# 15745 - Wave-Motion Phenomena

## Learning outcomes

At the end of the course, the student has acquired education to wave phenomena, their generic properties and some essential mathematical tools. He knows mechanical and electromagnetic waves and is able to solve simple problems.

## Course contents

Linear oscillations:

Free oscillations. Examples of pendulum, mass and springs systems, and of RLC circuits. Damped and forced harmonic oscillator. The phenomenon of resonance. Oscillation analysis with rotary vectors (phasors); Complex field analysis. Elastic and absorbing amplitudes. Linearity of equations of motion and superposition principle.

Fourier Analysis:

Mathematical tools for analyzing oscillatory phenomena. Fourier series and trigonometric series for periodic signal. Continuous Fourier transformation for non-periodic signals. Fourier series and transform calculations for simple signals.

Mechanical waves:

Introduction to wave phenomena. Propagation of physical perturbations. Progressive and stationary waves, scalar and vector waves, longitudinal and transverse waves, plane and spherical waves. Elastic waves in ropes and solids. D'Alembert's equation. Solution of D'Alembert's equation: progressive and regressive waves. Harmonic waves. Wavelength and frequency. Dispersion relation. Study of a progressive wave. Energy and power in a wave. Wave intensity. Impedance of a medium. Energy, reflection and transmission. Wave superposition. Beats. Phase and group speeds. Stationary waves. Rope with two constrained extremes. Ventrals and knots. Normal and harmonic oscillations. Harmonic frequencies. Stationary waves as harmonic series. Musical notes.

Sound propagation in the air. Sound speed. Power, intensity and energy delivered by sound waves. Decibels and human ear. Stationary waves in a gas column. Superposition principle and beats. Harmonic waves in 3 dimensions. Plane waves. D'Alembert's equation in space. Spherical and cylindrical waves. Group and phase velocity in dispersive media. Sound Doppler Effect.

Electromagnetic waves:

From Maxwell equations to the equation of the electromagnetic waves. Transversal character of electromagnetic waves. Speed of light in the vacuum and in media. Impedance. Representation of an electromagnetic wave. Linear, elliptical and circular polarization. Energy, intensity and impulse: Poynting vector, radiation pressure. Accelerated charges. Irradiation from oscillating charges. Spectrum of electromagnetic waves and visible light. Propagation in a dielectric: dispersion and absorption. Light propagation in transparent media. Reflection and refraction. Complex refractive index. Snell's Law. Total reflection and limit angle, evanescent wave. Fresnel formulas. Brewster's angle. Fermat's Principle and Snell's Law. Dispersion in a prism. Propagation of electromagnetic waves in a metal. Wave equation in a metal and corresponding solution.

Interference and diffraction:

Principle of Huygens-Fresnel. Introduction to interference. Interference of light and electromagnetic waves: Young's experience. Distribution of light intensity on the screen. Optical path. Polarization Conditions. Interference with lenses. Interference on thin glasses and on thin wedges. Interference from N coherent light sources. Primary and Secondary amplitudes. Diffraction phenomena: Fraunhofer and Fresnel diffraction. Intensity on a screen. Diffraction from circular holes and objects. Resolution power. Diffraction pattern. Resolution and dispersion power of a grid. Spectroscopy with diffraction pattern.

Optics:

Introduction to geometric optics. Light rays and laws of Descartes. Mirrors and diopters. Cromatism. Objects and images. Paraxial approximation. Properties of concave and convex mirrors: equation of the spherical diopter. Focus and focal distance. Transverse magnification. Flat mirror.  Thin lenses and their properties. Lens Equation. Converging and diverging lenses. Magnification. Aberrations. Optical instruments. The human eye.

------------------------------------------------

Prerequisites

Wave motion phenomena is a course present in the second semester of the second year in order to be able to observe an ideal training course path that, when followed closely, leads to a better overall preparation and a faster achievement of the degree.

For the content of the course, it is recommended to pass Analysis 1 and Mechanics and study the topics of Analysis 2, Thermal Phenomena and Electromagnetism before addressing the Wave motion phenomena exam.

There are several topics in common with the course of Electromagnetism and Optics Laboratory (same semester, same year) that are presented by the teachers with very different goals: one on the foundations, the other on laboratory and experimental aspects. The student will find much benefit in studying the two subjects together.

## Readings/Bibliography

Any book at the university level covering the subjects of the course. As an example:

Halliday and Resnick (+Walker), Fundamental of Physics,  Wiley

Young and Freedman, Universitary Physics (latest edition), Pearson, Addison Wesley.

Exercises and examples of previous written exams can be found on the campus site.

## Teaching methods

Traditional frontal lessons; blackboard and chalk. An overhead projector might be occasionally used.

## Assessment methods

Verification will consist of a written exam followed by an oral exam. For those attending the lessons, it will be possible to divide the script into two intermediate tests: oscillations, Fourier, mechanical waves in the first test and electromagnetic waves, diffraction, interference and optics in the second. The written tests will contain two / three exercises with open numeric answers and two open questions on theorems, demonstrations, phenomena and concepts seen during the course.

Results of the written test will be available through the teacher's alerts on the campus site. Positive outcomes allow for an oral examination and have a limited validity for the current session and the first call of the next session (this limit is clearly indicated in the outcomes).

For the written and oral tests, registration on AlmaEsami is compulsory and must be made no later than 7 days before the exam.

## Teaching tools

All the electronic material shown during lessons will be deposited on campus.

## Office hours

See the website of Mauro Villa

See the website of Lorenzo Rinaldi