Periodic phenomena and frequency analysis. How repetition creates structure — from simple oscillation to Fourier decomposition.
Waves (Part II: Hearing) Chapters: 4, 5, 6, 7 Plane Position: (-0.4, 0) radius 0.4 Primitives: 50
Periodic phenomena and frequency analysis. How repetition creates structure — from simple oscillation to Fourier decomposition.
Key Concepts: Simple Harmonic Motion, Frequency, Wave Function, Superposition Principle, Wave Equation
Simple Harmonic Motion (definition): Simple harmonic motion (SHM) is periodic motion where the restoring force is proportional to displacement: F = -kx. The solution is x(t) = Acos(omegat + phi) where omega = sqrt(k/m).
Frequency (definition): The frequency f of a periodic phenomenon is the number of complete cycles per unit time. f = 1/T where T is the period. Measured in hertz (Hz = cycles/second).
Wave Function (definition): The general sinusoidal wave function is y(x,t) = Asin(kx - omegat + phi), describing a traveling wave with amplitude A, wave number k, angular frequency omega, and phase offset phi.
Superposition Principle (theorem): For linear systems, the net response at a given point caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. For waves: y_total(x,t) = y_1(x,t) + y_2(x,t) + ...
Wave Equation (definition): The one-dimensional wave equation is the second-order partial differential equation: d^2u/dt^2 = c^2 * d^2u/dx^2, where c is the wave propagation speed and u(x,t) is the displacement field.
Harmonic Series (definition): The harmonic series of a fundamental frequency f_1 consists of integer multiples: f_n = n * f_1 for n = 1, 2, 3, ... The nth harmonic has frequency n times the fundamental.
Fundamental Frequency (definition): The fundamental frequency f_1 is the lowest resonant frequency of a vibrating system. For a string of length L with wave speed v: f_1 = v/(2L). All higher harmonics are integer multiples of f_1.
Separation of Variables for Waves (technique): Separation of variables assumes the solution to a PDE is a product of functions of individual variables: u(x,t) = X(x)T(t). Substituting into the wave equation and dividing by XT yields two ODEs: X''/X = T''/(c^2*T) = -lambda (separation constant).
Standing Wave (definition): A standing wave is a wave pattern that does not propagate through space but oscillates in place. It is formed by the superposition of two identical waves traveling in opposite directions: 2A*sin(kx)cos(omegat).
Period (definition): The period T of a periodic function f is the smallest positive value such that f(t + T) = f(t) for all t. T is the duration of one complete cycle.