Name: modeling waves with space and time
Uploaded: 2026-06-08T09:39:00.655Z
Duration: 1 min 21 s
Channel: JoVE

想象一块石头被扔进平静的池塘；波纹以圆形图案向外扩散。为了描述这种运动，必须研究水位高程如何依赖于两个变量：时间和空间。在池塘的任意一点，随着时间的推移，水位会上升和下降。这种运动涉及该点处波的加速度，由对时间的二阶导数给出。如果系统仅随一个变量（例如时间）变化，则使用普通导数即可。但它也随距离而变化。在任一固定时刻，水面形成平滑的曲线波；该点处的空间曲率由对空间的二阶导数给出。其目的是同时观察时间和空间。偏微分方程通过使用偏导数，将时间与空间中的变化关联起来，描述当多个变量共同作用时系统如何演化。偏微分方程将多个变量（如时间与空间）关联起来，使我们能够描述当这些变量相互作用时系统如何变化和表现。

偏微分方程

将一块石头投入平静的池塘中，会激起以圆形模式向外传播的波纹，形成一个动态的水面，其高度取决于位置和时间。在任意给定位置，当波经过时，水位会发生振荡；而在任意固定时刻，水面则呈现出在空间中延展的平滑曲线结构。这种双重依赖关系需要一种能够同时描述多个变量变化的数学表达方式。在水面的固定位置，随着时间推移，水面高度持续变化。这种时间上的变化不仅包括水面上升和下降的速率，还包括该速率本身的变化，对应于该点处波浪运动的加速度。此类行为反映了介质对初始扰动的动力学响应。除了时间上的变化外，该波浪还表现出空间结构。在某一瞬间，水面形成一系列波峰和波谷，这些波峰和波谷在池塘表面平滑地变化。任意一点处水面的曲率反映了该点相对于邻近点的高度变化情况，从而捕捉到波浪向外传播时的几何形态。由于该系统同时依赖于时间和空间，因此无法通过仅考虑单一变量的方法进行充分描述。相反，偏微分方程通过将时间变化与空间变化关联起来，为分析此类系统提供了框架。这些方程描述了波的时间演化如何受其空间构型的影响，从而能够全面理解连续介质中的波传播过程。

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Education

教育

JoVE Core

Calculus

微积分

Unit

单元

Partial Derivatives and Gradients

偏导数与梯度

KEY TERMS AND CONCEPTS

关键术语与概念

Partial Differential Equations

modeling waves with space and time

Partial differential equations help describe waves that change across both space and time. A stone dropped into a still pond creates circular waves that spread outward. The water surface rises and falls at each point as the wave passes, while the shape of the surface also changes across the pond at any fixed moment.

This kind of motion needs a mathematical model that tracks more than one variable at once. At one location, the water level changes as time moves forward. That change includes the rate of rise and fall, and it also includes how that rate changes over time. This captures the acceleration of the wave motion at that point.

The wave also has a clear spatial pattern. At a single instant, the pond surface shows smooth crests and troughs that vary from place to place. The curvature of the surface at any point shows how the height changes compared with nearby points. This helps describe the geometry of the wave as it spreads.

Because the system depends on both time and space, a one-variable method is not enough. Partial differential equations link temporal change with spatial variation. They provide a way to study how the wave’s motion depends on its shape across the medium.

Modeling Waves with Space and Time

functions of two variables

Graphs of Functions in Three Dimensions

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graphs of two-variable functions

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functions of three or more variables

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properties of limits in multivariable calculus

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Limit Laws for Multivariable Functions

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continuity for functions of multiple variables

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Testing Continuity with Paths in Two Variables

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introduction to partial derivatives

Partial Derivatives in Real-World Models

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interpretations of partial derivatives

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multivariable functions and higher derivatives

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Partial Derivatives and Surface Curvature

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partial differential equations

Partial differential equations help describe waves that change across both space and time. A stone dropped into a still pond creates circular waves that spread outward. The water surface rises and falls at each point as the wave passes, while the shape of the surface also changes across the pond at any fixed moment. This kind of motion needs a mathematical model that tracks more than one variable at once. At one location, the water level changes as time moves forward. That change includes the...

tangent planes to surfaces

Using Partial Derivatives for Tangent Planes

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linear approximations

Tangent Plane Estimates for Two Variables

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multivariable chain rule

Total Derivatives in Connected Variables

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implicit differentiation with partial derivatives

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Finding Tangent Slopes from Implicit Equations

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directional derivatives

Directional Derivatives in Any Direction

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gradient vectors and their applications

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Gradient Direction and Rate of Change

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tangent planes to level surfaces

Gradient Vectors and Tangent Planes

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significance of the gradient vector

Gradient Vector and Direction of Steepest Rise

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local maximum and minimum values

Finding Critical Points in Multivariable Calculus

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lagrange multipliers: two constraints

Lagrange Multipliers on an Intersection Curve

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lagrange multipliers: problem solving

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key terms and concepts

Learn how partial differential equations describe pond waves by linking changes in time with the surface shape across space in a simple high school context.

Modeling Waves with Space and Time | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Modeling Waves with Space and Time

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