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

Imagine a stone being thrown into a still pond; waves spread outward in circular patterns.To describe this motion, one must study how the water surface elevation is dependent upon two variables: time and space.At any single spot in the pond, the water level rises and falls as time passes. This motion involves the acceleration of the wave at that point, given by the second derivative with respect to time.If the system changed only with one variable, like time, an ordinary derivative would be enough.But it varies across distance also. At any fixed moment, the water forms smooth, curved waves; this spatial curvature at a point is given by the second derivative with respect to space.The idea is to observe time and space together. A partial differential equation connects these changes in time and space, describing how the system evolves when multiple variables act together by using partial derivatives.Partial differential equations link multiple variables, such as time and space, allowing us to describe how a system changes and behaves when these variables interact.

partial differential equations

A stone dropped into a still pond generates waves that propagate outward in circular patterns, creating a dynamic surface whose elevation depends on both position and time. At any given location, the water level oscillates as the wave passes, while at any fixed moment, the surface exhibits smooth, curved structures extending across space. This dual dependence requires a mathematical description that accounts for variation in multiple variables simultaneously.At a fixed point on the water surface, the elevation changes continuously as time progresses. This temporal variation includes not only the rate at which the water rises and falls but also how that rate itself changes, corresponding to the acceleration of the wave motion at that point. Such behavior reflects the dynamic response of the medium to the initial disturbance.In addition to temporal variation, the wave exhibits spatial structure. At a single instant, the water surface forms a pattern of crests and troughs that vary smoothly across the pond. The curvature of this surface at any point reflects how the elevation changes relative to neighboring points, capturing the geometry of the wave as it spreads outward.Because the system depends on both time and space, it cannot be adequately described using methods that consider only a single variable. Instead, partial differential equations provide a framework for analyzing such systems by linking changes in time with variations in space. These equations describe how the temporal evolution of the wave is influenced by its spatial configuration, enabling a comprehensive understanding of wave propagation in continuous media.

Partial Differential Equations

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Education

JoVE Core

Calculus

Kalkülüs

Unit

Birim

Partial Derivatives and Gradients

Kısmi Türevler ve Gradyanlar

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

Kısmi Diferansiyel Denklemler

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

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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...

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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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