# Solving wave equation using Fourier Transform

• MHB
• spideyjj1
In summary, the conversation discusses the inverse Fourier transform and how it relates to the Dirac delta function. It explains that for a given function, its Fourier transform is defined by an integral and how this integral can result in the Dirac delta function for certain values of s. It also provides an example of how the Fourier transform of the cosine function can be expressed in terms of the Dirac delta function.
spideyjj1
I am having trouble with doing the inverse Fourier transform. Although I can find some solutions online, I don't really understand what was going on, especially the part that inverse Fourier transform of cosine function somehow becomes some dirac delta. I've been stuck on it for 2 hrs...

The Fourier transform of f is defined by $F(s)=\int_{-\infty}^{+\infty}f(t)e^{-i2\pi st}dt$. if f(t)=1 let $F_{1}$ be it's Fourier transform for $s\neq0$ you get
$F_{1}(s)=\int_{-\infty}^{+\infty}e^{-i2\pi st}dt=0$ ( odd function ). And for s=0
$F_{1}(0)=\int_{-\infty}^{+\infty}1dt=+\infty$ so
$F_{1}$ is then defined by $F_{1}(s)=0$ if $s\neq0$ and $F_{1}(0)=+\infty$. $F_{1}$ is the Dirac delta "function" ( It's a distribution ). $F_{1}(s)=\delta(s)$
Now if $f(t)= cos(2\pi \omega t)$ then $f(t)=\frac{1}{2}(e^{i2\pi \omega t}+e^{-i2\pi\omega t})$. The Fourier transform of f is then $F(s)= \frac{1}{2}\int_{-\infty}^{+\infty}e^{-i2\pi (s-\omega)t}dt+ \frac{1}{2}\int_{-\infty}^{+\infty}e^{-i2\pi (s+\omega)t}dt= \frac{1}{2}F_{1}(s-\omega)+ \frac{1}{2}F_{1}(s+\omega)= \frac{1}{2}\delta(s-\omega)+ \frac{1}{2}\delta(s+\omega)$

## 1. What is the wave equation?

The wave equation is a mathematical formula that describes how waves propagate through a medium. It is a second-order partial differential equation that relates the second derivative of a wave function to its spatial and temporal coordinates.

## 2. What is the Fourier Transform?

The Fourier Transform is a mathematical operation that decomposes a function into its constituent frequencies. It converts a function from its original domain (usually time or space) to its frequency domain, allowing us to analyze and manipulate the different frequency components of a signal.

## 3. How is the Fourier Transform used to solve the wave equation?

The Fourier Transform is used to solve the wave equation by transforming the equation from its original domain (usually time or space) to its frequency domain. This allows us to express the wave function as a sum of sinusoidal functions, making it easier to solve for the different frequency components and their corresponding amplitudes and phases.

## 4. What are the advantages of using the Fourier Transform to solve the wave equation?

Using the Fourier Transform to solve the wave equation has several advantages. It allows us to break down complex wave functions into simpler components, making it easier to analyze and understand. It also allows us to easily manipulate the different frequency components, which can be useful in applications such as signal processing and image reconstruction.

## 5. Are there any limitations to using the Fourier Transform to solve the wave equation?

While the Fourier Transform is a powerful tool for solving the wave equation, it does have some limitations. It assumes that the wave function is continuous and has a finite energy, which may not always be the case in real-world scenarios. It also requires that the wave function be periodic, which may not always be true for all types of waves.

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