Oscillations and Waves

An Introduction, Second Edition

Richard Fitzpatrick

• Publisher: CRC Press
• Year: 2018
• ISBN:
  9781351063081
• Edition: 2
• Topic: Scienze fisiche
• Subtopic: Fisica

• Fully updated throughout and featuring new widgets, animations, and end of chapter exercises to enhance understanding
• Offers complete coverage of advanced topics in waves, such as electromagnetic wave propagation through the ionosphere
• Includes examples from mechanical systems, elastic solids, electronic circuits, optical systems, and other areas

CHAPTER 1

Simple Harmonic Oscillation

1.1 INTRODUCTION

The aim of this chapter is to investigate a particularly straightforward type of motion known as simple harmonic oscillation, and also to introduce the differential equation that governs such motion, which is known as the simple harmonic oscillator equation. We shall discover that simple harmonic oscillation always involves a back and forth flow of energy between two different energy types, with the total energy remaining constant in time. We shall also learn that the linear nature of the simple harmonic oscillator equation greatly facilitates its solution. In this chapter, examples are drawn from simple mechanical and electrical systems.

1.2 MASS ON SPRING

Consider a compact mass m that slides over a frictionless horizontal surface. Suppose that the mass is attached to one end of a light horizontal spring whose other end is anchored in an immovable wall. See Figure 1.1. At time t, let x(t) be the extension of the spring; that is, the difference between the spring’s actual length and its unstretched length. x(t) can also be used as a coordinate to determine the instantaneous horizontal displacement of the mass.

The equilibrium state of the system corresponds to the situation in which the mass is at rest, and the spring is unextended (i.e., x = ẋ = 0, where ̇. ≡ d/dt). In this state, zero horizontal force acts on the mass, and so there is no reason for it to start to move. However, if the system is perturbed from its equilibrium state (i.e., if the mass is displaced horizontally, such that the spring becomes extended) then the mass experiences a horizontal force given by Hooke’s law,

$$f\left(x\right)=-kx.$$ (1.1)

FIGURE 1.1 Mass on a spring.

Here, k > 0 is the so-called force constant of the spring. The negative sign in the preceding expression indicates that f(x) is a so-called restoring force that always acts to return the displacement, x, to its equilibrium value, x = 0 (i.e., if the displacement is positive then the force is negative, and vice versa). Note that the magnitude of the restoring force is directly proportional to the displacement of the mass from its equilibrium position (i.e., | f | ∝ x). Hooke’s law only holds for relatively small spring extensions. Hence, the mass’s displacement cannot be made too large, otherwise Equation (1.1) ceases to be valid. Incidentally, the motion of this particular dynamical ...