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创新世界 偏微分 equation that governs the propagation of various types of waves. In its simplest form, it is written as [latex]frac{partial^2 u}{partial t^2} = c^2 nabla^2 u[/latex], where [latex]u(vec{x},t)[/latex] is the amplitude of the wave, [latex]c[/latex] is the wave speed, and [latex]nabla^2[/latex] is the 拉普拉斯 operator. It models phenomena like vibrating strings, sound waves, and light waves.
The wave equation is the archetypal hyperbolic PDE. Unlike the 热方程, it is second-order in time, which gives rise to its oscillatory, wave-like solutions. The presence of the [latex]frac{partial^2 u}{partial t^2}[/latex] term implies that acceleration is proportional to the local curvature of the function, a relationship characteristic of restorative forces like tension in a string. The constant [latex]c[/latex] represents the finite speed at which disturbances propagate through the medium.
A crucial feature of the wave equation is the principle of causality and finite propagation speed. A disturbance at a point [latex]vec{x}_0[/latex] at time [latex]t_0[/latex] can only affect points [latex]vec{x}[/latex] at a later time [latex]t[/latex] that are within a distance of [latex]c(t-t_0)[/latex]. This region is known as the ‘cone of influence’. Conversely, the value of the solution at [latex](vec{x}, t)[/latex] depends only on the initial data within its ‘domain of dependence’. This contrasts sharply with the infinite propagation speed of the heat equation.
In one spatial dimension, the equation [latex]u_{tt} = c^2 u_{xx}[/latex] has a remarkably simple general solution, discovered by d’Alembert: [latex]u(x,t) = F(x-ct) + G(x+ct)[/latex]. This represents the superposition of two waves traveling in opposite directions with speed [latex]c[/latex]. The shapes of these waves, determined by the functions [latex]F[/latex] and [latex]G[/latex], are preserved as they propagate.
波动方程（物理学）
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