Wave Equation

MIT OpenCourseWare
6 May 201615:13
EducationalLearning
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TLDRThis video script delves into the world of partial differential equations, focusing on the wave equation, the third in a series after Laplace's and the heat equation. It highlights the differences between these equations, particularly how the wave equation, being second-order in time, allows for finite signal propagation at speed 'c'. The script provides insights into solutions for both the wave and heat equations, starting from a delta function, and contrasts their behaviors. It also touches on solving the wave equation using Fourier series for finite domains like a violin string and infinite domains, like space, showcasing the method of separation of variables as a powerful tool in finding solutions.

Takeaways
  • ๐Ÿ“š Laplace's equation is elliptic, the heat equation is parabolic, and the wave equation is hyperbolic.
  • ๐ŸŒ€ The wave equation is second-order in time, represented by dยฒu/dtยฒ, and matches the second derivative in space with a velocity coefficient cยฒ.
  • ๐ŸŒŠ In the wave equation, the signal travels with finite velocity, unlike the heat equation where the signal travels infinitely fast.
  • ๐Ÿ“ˆ A delta function as the initial condition in the heat equation results in an immediate, highly damped response.
  • ๐Ÿ”Š For the wave equation with a delta function, the signal splits into two waves traveling in opposite directions at speed c.
  • ๐ŸŒŠ Tsunamis and sound waves can be modeled using the wave equation due to their finite travel times.
  • ๐ŸŽป In solving the wave equation for a violin string, initial conditions include both the displacement and velocity of the string.
  • ๐Ÿ“ The solution to the wave equation on a finite string involves functions of x ยฑ ct and can be expressed using separation of variables.
  • ๐Ÿงฎ D'Alembert's formula provides a method to solve the wave equation using initial conditions to find the functions f and g.
  • ๐ŸŽถ For a violin string, the solution uses a series of sine and cosine functions to represent the displacement over time, with sine functions representing the fixed endpoints.
Q & A
  • What are the three types of partial differential equations mentioned in the script?

    -The three types of partial differential equations mentioned are Laplace's equation, the heat equation, and the wave equation.

  • How are these equations classified, and what are their geometric analogs?

    -Laplace's equation is classified as elliptic, the heat equation as parabolic, and the wave equation as hyperbolic. These classifications correspond to the geometric shapes of ellipses, parabolas, and hyperbolas.

  • What is the key difference between the heat equation and the wave equation in terms of signal propagation?

    -The key difference is that under the heat equation, the signal travels infinitely fast, whereas under the wave equation, the signal travels with a finite velocity, denoted by the speed 'c'.

  • What happens to a delta function under the heat equation?

    -Under the heat equation, a delta function quickly spreads out, and its effect is immediately felt everywhere, but with very small intensity, as indicated by the exponential decay e^(-x^2/4t).

  • How does a delta function behave under the wave equation?

    -Under the wave equation, a delta function splits into two waves, each carrying half of the original delta function, and these waves travel in opposite directions at the speed 'c'.

  • What initial conditions are needed to solve the wave equation?

    -To solve the wave equation, you need to specify the initial displacement 'u(x, 0)' and the initial velocity 'du/dt(x, 0)' for all positions 'x'.

  • How can Fourier series be used to solve the wave equation on a finite domain?

    -Fourier series can be used to solve the wave equation on a finite domain, such as a violin string, by expressing the solution as a sum of sine and cosine functions that satisfy the boundary conditions.

  • What is d'Alembert's formula, and when is it used?

    -D'Alembert's formula is used to solve the wave equation in one-dimensional space, and it expresses the solution as a sum of two functions, each representing a wave traveling in opposite directions.

  • How does the wave equation solution differ on an infinite line compared to a finite string?

    -On an infinite line, the wave equation solution consists of waves that travel indefinitely without boundaries. On a finite string, the solution involves standing waves that reflect back and forth due to the fixed boundary conditions.

  • What is the method of separation of variables, and why is it important?

    -The method of separation of variables is a technique used to solve partial differential equations by separating the variables, typically 'x' and 't', into independent functions. It is important because it simplifies solving complex equations by reducing them to simpler ordinary differential equations.

Outlines
00:00
๐Ÿ“š Introduction to Partial Differential Equations

This paragraph introduces the three main types of partial differential equations (PDEs): Laplace's equation, the heat equation, and the wave equation. It explains that Laplace's equation is elliptic and is solved within a closed region, while the heat and wave equations are parabolic and hyperbolic, respectively, and involve time as a variable. The heat equation is first-order in time, whereas the wave equation is second-order, with the second derivative in time matching the second derivative in space, scaled by the velocity coefficient c squared. The paragraph emphasizes the differences between the heat and wave equations, particularly the infinite speed of signal propagation in the heat equation versus the finite speed in the wave equation, which is exemplified by the speed of sound.

05:03
๐ŸŒŠ Exploring the Wave Equation and Its Solutions

This paragraph delves deeper into the wave equation, contrasting it with the heat equation by examining the propagation of signals over time. It uses the example of a delta function as an initial condition to illustrate the differences in solutions. The heat equation spreads the signal infinitely fast, while the wave equation propagates the signal with a finite speed, which is crucial for phenomena like tsunamis. The paragraph also discusses the form of the solution to the wave equation in one-dimensional space, highlighting the rightward and leftward traveling waves originating from a point source, each carrying half of the initial impulse.

10:03
๐ŸŽป Solving the Wave Equation with Fourier Series

The final paragraph discusses methods for solving the wave equation under different conditions. It starts by considering the solution in free space, suggesting a general form involving functions of (x-ct) and (x+ct), which are reminiscent of the delta function solutions. The paragraph then shifts focus to solving the wave equation for a finite string, such as a violin string, using the method of separation of variables. This method leads to a solution involving a sum of cosine functions of time, multiplied by sine functions of space, with coefficients that depend on the string's initial conditions. The importance of the separation of variables method is highlighted, as it is a powerful tool for solving many types of PDEs.

Mindmap
Keywords
๐Ÿ’กLaplace's equation
Laplace's equation is an elliptic partial differential equation. It is used to describe the behavior of electric potential and gravitational potential in a given region. In the video, it is referred to as one of the three major partial differential equations, often solved within a closed region like a circle.
๐Ÿ’กheat equation
The heat equation is a parabolic partial differential equation that describes how heat diffuses through a given region over time. It is first order in time, meaning it involves the first time derivative of the temperature. The video contrasts it with the wave equation, noting that heat signals travel infinitely fast.
๐Ÿ’กwave equation
The wave equation is a hyperbolic partial differential equation that describes the propagation of waves, such as sound or light waves. It is second order in time, involving the second time derivative of the displacement. The video explains that wave signals travel with a finite velocity, unlike heat signals.
๐Ÿ’กdelta function
A delta function, or Dirac delta function, is a mathematical function that represents an infinitely small and infinitely high spike at a single point. It is used in the video to describe initial conditions for both the heat and wave equations, representing a point source of heat or sound.
๐Ÿ’กfinite velocity
Finite velocity refers to the limited speed at which waves propagate through a medium. In the video, this concept is contrasted with the infinite speed of heat propagation, emphasizing that sound and light waves travel at a measurable speed determined by the wave equation.
๐Ÿ’กFourier series
A Fourier series is a way to represent a function as the sum of simple sine and cosine waves. It is used in the video as a method to solve partial differential equations by breaking down complex functions into simpler components, particularly useful in problems involving a finite region like a violin string.
๐Ÿ’กseparation of variables
Separation of variables is a mathematical method used to solve partial differential equations by separating the variables involved into distinct functions. In the video, this technique is described as highly important and is demonstrated in the context of solving the wave equation for a finite string.
๐Ÿ’กboundary conditions
Boundary conditions are constraints necessary to solve differential equations, specifying the behavior of a function on the boundaries of the domain. The video discusses how the wave equation is solved with boundary conditions, such as a violin string being held at its ends, resulting in sine functions.
๐Ÿ’กinitial condition
An initial condition specifies the state of a system at the beginning of a time period for solving differential equations. The video mentions initial conditions for both the heat and wave equations, such as the starting position and velocity of a plucked violin string.
๐Ÿ’กd'Alembert's formula
D'Alembert's formula is a solution to the one-dimensional wave equation, representing the displacement of a wave as a combination of two traveling waves. The video references this formula to explain the behavior of waves originating from a delta function, showing how sound propagates in both directions.
Highlights

Introduction of the three great types of partial differential equations: elliptic (Laplace's equation), parabolic (heat equation), and hyperbolic (wave equation).

Differences in the nature of solutions for Laplace's, heat, and wave equations based on their geometric analogies to ellipses, parabolas, and hyperbolas.

Laplace's equation is solved within a closed region, while heat and wave equations involve time progression.

Heat equation is first-order in time, whereas the wave equation is second-order in time.

Wave equation's second derivative in time matches the second derivative in space with a velocity coefficient c squared.

In three dimensions, second derivatives in all space directions are involved for sound and light waves.

Fundamental differences between heat and wave equations: heat signals travel infinitely fast, while wave signals have a finite velocity c.

Example of a delta function as an initial condition for both heat and wave equations and their distinct outcomes.

Heat equation's solution from a delta function involves rapid damping and immediate, albeit minimal, spread.

Wave equation's solution from a delta function results in waves traveling in both directions with speed c.

Explanation of d'Alembert's formula for solving the wave equation in free space.

Application of Fourier series to solve the wave equation for a finite string, like a violin string.

The importance of initial conditions for the wave equation, including the initial distribution and velocity.

Introduction to separation of variables as a method for solving partial differential equations.

The form of the solution for a finite string using separation of variables, involving cosine and sine functions.

Explanation of how the initial condition of the violin string at rest affects the solution's form.

The significance of the sine function in matching the boundary conditions of the violin string.

Emphasis on the need for a proper explanation of the separation of variables method in future content.

Transcripts
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