 y}{\partial t^2} \]
\[ v = \sqrt{\frac{T}{\mu}} \]

Derivation of Wave Equation

🧮 General Wave Equation

Step 1: Start with Traveling Wave

\[ y(x, t) = f(x - vt) \]

Step 2: First Partial Derivatives

Let u = x - vt, then:
\[ \frac{\partial y}{\partial x} = \frac{df}{du} \frac{\partial u}{\partial x} = f'(u) \]
\[ \frac{\partial y}{\partial t} = \frac{df}{du} \frac{\partial u}{\partial t} = -v f'(u) \]

Step 3: Second Partial Derivatives

\[ \frac{\partial^2 y}{\partial x^2} = f''(u) \]
\[ \frac{\partial^2 y}{\partial t^2} = v^2 f''(u) \]

Step 4: Wave Equation

Combining the results:
\[ \frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 y}{\partial t^2} \]

This is the one-dimensional wave equation that describes wave motion in many physical systems.

Sample Problem 2: Wave Speed Calculation

A string of length 2 m and mass 0.1 kg is fixed at both ends. The tension in the string is 50 N.

(a) What is the linear mass density of the string?

(b) What is the speed of transverse waves on the string?

(c) What is the fundamental frequency of vibration?

Given:
\[ L = 2 \, \text{m} \]
\[ m = 0.1 \, \text{kg} \]
\[ T = 50 \, \text{N} \]
(a) Linear mass density:
\[ \mu = \frac{m}{L} = \frac{0.1}{2} = 0.05 \, \text{kg/m} \]
(b) Wave speed:
\[ v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{50}{0.05}} = \sqrt{1000} = 31.62 \, \text{m/s} \]
(c) Fundamental frequency:
For a string fixed at both ends, the fundamental wavelength is:
\[ \lambda_1 = 2L = 4 \, \text{m} \]
\[ f_1 = \frac{v}{\lambda_1} = \frac{31.62}{4} = 7.905 \, \text{Hz} \]

Power and Intensity in Wave Motion

💪 Power in Wave Motion

Waves transport energy. The rate at which energy is transported by a wave is called the power of the wave.

For a sinusoidal wave on a string, the average power is given by:

\[ P_{avg} = \frac{1}{2} \mu \omega^2 A^2 v \]

where μ is the linear mass density, ω is the angular frequency, A is the amplitude, and v is the wave speed.

🧮 Derivation of Wave Power

Step 1: Instantaneous Power

\[ P = F \cdot v \]
For a transverse wave on a string:
\[ P = -T \frac{\partial y}{\partial x} \frac{\partial y}{\partial t} \]

Step 2: Sinusoidal Wave

\[ y(x, t) = A \sin(kx - \omega t) \]
\[ \frac{\partial y}{\partial x} = Ak \cos(kx - \omega t) \]
\[ \frac{\partial y}{\partial t} = -A\omega \cos(kx - \omega t) \]

Step 3: Substitute into Power Equation

\[ P = -T [Ak \cos(kx - \omega t)] [-A\omega \cos(kx - \omega t)] \]
\[ = TA^2 k \omega \cos^2(kx - \omega t) \]

Step 4: Average Power

Average of cos²(θ) over one cycle is 1/2:
\[ P_{avg} = \frac{1}{2} TA^2 k \omega \]
Using v = ω/k and v² = T/μ:
\[ P_{avg} = \frac{1}{2} \mu \omega^2 A^2 v \]

💡 Key Insight

The power carried by a wave is proportional to the square of its amplitude and the square of its frequency. This is a general result that applies to many types of waves, including sound and light waves.

Principle of Superposition

➕ Principle of Superposition

"When two or more waves overlap in space, the resultant displacement at any point is the algebraic sum of the displacements of the individual waves at that point."

Mathematically, if y₁(x,t) and y₂(x,t) are two waves, then the resultant wave is:

\[ y(x,t) = y_1(x,t) + y_2(x,t) \]

This principle applies to all linear waves, where the wave equation is linear.

Interference of Waves

🔀 Wave Interference

When two waves of the same frequency travel in the same direction and overlap, they interfere with each other. The resultant wave depends on the phase difference between the two waves.

Type of Interference Phase Difference Path Difference Resultant Amplitude
Constructive φ = 0, 2π, 4π, ... Δx = 0, λ, 2λ, ... A = A₁ + A₂
Destructive φ = π, 3π, 5π, ... Δx = λ/2, 3λ/2, ... A = |A₁ - A₂|
General Case φ Δx = (φ/2π)λ A = √(A₁² + A₂² + 2A₁A₂cosφ)
Sample Problem 3: Wave Interference

Two sinusoidal waves of the same frequency travel in the same direction along a string. Their amplitudes are 4 mm and 3 mm, and they have a phase difference of 60°.

(a) What is the amplitude of the resultant wave?

(b) What should be the phase difference for maximum destructive interference?

Given:
\[ A_1 = 4 \, \text{mm} \]
\[ A_2 = 3 \, \text{mm} \]
\[ \phi = 60^\circ = \frac{\pi}{3} \, \text{radians} \]
(a) Resultant amplitude:
\[ A = \sqrt{A_1^2 + A_2^2 + 2A_1A_2\cos\phi} \]
\[ = \sqrt{4^2 + 3^2 + 2 \times 4 \times 3 \times \cos(60^\circ)} \]
\[ = \sqrt{16 + 9 + 24 \times 0.5} \]
\[ = \sqrt{16 + 9 + 12} \]
\[ = \sqrt{37} \]
\[ = 6.08 \, \text{mm} \]
(b) For maximum destructive interference:
\[ \phi = 180^\circ = \pi \, \text{radians} \]
Resultant amplitude would be |4 - 3| = 1 mm

Standing Waves

📏 Standing Waves

When two waves of the same frequency and amplitude travel in opposite directions and interfere, they form a standing wave. In a standing wave:

  • Certain points called nodes have zero amplitude
  • Points of maximum amplitude are called antinodes
  • The wave pattern does not move through space
  • Energy oscillates between kinetic and potential forms

🧮 Mathematical Description of Standing Waves

Step 1: Two Opposing Waves

\[ y_1(x,t) = A \sin(kx - \omega t) \]
\[ y_2(x,t) = A \sin(kx + \omega t) \]

Step 2: Apply Superposition

\[ y(x,t) = y_1 + y_2 \]
\[ = A \sin(kx - \omega t) + A \sin(kx + \omega t) \]

Step 3: Use Trigonometric Identity

Using sinα + sinβ = 2 sin[(α+β)/2] cos[(α-β)/2]:
\[ y(x,t) = 2A \sin(kx) \cos(\omega t) \]

Step 4: Interpret the Result

\[ y(x,t) = [2A \sin(kx)] \cos(\omega t) \]

This represents a standing wave with:

  • Spatial amplitude: 2A sin(kx)
  • Temporal oscillation: cos(ωt)

📈 Standing Wave Pattern

[Graph: Standing wave showing nodes and antinodes]

In a standing wave, nodes occur where sin(kx) = 0, i.e., at x = 0, λ/2, λ, 3λ/2, ... Antinodes occur where |sin(kx)| = 1, i.e., at x = λ/4, 3λ/4, 5λ/4, ...

Resonance and Natural Frequencies

🎵 Resonance in Wave Systems

Resonance occurs when a system is driven at one of its natural frequencies, resulting in large amplitude oscillations. For waves on strings and in pipes, the natural frequencies are determined by boundary conditions.

System Boundary Conditions Wavelengths Frequencies
String fixed at both ends Nodes at both ends λₙ = 2L/n fₙ = nv/(2L)
String fixed at one end Node at fixed end, antinode at free end λₙ = 4L/(2n-1) fₙ = (2n-1)v/(4L)
Pipe open at both ends Antinodes at both ends λₙ = 2L/n fₙ = nv/(2L)
Pipe closed at one end Node at closed end, antinode at open end λₙ = 4L/(2n-1) fₙ = (2n-1)v/(4L)
Sample Problem 4: Standing Waves on a String

A string of length 1.2 m is fixed at both ends. The wave speed on the string is 48 m/s.

(a) Find the wavelengths of the first three normal modes.

(b) Find the frequencies of these modes.

Given:
\[ L = 1.2 \, \text{m} \]
\[ v = 48 \, \text{m/s} \]
For a string fixed at both ends:
\[ \lambda_n = \frac{2L}{n} \]
\[ f_n = \frac{nv}{2L} \]
(a) Wavelengths:
\[ \lambda_1 = \frac{2 \times 1.2}{1} = 2.4 \, \text{m} \]
\[ \lambda_2 = \frac{2 \times 1.2}{2} = 1.2 \, \text{m} \]
\[ \lambda_3 = \frac{2 \times 1.2}{3} = 0.8 \, \text{m} \]
(b) Frequencies:
\[ f_1 = \frac{1 \times 48}{2 \times 1.2} = 20 \, \text{Hz} \]
\[ f_2 = \frac{2 \times 48}{2 \times 1.2} = 40 \, \text{Hz} \]
\[ f_3 = \frac{3 \times 48}{2 \times 1.2} = 60 \, \text{Hz} \]

🎸 Musical Instruments

String instruments like guitars and violins use standing waves on strings to produce musical notes. The fundamental frequency and harmonics determine the pitch and timbre of the sound.

🎵 Wind Instruments

Wind instruments like flutes and trumpets use standing waves in air columns. The length of the air column and whether ends are open or closed determine the resonant frequencies.

🏗️ Structural Engineering

Understanding wave motion and resonance is crucial in designing buildings and bridges to avoid catastrophic failure due to resonance with earthquakes or wind.

Frequently Asked Questions

❓ What is the difference between transverse and longitudinal waves?

The key difference lies in the direction of particle vibration relative to wave propagation:

  • Transverse waves: Particles vibrate perpendicular to the direction of wave propagation. Examples include waves on strings and electromagnetic waves.
  • Longitudinal waves: Particles vibrate parallel to the direction of wave propagation. Examples include sound waves in air and waves in springs.

Some waves, like water waves, have characteristics of both transverse and longitudinal waves.

❓ Why is the wave speed independent of frequency and wavelength?

The wave speed is determined by the properties of the medium, not by the characteristics of the wave itself. For mechanical waves:

  • On a string: v = √(T/μ) depends only on tension and linear density
  • In a solid: v = √(Y/ρ) depends only on Young's modulus and density
  • In a fluid: v = √(B/ρ) depends only on bulk modulus and density

This means that in a given medium, all waves of the same type travel at the same speed, regardless of their frequency or wavelength. This is why we have the fundamental relationship v = fλ.

❓ What is the physical significance of the wave equation?

The wave equation:

\[ \frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 y}{\partial t^2} \]

has several important physical interpretations:

  • It describes how a disturbance propagates through space with constant speed v
  • It shows that the curvature of the wave in space is proportional to its acceleration in time
  • It's a linear equation, which means the principle of superposition applies
  • Any function of the form f(x ± vt) is a solution, representing waves traveling in the ±x direction

The wave equation appears in many areas of physics, including electromagnetism, quantum mechanics, and general relativity.

📚 Master Waves and Oscillations

Understanding wave motion is fundamental to physics, engineering, and many other scientific disciplines. The concepts covered in this guide form the basis for understanding more advanced topics like quantum mechanics, electromagnetism, and signal processing.

Read More: Physics HRK Notes of Thermodynamics

© House of Physics | HRK Physics: Waves and Oscillations - Wave Motion

Based on Halliday, Resnick, and Krane's "Physics" with additional insights from university physics curriculum

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