# Solving the wave equation numerically using the Fast Fourier Transform

• tjackson3
In summary, the conversation is discussing a homework statement about writing a program to calculate the evolution of the Wave Equation for a bound string using Matlab's fft. The relevant equation is u_{tt} = a^2u_{xx} and the program should be tested on base eigenfunctions, such as sinusoids. The person is seeking advice and resources for understanding how to use fft for the wave equation.
tjackson3

## Homework Statement

According to the website, the statement is as follows:

Write a program which will calculate the evolution of the Wave Equation,
in the case of a bound string. Test this program on the base eigenfunctions,
i.e. the sinusoids, and on more interesting combinations. You should use
matlab’s fft to do this.

## Homework Equations

The only really relevant equation here is the wave equation, $$u_{tt} = a^2u_{xx}$$

## The Attempt at a Solution

Specifically, I'm trying to numerically solve the wave equation for a bound string of length L using a fast Fourier transform. The only thing is, I have absolutely no idea what I'm supposed to be FFT'ing. According to the instructions on my project, I'm supposed to try it on the "base eigenfunctions; i.e., the sinusoids." Where do I start with this? Nothing I can find really gives me a clue as to where to start; that's all I'm looking for.

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I would also appreciate any advice or resources that could help me better understand the concept of FFT'ing the wave equation.

## 1. What is the wave equation and why is it important?

The wave equation is a mathematical formula that describes how waves propagate through a medium. It is important because it is used in many fields of science and engineering, such as acoustics, electromagnetism, and quantum mechanics, to model and understand wave phenomena.

## 2. What is the Fast Fourier Transform (FFT) and how does it relate to the wave equation?

The Fast Fourier Transform is an algorithm used to efficiently compute the discrete Fourier transform, which decomposes a signal into its constituent frequencies. It is often used in solving the wave equation numerically because it allows for a faster and more accurate calculation of the spatial and temporal components of the wave.

## 3. What are the advantages of using the FFT in solving the wave equation?

Using the FFT in solving the wave equation offers several advantages. It is more computationally efficient, allowing for faster calculations and simulations. It is also more accurate, as it reduces the truncation errors that can occur in other numerical methods. Additionally, it is a versatile tool that can be applied to a wide range of wave equations and boundary conditions.

## 4. Are there any limitations to using the FFT in solving the wave equation?

While the FFT is a powerful tool, it does have some limitations in solving the wave equation. It is most effective for linear, time-invariant systems and may not accurately model nonlinear or time-varying systems. It also requires a uniform grid and periodic boundary conditions, which may not be suitable for all wave equations.

## 5. What are some practical applications of solving the wave equation numerically using the FFT?

The FFT is used in a wide range of practical applications, such as simulating wave propagation in seismology and predicting acoustic properties in architectural design. It is also used in medical imaging, signal processing, and telecommunications. Essentially, any field that involves the analysis or manipulation of wave phenomena can benefit from using the FFT to solve the wave equation numerically.

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