Waves and Oscillations: Complete Guide to SHM, Lissajous Figures, Damped & Forced Oscillations | Physics HRK Notes

House of Physics October 05, 2025
Complete Guide to Waves and Oscillations: SHM, Lissajous Figures, Damped and Forced Oscillations
Master Simple Harmonic Motion, Applications, Lissajous Figures with All Cases, Damped and Forced Harmonic Oscillations
Simple Harmonic Motion Lissajous Figures Damped Oscillations Forced Oscillations Physics HRK Reading Time: 30 min

📋 Table of Contents

📜 Historical Background

The study of oscillations and waves has been fundamental to physics since ancient times:

  • Galileo Galilei (1583): Studied pendulum motion and discovered isochronism
  • Robert Hooke (1678): Formulated Hooke's Law for springs
  • Leonhard Euler (1740s): Developed mathematical theory of oscillations
  • Jules Antoine Lissajous (1857): Discovered Lissajous figures
  • Lord Kelvin (late 1800s): Developed theory of damped oscillations

These developments laid the foundation for modern physics, engineering, and technology applications.

Introduction to Oscillations

🔬 What are Oscillations?

Oscillatory motion is a repetitive back-and-forth movement about an equilibrium position. We encounter oscillatory motion daily in various forms:

  • Mechanical oscillations: Swinging pendulum, vibrating guitar string, vibrating atoms in quartz crystals
  • Electromagnetic oscillations: Electrons surging back and forth in circuits for radio/TV signals
  • Acoustic oscillations: Vibrating air molecules transmitting sound waves

Despite their different physical natures, all oscillatory systems share a common mathematical formulation using sine or cosine functions.

📝 Common Features of Oscillatory Systems

All oscillatory systems, regardless of their physical nature, share these characteristics:

  • They have an equilibrium position where net force is zero
  • They experience a restoring force proportional to displacement
  • Their motion can be described by sine or cosine functions
  • They possess inertia that carries them past the equilibrium position
  • They exchange energy between different forms (kinetic and potential)

Simple Harmonic Oscillator (SHO)

🔥 What is a Simple Harmonic Oscillator?

A simple harmonic oscillator is defined as:

"An oscillator whose acceleration is directly proportional to the displacement and acceleration is directed towards the mean position."

The simplest example is a mass-spring system where a mass m is attached to a spring with spring constant k, moving on a frictionless surface.

Derivation of SHO Equation

🧮 Mathematical Derivation

Step 1: Restoring Force

According to Hooke's Law, the restoring force is proportional to displacement:

\[ F \propto -x \]
\[ F = -kx \quad \text{(1)} \]

where k is the spring constant (stiffness factor) and x is displacement from equilibrium.

Step 2: Newton's Second Law

From Newton's second law of motion:

\[ F = ma \]
\[ F = m\ddot{x} \quad \text{(2)} \]

Step 3: Equating Forces

Equating equations (1) and (2):

\[ -kx = m\ddot{x} \]
\[ \ddot{x} = -\frac{k}{m}x \]
\[ \ddot{x} + \frac{k}{m}x = 0 \]
\[ \frac{d^2x}{dt^2} + \frac{k}{m}x = 0 \quad \text{(3)} \]

Step 4: Angular Frequency

Let \( \frac{k}{m} = \omega^2 \), where ω is the angular frequency:

\[ \frac{d^2x}{dt^2} + \omega^2 x = 0 \quad \text{(4)} \]

This is the standard differential equation for simple harmonic motion.

Solution of SHO Equation

🧮 Solving the Differential Equation

Step 1: General Solution

The general solution to equation (4) can be written as:

\[ x = x_m \sin(\omega t + \varphi) \quad \text{(5)} \]

or equivalently:

\[ x = x_m \cos(\omega t + \varphi) \quad \text{(6)} \]

where \( x_m \) is the amplitude and φ is the phase constant.

Step 2: Verification

Let's verify that equation (5) satisfies the differential equation:

\[ x = x_m \sin(\omega t + \varphi) \]
\[ \dot{x} = \omega x_m \cos(\omega t + \varphi) \]
\[ \ddot{x} = -\omega^2 x_m \sin(\omega t + \varphi) = -\omega^2 x \]
\[ \ddot{x} + \omega^2 x = -\omega^2 x + \omega^2 x = 0 \]

This confirms that equation (5) is indeed a solution.

Parameters of Oscillator

📏 Amplitude

The maximum displacement from the mean position:

\[ x_m = \frac{\sqrt{2B}}{\omega} \]

where B is a constant from the energy equation.

⏱️ Time Period

Time taken to complete one vibration:

\[ T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{m}{k}} \]

Depends only on mass and spring constant, not amplitude.

📊 Frequency

Number of vibrations per second:

\[ f = \frac{1}{T} = \frac{\omega}{2\pi} = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \]

Independent of amplitude.

🔄 Phase

The term \( (\omega t + \varphi) \) determines displacement and direction:

\[ \text{Phase} = \omega t + \varphi \]

φ is the initial phase at t = 0.

Energy in Simple Harmonic Motion

⚡ Energy Conservation in SHM

In simple harmonic motion, the total mechanical energy remains constant in the absence of dissipative forces. The energy continuously transforms between kinetic and potential forms.

Potential and Kinetic Energy

🧮 Energy Derivation

Step 1: Potential Energy

Potential energy stored in the spring:

\[ U = \frac{1}{2} kx^2 \]
\[ = \frac{1}{2} k x_m^2 \sin^2(\omega t + \varphi) \quad \text{(7)} \]

Step 2: Kinetic Energy

Kinetic energy of the mass:

\[ K = \frac{1}{2} m\dot{x}^2 \]
\[ = \frac{1}{2} m \omega^2 x_m^2 \cos^2(\omega t + \varphi) \]
\[ = \frac{1}{2} k x_m^2 \cos^2(\omega t + \varphi) \quad \text{(8)} \]

since \( \omega^2 = k/m \).

Total Energy and Conservation

🧮 Total Energy Calculation

Step 1: Total Energy

Total mechanical energy:

\[ E = K + U \]
\[ = \frac{1}{2} k x_m^2 \cos^2(\omega t + \varphi) + \frac{1}{2} k x_m^2 \sin^2(\omega t + \varphi) \]
\[ = \frac{1}{2} k x_m^2 [\cos^2(\omega t + \varphi) + \sin^2(\omega t + \varphi)] \]
\[ = \frac{1}{2} k x_m^2 \quad \text{(9)} \]

Step 2: Energy Conservation

Since \( \frac{1}{2} k x_m^2 \) is constant, the total energy remains constant:

\[ E = \frac{1}{2} k x_m^2 = \text{constant} \]

This confirms energy conservation in SHM.

📈 Energy vs. Displacement Graph

[Graph: Energy vs. Displacement showing potential, kinetic, and total energy curves]

The graph shows how energy transforms between potential and kinetic forms while the total energy remains constant.

Applications of SHM

🔧 Practical Applications

Simple harmonic motion principles apply to various physical systems beyond the basic mass-spring system. These include torsional oscillators, pendulums, and many other mechanical systems.

Torsional Oscillator

⚙️ Torsional Oscillator Setup

Torsional Oscillator
[Diagram: Disk suspended by wire]
Disk with moment of inertia I
↓
Torsion wire with constant κ

Working Principle: A torsional oscillator consists of a disk suspended by a wire. When twisted, the wire exerts a restoring torque:

\[ \tau = -\kappa \theta \]

where κ is the torsion constant and θ is the angular displacement.

Equation of Motion:

\[ I \ddot{\theta} = -\kappa \theta \]
\[ \ddot{\theta} + \frac{\kappa}{I} \theta = 0 \]

Angular Frequency:

\[ \omega = \sqrt{\frac{\kappa}{I}} \]

Time Period:

\[ T = 2\pi \sqrt{\frac{I}{\kappa}} \]

Simple Pendulum

⚙️ Simple Pendulum Setup

Simple Pendulum
[Diagram: Pendulum bob suspended by string]
Bob mass m
↓
String length L

Working Principle: For small angular displacements (θ < 10°), the restoring force is approximately:

\[ F = -mg \sin\theta \approx -mg\theta \]

Equation of Motion:

\[ m L \ddot{\theta} = -mg\theta \]
\[ \ddot{\theta} + \frac{g}{L} \theta = 0 \]

Angular Frequency:

\[ \omega = \sqrt{\frac{g}{L}} \]

Time Period:

\[ T = 2\pi \sqrt{\frac{L}{g}} \]

Important Note: The period is independent of the mass of the bob and the amplitude (for small angles).

Physical Pendulum

⚙️ Physical Pendulum Setup

Physical Pendulum
[Diagram: Rigid body pivoted at point O]
Center of mass at C
Distance OC = h

Working Principle: A physical pendulum is any rigid body suspended from a fixed point that oscillates under gravity.

Restoring Torque:

\[ \tau = -mgh \sin\theta \approx -mgh\theta \]

Equation of Motion:

\[ I \ddot{\theta} = -mgh\theta \]
\[ \ddot{\theta} + \frac{mgh}{I} \theta = 0 \]

Angular Frequency:

\[ \omega = \sqrt{\frac{mgh}{I}} \]

Time Period:

\[ T = 2\pi \sqrt{\frac{I}{mgh}} \]

where I is the moment of inertia about the pivot point.

SHM and Uniform Circular Motion

🔄 Geometric Relationship

There is a fundamental relationship between simple harmonic motion and uniform circular motion. The projection of uniform circular motion onto any diameter executes simple harmonic motion.

🧮 Mathematical Connection

Step 1: Circular Motion

Consider a particle moving with uniform angular velocity ω in a circle of radius A:

\[ \theta = \omega t + \varphi \]

where φ is the initial phase.

Step 2: Projection onto Diameter

The x-coordinate of the particle is:

\[ x = A \cos(\omega t + \varphi) \]

This is exactly the equation for SHM with amplitude A.

Step 3: Velocity and Acceleration

The x-component of velocity:

\[ v_x = -A\omega \sin(\omega t + \varphi) \]

The x-component of acceleration:

\[ a_x = -A\omega^2 \cos(\omega t + \varphi) = -\omega^2 x \]

This confirms the SHM relationship \( a = -\omega^2 x \).

📈 Reference Circle for SHM

[Graph: Reference circle showing projection of uniform circular motion onto x-axis]

The reference circle provides a geometric interpretation of SHM parameters and is particularly useful for understanding phase relationships.

Lissajous Figures

🎨 What are Lissajous Figures?

Lissajous figures are patterns produced when two perpendicular simple harmonic motions with different frequencies and phase differences are combined. They are named after Jules Antoine Lissajous, who studied them in detail.

The parametric equations for Lissajous figures are:

\[ x = A \sin(\omega_x t + \varphi_x) \]
\[ y = B \sin(\omega_y t + \varphi_y) \]

⚙️ Generating Lissajous Figures

Lissajous Figure Generator
[Diagram: Two perpendicular oscillators connected to oscilloscope]
X-input: \( A \sin(\omega_x t + \varphi_x) \)
Y-input: \( B \sin(\omega_y t + \varphi_y) \)

Experimental Setup: Lissajous figures are typically generated using an oscilloscope with two signal generators connected to the X and Y inputs.

Key Parameters:

  • Amplitudes A and B
  • Frequencies ω_x and ω_y
  • Phase difference Δφ = φ_y - φ_x

Applications:

  • Frequency comparison and measurement
  • Phase difference measurement
  • Signal analysis in electronics

Special Cases of Lissajous Figures

Frequency Ratio Phase Difference Resulting Figure
1:1 0° Straight line with positive slope
1:1 90° Circle (if A = B) or ellipse
1:1 180° Straight line with negative slope
1:2 0° Figure-8 pattern
1:2 90° Asymmetric figure-8
2:3 0° More complex pattern with loops

📈 Common Lissajous Figures

[Graph: Collection of common Lissajous figures for different frequency ratios and phase differences]

Lissajous figures provide a visual method for analyzing the relationship between two oscillatory signals.

Damped Harmonic Oscillator

💨 What is Damped Oscillation?

In real physical systems, oscillations are not perpetual due to dissipative forces like friction, air resistance, or electrical resistance. These forces cause the amplitude to decrease over time - this is called damped oscillation.

The equation of motion for a damped harmonic oscillator is:

\[ m\ddot{x} + b\dot{x} + kx = 0 \]

where b is the damping coefficient.

🧮 Solving the Damped Oscillator Equation

Step 1: Standard Form

Divide the equation by m:

\[ \ddot{x} + \frac{b}{m}\dot{x} + \frac{k}{m}x = 0 \]

Let \( \gamma = \frac{b}{2m} \) (damping constant) and \( \omega_0 = \sqrt{\frac{k}{m}} \) (natural frequency):

\[ \ddot{x} + 2\gamma\dot{x} + \omega_0^2 x = 0 \quad \text{(10)} \]

Step 2: Trial Solution

Assume a solution of the form:

\[ x = e^{rt} \]

Substitute into equation (10):

\[ r^2 e^{rt} + 2\gamma r e^{rt} + \omega_0^2 e^{rt} = 0 \]
\[ r^2 + 2\gamma r + \omega_0^2 = 0 \quad \text{(11)} \]

Step 3: Characteristic Equation

Solve the quadratic equation (11):

\[ r = -\gamma \pm \sqrt{\gamma^2 - \omega_0^2} \]

Parameters of Damped Oscillator

📉 Underdamped Case

When \( \gamma < \omega_0 \):

\[ x = A e^{-\gamma t} \cos(\omega_d t + \varphi) \]

where \( \omega_d = \sqrt{\omega_0^2 - \gamma^2} \) is the damped frequency.

Oscillations with exponentially decaying amplitude.

⚖️ Critically Damped Case

When \( \gamma = \omega_0 \):

\[ x = (A + Bt) e^{-\gamma t} \]

Fastest return to equilibrium without oscillation.

Used in car suspensions and door closers.

📈 Overdamped Case

When \( \gamma > \omega_0 \):

\[ x = A e^{r_1 t} + B e^{r_2 t} \]

where \( r_{1,2} = -\gamma \pm \sqrt{\gamma^2 - \omega_0^2} \)

Slow return to equilibrium without oscillation.

📈 Damped Oscillation Types

[Graph: Comparison of underdamped, critically damped, and overdamped oscillations]

The graph shows how different damping levels affect the oscillator's return to equilibrium.

Energy of Damped Oscillator

🧮 Energy Dissipation

Step 1: Energy Equation

For an underdamped oscillator, the total energy decreases exponentially:

\[ E(t) = \frac{1}{2} k A^2 e^{-2\gamma t} \]

where A is the initial amplitude.

Step 2: Quality Factor

The quality factor Q measures how underdamped an oscillator is:

\[ Q = \frac{\omega_0}{2\gamma} \]

Higher Q means slower energy dissipation.

Step 3: Logarithmic Decrement

The logarithmic decrement δ measures the rate of amplitude decay:

\[ \delta = \ln\left(\frac{x_n}{x_{n+1}}\right) = \gamma T_d \]

where \( T_d = \frac{2\pi}{\omega_d} \) is the damped period.

Forced Harmonic Oscillator

🎯 What is Forced Oscillation?

When an external periodic force drives an oscillator, it's called forced oscillation. The equation of motion becomes:

\[ m\ddot{x} + b\dot{x} + kx = F_0 \cos(\omega t) \]

where \( F_0 \) is the amplitude of the driving force and ω is the driving frequency.

🧮 Solving the Forced Oscillator Equation

Step 1: Standard Form

Divide by m:

\[ \ddot{x} + 2\gamma\dot{x} + \omega_0^2 x = \frac{F_0}{m} \cos(\omega t) \]

Step 2: Steady-State Solution

The steady-state solution (after transients die out) is:

\[ x = A \cos(\omega t - \delta) \]

where A is the amplitude and δ is the phase difference between driving force and displacement.

Step 3: Amplitude and Phase

The amplitude A is given by:

\[ A = \frac{F_0/m}{\sqrt{(\omega_0^2 - \omega^2)^2 + (2\gamma\omega)^2}} \]

The phase difference δ is:

\[ \tan\delta = \frac{2\gamma\omega}{\omega_0^2 - \omega^2} \]

Resonance Phenomenon

🎵 What is Resonance?

Resonance occurs when the driving frequency matches the natural frequency of the system, causing a dramatic increase in amplitude. This phenomenon is observed in many physical systems.

🧮 Resonance Conditions

Step 1: Resonance Frequency

The amplitude A is maximum when the denominator is minimum:

\[ \frac{d}{d\omega}[(\omega_0^2 - \omega^2)^2 + (2\gamma\omega)^2] = 0 \]
\[ -4\omega(\omega_0^2 - \omega^2) + 8\gamma^2\omega = 0 \]
\[ \omega_r = \sqrt{\omega_0^2 - 2\gamma^2} \]

This is the resonance frequency.

Step 2: Maximum Amplitude

At resonance, the maximum amplitude is:

\[ A_{max} = \frac{F_0/m}{2\gamma\sqrt{\omega_0^2 - \gamma^2}} \]

For light damping (\( \gamma \ll \omega_0 \)), this simplifies to:

\[ A_{max} \approx \frac{F_0/m}{2\gamma\omega_0} \]

Step 3: Phase at Resonance

At resonance (\( \omega = \omega_r \)), the phase difference is:

\[ \tan\delta = \frac{2\gamma\omega_r}{\omega_0^2 - \omega_r^2} \]

For light damping, \( \omega_r \approx \omega_0 \), so:

\[ \tan\delta \approx \frac{2\gamma\omega_0}{2\gamma^2} = \frac{\omega_0}{\gamma} \]

Since \( \omega_0 \gg \gamma \), \( \delta \approx \frac{\pi}{2} \).

📈 Resonance Curves

[Graph: Amplitude vs. frequency for different damping levels, showing resonance peaks]

The graph shows how the resonance peak becomes sharper and higher as damping decreases.

🎵 Musical Instruments

String and wind instruments use resonance to amplify sound. The body of a guitar or violin resonates with the vibrating strings to produce louder sound.

🌉 Structural Engineering

Buildings and bridges are designed to avoid resonance with earthquakes or wind to prevent catastrophic failure. The Tacoma Narrows Bridge collapse is a famous example of resonance disaster.

📡 Radio and TV Tuners

LC circuits in radios and TVs use resonance to select specific frequencies from the many signals received by the antenna.

🔬 Magnetic Resonance Imaging (MRI)

MRI machines use nuclear magnetic resonance to create detailed images of the human body by detecting the resonance of hydrogen atoms in magnetic fields.

Frequently Asked Questions

❓ Why is the period of a simple pendulum independent of mass?

The period of a simple pendulum is given by \( T = 2\pi\sqrt{\frac{L}{g}} \). Notice that the mass m does not appear in this equation. This is because:

  • The restoring force is \( F = -mg\sin\theta \approx -mg\theta \)
  • The inertia is proportional to mass m
  • When we write Newton's second law \( F = ma \), the mass cancels out

This means that heavier and lighter pendulums of the same length have the same period, as demonstrated by Galileo's experiments.

❓ What is the difference between angular frequency and frequency?

Angular frequency (ω) and frequency (f) are related but distinct concepts:

  • Frequency (f): Number of complete oscillations per second, measured in Hertz (Hz)
  • Angular frequency (ω): Rate of change of phase angle, measured in radians per second

They are related by:

\[ \omega = 2\pi f \]

For example, if a pendulum completes 2 oscillations per second:

\[ f = 2 \, \text{Hz}, \quad \omega = 4\pi \, \text{rad/s} \]

Angular frequency is more convenient for mathematical analysis of oscillatory motion.

❓ Why does critical damping provide the fastest return to equilibrium?

Critical damping provides the fastest return to equilibrium without oscillation because:

  • In underdamped systems, the oscillator overshoots the equilibrium multiple times before settling
  • In overdamped systems, the return is slowed by excessive damping
  • Critical damping represents the perfect balance where the system returns to equilibrium as quickly as possible without overshooting

Mathematically, for critical damping, the characteristic equation has a repeated root, leading to a solution of the form \( x = (A + Bt)e^{-\gamma t} \). This represents the fastest possible decay without oscillation.

This property is exploited in many engineering applications like car suspensions and door closers, where we want the system to return to equilibrium quickly without oscillating.

❓ What happens at resonance in a forced oscillator?

At resonance in a forced oscillator, several important phenomena occur:

  • Maximum amplitude: The amplitude of oscillation reaches its maximum value
  • Phase relationship: The displacement lags the driving force by 90° (π/2 radians)
  • Energy transfer: Maximum energy is transferred from the driving force to the oscillator
  • Velocity in phase with force: The velocity of the oscillator is in phase with the driving force, ensuring optimal energy transfer

The sharpness of the resonance peak depends on the damping: lighter damping produces a sharper, higher resonance peak.

Resonance can be beneficial (as in musical instruments) or destructive (as in collapsing bridges), depending on the context.

📚 Master Waves and Oscillations

Understanding oscillations is fundamental to physics, engineering, and many technological applications. From the precise timekeeping of pendulum clocks to the sophisticated imaging of MRI machines, oscillatory principles underpin much of modern technology.

Continue your exploration of physics with our comprehensive guides on waves, electromagnetism, and quantum mechanics.

Explore More Physics Topics

© House of Physics | HRK Physics: Waves and Oscillations

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

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