wave equation

波方程

Analysis, Measure & PDE · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

The constant-speed wave equation is partial_t^2 u=c^2 Delta_x u with c>0, classically for a twice differentiable space-time function; an initial value problem normally specifies both u and partial_t u at time zero.

常速波方程为 u_tt=c²Δu，其中 c>0；经典解要求相应二阶导数存在，初值问题通常同时给定初始位移和初始速度。

Example

$u(x,t)=\cos(3x)\cos(3ct)$ solves it since $u_{tt}=-9c^2u=c^2u_{xx}$.

Usage and distinctions

Prescribing displacement alone does not generally determine the solution; an independent initial velocity is needed.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. John K. Hunter — Notes on Partial Differential Equations§7.1, equations (7.1)–(7.2), printed p.211; general speed obtained by time rescaling · Checked 2026-10-14

Translation sources

1. 北京大学数学科学学院 — 数学分析（实验班）课程介绍第三部分，级数理论：五、Fourier分析初步，波方程和热方程简介 · Checked 2026-10-14

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