Mechanical and Electromagnetic Vibrations and Waves

Chapter 1

Free Oscillations

In this chapter we introduce the basic notions of free oscillations. Starting with a study of the differential equation governing the undamped vibrations, its general solution and its trigonometric, complex and phasor representations, we then progress to the equation of damped oscillations and its solutions. We analyze some simple oscillating systems with one degree of freedom by emphasizing the notion of energy which, in modern physics, is considered to be a more fundamental quantity than forces. We generalize these results to systems undergoing small displacements or variations of the state “back-and-forth” near an equilibrium position. Afterwards, we analyze systems with two or several degrees of freedom.

1.1. Oscillations and waves, period and frequency

Vibrations or oscillations are motions or changes in the state of physical systems back-and-forth on both sides of an equilibrium position that are repeated more or less regularly in time. Waves are vibrations that propagate from one region to another. We encounter vibratory and wave phenomena in almost all branches of physics: mechanics, geophysics, electromagnetism, optics, quantum physics, etc. We consider in this book two kinds of vibrations: mechanical vibrations (of a pendulum, a string, etc.) and electromagnetic vibrations (of electric circuits, radio waves, etc.).

Vibrations are free if, after an initial excitation, the system oscillates subject to its own internal forces but no-external ...