--- title: "The Wave Equation and Its Characteristics" description: "Along two families of lines the wave equation reduces to an ordinary derivative, and the solution reads as two fixed shapes travelling in opposite directions. A discontinuity in the data is carried al" canonical: https://lightmysky.com/learn/mathematics/the-wave-equation-and-its-characteristics-mt_-_QCiBWg1w source: https://lightmysky.com/learn/mathematics/the-wave-equation-and-its-characteristics-mt_-_QCiBWg1w.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # The Wave Equation and Its Characteristics Along two families of lines the wave equation reduces to an ordinary derivative, and the solution reads as two fixed shapes travelling in opposite directions. A discontinuity in the data is carried along, not smoothed away. Subject: Mathematics · Area: Differential Equations · Ages 22 to 23 Page: https://lightmysky.com/learn/mathematics/the-wave-equation-and-its-characteristics-mt_-_QCiBWg1w ## Ready when they can - Draw the characteristics through a point and mark its domain of dependence - Apply d'Alembert's formula to given initial displacement and velocity - Contrast a travelling kink here with the same kink in the heat equation ## Lesson: Waves that carry their shape along fixed paths Write the wave equation as u_tt equals c squared u_xx, where c is the signal speed. Along the lines x minus ct equals const and x plus ct equals const, the equation reduces to an ordinary derivative. These lines are the characteristics: every signal and every kink rides along them. Tracing both characteristics backward from a point (x,t) lands at x minus ct and x plus ct on the initial line. Only data between those feet can reach the point, and that interval is its domain of dependence. **Example.** Take speed 1 with zero initial displacement and constant velocity 2. Then d'Alembert keeps only the velocity integral: u(0,1) is half the integral of 2 from negative 1 to 1, which is half of 4, giving 2. With displacement 3x and zero velocity instead, the solution averages f at the two feet: u(2,1) is the average of f(1) equals 3 and f(3) equals 9, giving 6. In general at speed 1, u is the average of f at x minus t and x plus t, plus half the integral of g between them. A sharp corner in the data splits into left and right travelling copies that keep their shape, corners included. The heat equation does the opposite: its modes decay, with sharp features dying fastest, so a kink rounds off at once. The dependence interval also widens with time, since the feet x minus ct and x plus ct spread apart linearly. Later points therefore sample wider slices of initial data. **Tip.** A string pinned at both ends cannot keep travelling pulses for long, because reflections force standing shapes. Only sine shapes that vanish at both ends survive, each ringing at its own frequency. The second mode, shaped like sin(2 pi x) on a unit string, has exactly one interior node at x equals one half. Fixed ends turn travelling waves into a sum of these standing modes. **Recap.** Characteristics carry the initial shape unchanged, and each point sees only the interval between its backward feet. ## Practice 18 questions on this page, each with its working shown. ## Needs first - [Direction Fields and Euler's Method](https://lightmysky.com/learn/mathematics/direction-fields-and-eulers-method-mt_Ak-TZOX3GR) - [Separation of Variables and the Heat Equation](https://lightmysky.com/learn/mathematics/separation-of-variables-and-the-heat-equation-mt_XnPDL0X1TL) ## Opens up - [From Coordinates to Fields: the Lagrangian Density](https://lightmysky.com/learn/science/from-coordinates-to-fields-the-lagrangian-density-mt_fRZN3ZByjr) - [Laplace's Equation and the Maximum Principle](https://lightmysky.com/learn/mathematics/laplaces-equation-and-the-maximum-principle-mt_jSEkaTav1V)