Slightly Disturbed A Mathematical Approach to Oscillations and Waves Brooks Thomas Lafayette College Second Edition 2017 Contents 1 Simple Harmonic Motion 4 1.1 Equilibrium, Restoring Forces, and Periodic Motion . ........... 4 1.2 Simple Harmonic Oscillator . ...... 5 1.3 Initial Conditions . ....... 8 1.4 Relation to Cirular Motion . ...... 9 1.5 Simple Harmonic Oscillators in Disguise . ....... 9 1.6 StateSpace ....................................... ....... 11 1.7 Energy in the Harmonic Oscillator . ........ 12 2 Simple Harmonic Motion 16 2.1 Motivational Example: The Motion of a Simple Pendulum . .......... 16 2.2 Approximating Functions: Taylor Series . ........... 19 2.3 Taylor Series: Applications . ........ 20 2.4 TestsofConvergence............................... .......... 21 2.5 Remainders ........................................ ...... 22 2.6 TheHarmonicApproximation. ........ 23 2.7 Applications of the Harmonic Approximation . .......... 26 3 Complex Variables 29 3.1 ComplexNumbers .................................... ...... 29 3.2 TheComplexPlane .................................... ..... 31 3.3 Complex Variables and the Simple Harmonic Oscillator . ......... 32 3.4 Where Making Things Complex Makes Them Simple: AC Circuits . ......... 33 3.5 ComplexImpedances................................. ........ 35 4 Introduction to Differential Equations 37 4.1 DifferentialEquations ................................ ........ 37 4.2 SeparationofVariables............................... ......... 39 4.3 First-Order Linear Differential Equations . ............ 43 4.4 General Solutions from Solutions to the Complementary Equation ............... 48 5 Second-Order Differential Equations and Damped Oscillations 51 5.1 Second-Order Homogeneous Linear Differential Equations . ................ 51 5.2 ReductionofOrder.................................. ........ 52 5.3 Finding Roots: Equations with Constant Coefficients . ............. 53 5.4 The Damped Harmonic Oscillator . ....... 54 5.5 Underdamping, Overdamping, and Critical Damping . ............ 54 5.5.1 UnderdampedMotion ................................ .... 55 5.5.2 OverdampedMotion................................. .... 58 5.5.3 Critically-Damped Motion . .... 60 5.6 TheEnergeticsofDampedHarmonicMotion . .......... 60 5.7 The State-Space Picture of Damped Harmonic Motion . ............. 65 5.8 FrictionalDamping................................... ....... 65 2 CONTENTS 3 6 Driven Oscillations and Resonance 71 6.1 Second-Order Inomogeneous Linear Differential Equations . ................. 71 6.2 TheMethodofUndeterminedCoefficients . ........... 72 6.3 Solving the Driven Harmonic Oscillator Equation . .......... 75 6.4 Resonance........................................ ....... 76 6.5 Energy in a Driven-Oscillator System . .......... 78 6.6 Complexification ..................................... ...... 80 6.7 The Principle of Superposition . ........ 84 7 Fourier Analysis 88 7.1 Superposition and the Decomposition of Functions . ............. 88 7.2 FourierSeries...................................... ....... 88 7.3 OrthogonalFunctions ............................... ......... 91 7.4 Determining the Fourier Coefficients . .......... 93 7.5 Fourier Series in Terms of Complex Exponentials . ........... 97 7.6 Solving Differential Equations with Fourier Series . ............ 98 7.7 FourierTransforms................................. ......... 100 8 Impulses and Green’s Functions 107 8.1 IntroductionandMotivation. .......... 107 8.2 TheDiracDeltaFunction ............................... ....... 107 8.3 Impulses.......................................... ...... 110 8.4 Green’sFunctions ................................... ....... 112 8.5 Determining the Green’s Functions . .......... 116 9 Coupled Oscillations and Linear Systems of Equations 120 9.1 SystemsofDifferentialEquations . ........... 120 9.2 Our First Coupled System: Two Oscillators . .......... 120 9.3 Our Second Coupled System: Charged Particle in a Magnetic Field . ............. 124 9.4 VectorSpaces..................................... ........ 126 9.5 TheInnerProductandInnerProductSpaces . .............. 131 9.6 Matrices.......................................... ...... 134 9.7 OperationsonMatrices ............................... ........ 138 9.8 MatricesandTransformationsofVectors. .............. 141 9.9 EigenvaluesandEigenvectors . .......... 144 9.10 Coupled Differential Equations as Matrix Equations . ............. 147 9.11 From Oscillations to Waves . ........ 152 Index 158 Chapter 1 Simple Harmonic Motion The Physics: Stable and unstable equilibrium, springs and simple harmonic motion, state space, LC • circuits, the energetics of oscillation The Math: Differential equations, initial conditions • . 1.1 Equilibrium, Restoring Forces, and Periodic Motion Periodic motion — motion that repeats itself in finite time — is ubiquitous in nature. Objects shake back and forth when we bump into them; automobile engines, cell phones, drum heads, hummingbirds’ wings, and audio speakers vibrate; rocking chairs rock; pendulums swing back and forth; and coffee sloshes back and forth in a cup after that cup is set down on a table. The electrical current used to power anything you might plug into a wall oscillates back and forth in a sinusoidal pattern 60 times each second. Nearly every medium of communication over distance relies on some sort of wave to transmit information, from direct speech (sound waves) to radio, television, and cell phones (electromagnetic waves). Why is periodic motion such a common phenomenon? The reason is that oscillations are what generically happens when a system in a stable equilibrium state gets disturbed a little bit. In order to clarify precisely what this statement means, however, we’re going to have to go into a little bit more detail about what we mean by “stable,” “equilibrium,” and “disturbed.” First, let’s review what we mean by equilibrium. A rigid body is said to be in mechanical equilibrium if it is not accelerating. The acceleration of such a body is related to the force acting on it by Newton’s Second Law. For example, in a one-dimensional system, this relationship takes the form d2x F = ma = m , (1.1) dt2 where x is the position of the body, m is its mass, and t is time. A body in mechanical equilibrium is therefore one on which the net force is zero.1 Two examples of systems in mechanical equilibrium are illustrated in Fig. 1.1. The diagram on the left shows a ball at rest at the bottom of a valley, while the diagram on the right shows a similar ball at rest at the top of a hill. In both of these situations, the gravitational force F = mg and the normal force g − FN = mg are equal and opposite at the point where the ball is located, so the net force acting on the ball is zero. Now let’s review what we mean by “disturbance” and “stable.” Nature is full of effects that disturb physical systems away from equilibrium. For example, in each of the situations illustrated in Fig. 1.1, a gust of wind might blow, an insect might land on the ball, a leaf might fall on it, raindrops might strike it, the ground might shift a little bit, someone or something might jostle it accidentally, and so on. However, the 1In the generalization of this principle to motion in more than one dimension, in which case rotation is possible, the net torque on the body must also vanish. 4 1.2. SIMPLE HARMONIC OSCILLATOR 5 Figure 1.1: Examples of stable and unstable equilibrium. The diagram on the left shows a stable equilibrium state. In this case, the net force which acts on the ball when it moves away from its equilibrium position acts to drive it back toward that equilibrium position. The diagram on the right shows an unstable equilibrium state. In this case, the net force which acts on the ball when it moves away from its equilibrium position serves to drive it farther away from that equilibrium position. two systems in this figure respond to these disturbances in very different ways. In the diagram on the left, the net force which acts on the ball when it is displaced slightly from its equilibrium position always serves to accelerate it back toward that equilibrium position. A force that serves to “correct” for any departure from equilibrium in this way is called a restoring force, an equilibrium state which is robust against small disturbances because of such “corrections” is called stable.2 By contrast, in the diagram on the right, the force which acts on the ball when it is displaced from its equilibrium position accelerates it in a direction further away from that position. This precarious situation is an example of an system with an unstable equilibrium state. While a restoring force in a system with a stable equilibrium state acts drive the system back toward that equilibrium state, that doesn’t mean that it causes the system to return promptly to that equilibrium state and stop. For example, when the ball in the left diagram of Fig. 1.1 rolls back toward its equilibrium position at the bottom of the valley under the influence of the restoring force, it acquires momentum in the process. As a result, the ball will “overshoot” and roll right past the equilibrium point a the bottom of the valley because of inertia — i.e., the tendency of an object to oppose change in its motion. It will continue rolling up the other side of the valley until the restoring force overcomes that inertial tendency and drives it back down toward the equilibrium position again, and the process repeats itself. This interplay between the action of the restoring