Our review
This skill helps analyze and understand periodic phenomena and frequency decomposition, from simple oscillations to Fourier transforms.
Strengths
- Covers fundamental concepts like simple harmonic motion, superposition, and the wave equation.
- Enables harmonic analysis and determination of resonant frequencies in physical systems.
- Includes rigorous mathematical techniques like separation of variables for solving the wave equation.
Limitations
- Requires a strong mathematical background (differential equations, trigonometry).
- Primarily limited to one-dimensional, linear systems.
- Does not provide built-in numerical implementations or visualization tools.
Use this skill when you need to model or explain wave phenomena, resonances, or spectral analysis.
Avoid this skill if you need a quick empirical formula without theoretical underpinning, or if the problem involves nonlinear waves or complex media.
Security analysis
SafeThis skill is a collection of definitions and concepts about waves and harmonic motion. It does not trigger any executable actions, and the allowed tools are read-only. There is no risk of data exfiltration, destruction, or unsafe operations.
No concerns found
Examples
Explain the superposition principle for two sinusoidal waves of different frequencies and amplitudes traveling in the same direction.What is the Fourier series representation of a square wave with amplitude 1 and period T? Derive the coefficients.Derive the one-dimensional wave equation for a vibrating string fixed at both ends, and find the fundamental frequency in terms of string length, tension, and linear density.name: mfe-waves description: "Periodic phenomena and frequency analysis. How repetition creates structure — from simple oscillation to Fourier decomposition." user-invocable: false allowed-tools: Read Grep Glob metadata: extensions: gsd-skill-creator: version: 1 createdAt: "2026-02-26" triggers: intents: - "wave" - "frequency" - "harmonic" - "oscillation" - "period" - "amplitude" - "resonance" - "standing wave" - "Fourier" - "spectrum" contexts: - "mathematical problem solving" - "math reasoning"
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
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