diegosouzapw 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