**As Taught in:**

2019/2020

**Level: **

Undergraduate

**Learning Resource Types:**

=> Problem Sets

=> Notes

=> Reading Resources

**Course Overview:**

Vibrations and waves are everywhere. If human take any system and disturb it from a stable equilibrium, the resultant motion will be waves and vibrations. Think of a guitar string—pluck the string, and it vibrates. The sound waves generated make their way to our ears, and we hear the string’s sound. Our eyes see what’s happening because they receive the electromagnetic waves of the light reflected from the guitar string, so that we can recognize the beautiful sinusoidal waves on the string. In fact, without vibrations and waves, human could not recognize the universe around us at all!

Indicate, what are vibrations? Many physical systems vibrate or oscillate when something disrupts their equilibrium. Think of what happens when you pull a mass attached to a spring or push a pendulum in a certain direction. The mass bounces up and down. The pendulum swings back and forth. But even these simple phenomena can respond in complicated and counterintuitive ways when subjected to forces known in physics as damping and driving. Combining several pendulums or springs can lead to even more unexpected motions. In this course you will see how physics can describe these motions mathematically.

The amazing thing is that human can describe many fascinating phenomena arising from very different physical systems with mathematics. This course will provide learner with the concepts and mathematical tools necessary to understand and explain a broad range of vibrations and waves. Learners will learn that waves come from many interconnected objects when they are vibrating together. We will discuss many of these phenomena, along with standards topic like Mechanical Vibrations and Waves, Simple Harmonic Motion, Normal Modes, Forced Vibrations, Resonance, Coupled Oscillations, Driven Coupled Oscillators, Vibrations of Continuous Systems, Reflection and Refraction, Phase and Group Velocity, Wave Solutions to Maxwell’s Equations, Polarizations, Rayleigh Scattering, Snell’s Law, Fresnel Equations, Interference, Thin Films, Huygens’s Principle, Fraunhofer Diffraction, and Gratings.

Then a variety of related interesting topics, including Sound Waves, Electromagnetic Waves, Optics, Musical Instruments, Skies, Glories, Haloes, Rainbow, Glassbow, Doppler Effect, Binary Stars, Neutron Stars, and Gravitational Waves.

**Prerequisites:**

Math – Multivariable Calculus

Physics – Electricity and Magnetism

In this course we also rely on fundamental mathematics principle taught in *Single Variable Calculus* and some aspect of *Linear Algebra*.

**Textbook:**

Georgi, Howard. The Physics of Waves. Benjamin Cummings, 1992. ISBN: 978136656210

French, A. P. Vibrations and Waves. New York: W.W. Norton, 1980. ISBN: 9780393099362.

Bekefi, George, and Alan H. Barrett. Electromagnetic Vibrations, Waves, and Radiation. Cambridge, MA: MIT Press, September 15, 1977. ISBN: 9780262520478.

**Course Goals**

This course has two complementary goals.

The first is to provide learner with the concepts and mathematical tools necessary to understand and explain a broad range of vibrations and waves. That will allow learner to gain a deeper appreciation for the true nature and beauty of phenomena like music and rainbows, which all of us observe or experience every day.

The second goal is to provide learner with the skill of using a broad range of techniques that can greatly simplify the analysis and solutions of complex systems. Those techniques include complex numbers, combinations of oscillatory and exponentially decaying functions, resonance, normal modes, Cramer’s rule for solving several equations in several variables, boundary conditions, general wave equations, Fourier decomposition, dispersive and non-dispersive media, phase and group velocities, sound cavities, and wave guides, polarization, Doppler Effect, reflection and refraction, Fresnel’s equation for transmission and reflection, total internal reflection, constructive an destructive interference, and diffraction.

List of the Major topic that will be covered are:

- Mechanical Vibrations and Waves
- Electromagnetic Waves
- Optics

These readings are suggested before the course and experiments:

Chapter 1 – Harmonic Oscillation >> Periodic Oscillations, Harmonic Oscillators.

Chapter 2 – Forced Oscillation and Resonance >> Damped Free Oscillators, Driven Oscillators, Transient Phenomena, Resonance.

Chapter 3 – Normal Modes >> Coupled Oscillators, Normal Modes, Beat Phenomena

Chapter 4 – Symmetries & Chapter 5 – Waves (till sections 5.2) >> Driven Oscillators, Resonance, Symmetry, Infinite Number of Coupled Oscillators.

Chapter 5 – Wave (Section 5.3 to end) – Chapter 6 – Continuum Limit and Fourier Series (Section 6.1) >> Translation Symmetry, Wave Equation, Standing Waves, Fourier Series.

Chapter 6 – Continuum Limit and Fourier Seres (section 6.2 to end), Chapter 7 – Longitudinal Oscillations and Sound, & Chapter 8 – Traveling Waves (until Section 8.2) >> Traveling Waves & Sound Waves

Chapter 8 – Traveling Waves (Section 8.3 to end) & Chapter 9 – The Boundary at Infinity (till Section 9.2) >> Maxwell’s Equation, Electromagnetic Waves, Dispersive Medium, Phase Velocity, Group Velocity

Chapter 10 – Signals and Fourier Analysis >> Fourier Transform, AM Radio, Uncertainty Principle, & 2D Waves

Chapter 11 – Two and Three Dimensions & Chapter 12 – Polarization (till section 12.3) >> 2D and 3D Waves, Snell’s Law, Polarization, and Polarizer.

Chapter 12 – Polarization (section 12.4 till end) >> Wave Plates, Radiation, and Waves in Medium.

Chapter 13 – Interference and Diffraction >> Interference, Soap Bubble, Phased Radar, Single Electron Interference, Diffraction, and Resolution.

Additional Notes >> Quantum Waves and Gravitational Waves.

The desktop experiments were demonstrated during the class sessions, with topics:

- Air Cart Between Springs
- Mass on a Spring
- Two Pendulums with Different Amplitudes
- Oscillating Steel Ball on a Track
- Physical Pendulum
- Driven Torsional Balance Oscillator
- Driven Cart on Air Track
- Driven Mechanical Oscillator
- Double Pendulum
- Two Rigid Pendulums Coupled with a Spring
- Two Pendulums Coupled with a Rod
- Wave Beats
- Coupled Tuning Forks
- Air Carts
- Bell Labs Wave Machine
- Vibrating Spring
- Rijke Tube
- Helium Balloon
- Standing Sound Waves in a Glass Tube
- Speaker Driven Figures
- Oscillating Soap Film
- Internal reflection
- Fiber Optic Bundle
- Polarization of Microwaves
- Polarizing Filters and Light
- Polarization in a Sugar
- Polarization of Radio Waves
- Brewster’s Angle
- Reflection off Soap Bubbles
- Reflection off a Soap Film
- Ripple Tank
- Moiré Pattern
- Single Slit Diffraction with a Laser.

These Problem Solving Slides provide step-by-step solutions to typical problems solved in an undergraduate course on Vibrations and Waves. Along with each slide, educator has included two sample problems for you to solve. Answers and hints are provided, but not full solutions.

- Simple Harmonic Motion and Introduction to Problem Solving
- Harmonic Oscillators with Damping
- Driven Harmonic Oscillators
- Coupled Oscillators without Damping
- Traveling Waves without Damping
- Standing Waves
- Electromagnetic Waves in a Vacuum
- Accelerated Charges Radiating Electromagnetic Waves
- Interference of Electromagnetic Waves

**Note**: These slides were originally produced as part of a physics course that is no longer available in mostly online learning.