Waves

Summary

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

Key Primitives

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).

• modeling back-and-forth motion of a pendulum or spring
• any system with a linear restoring force proportional to displacement

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).

• determining how many oscillations occur per second
• relating pitch of a sound to its physical frequency

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.

• describing a sinusoidal disturbance propagating through a medium
• modeling light, sound, or any traveling periodic signal

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) + ...

• adding together multiple wave sources to find the combined effect
• analyzing interference patterns from multiple coherent sources

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.

• modeling wave propagation in strings, air columns, or electromagnetic fields
• predicting how disturbances travel through a medium

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.

• determining the frequency content of a vibrating string or air column
• understanding why different instruments sound different even playing the same note

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.

• finding the lowest pitch produced by a vibrating string or air column
• tuning musical instruments to a specific pitch

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).

• solving the wave equation on a bounded domain with fixed or free boundary conditions
• finding the natural vibration modes of a physical system

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).

• analyzing vibration patterns on strings, membranes, or in cavities
• determining where resonant systems have maximum and minimum displacement

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.

• measuring the time for one complete oscillation cycle
• determining how long before a periodic system returns to its initial state

Composition Patterns

• Simple Harmonic Motion + waves-frequency -> Complete SHM description with temporal period and spatial amplitude (parallel)
• Frequency + waves-wavelength -> Wave speed: v = f * lambda, connecting temporal and spatial periodicity (parallel)
• Period + waves-frequency -> Complete temporal characterization: T = 1/f, f = 1/T (parallel)
• Angular Frequency + perception-radian-measure -> Natural sinusoidal parameterization: sin(omegat) cycles at frequency f = omega/(2pi) (nested)
• Wave Function + waves-wave-number -> Complete space-time wave description: y(x,t) = Asin(kx - omegat) (parallel)
• Wavelength + waves-frequency -> Wave speed relation: v = lambda * f (parallel)
• Sum-to-Product Formulas + waves-superposition-principle -> Analysis of combined waves: sum of two sinusoids reveals beat and carrier frequencies (sequential)
• Product-to-Sum Formulas + waves-sum-to-product -> Complete toolkit for converting between product and sum forms of trigonometric expressions (parallel)
• Superposition Principle + waves-constructive-destructive-interference -> Complete interference analysis: constructive when in-phase, destructive when out-of-phase (sequential)
• Phasor Representation + waves-superposition-principle -> Adding sinusoids by vector addition of their phasors (sequential)

Cross-Domain Links

• perception: Compatible domain for composition and cross-referencing
• change: Compatible domain for composition and cross-referencing
• reality: Compatible domain for composition and cross-referencing
• mapping: Compatible domain for composition and cross-referencing
• synthesis: Compatible domain for composition and cross-referencing

Activation Patterns

• wave
• frequency
• harmonic
• oscillation
• period
• amplitude
• resonance
• standing wave
• Fourier
• spectrum