Discretised wave equations

Can anyone please explain this to me? I can’t really figure out what the symbols mean.

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The wave equation is usually expressed in the form

$latex displaystyle partial_{tt} u – Delta u = 0&fg=000000$

where $latex {u colon {bf R} times {bf R}^d rightarrow {bf C}}&fg=000000$ is a function of both time $latex {t in {bf R}}&fg=000000$ and space $latex {x in {bf R}^d}&fg=000000$, with $latex {Delta}&fg=000000$ being the Laplacian operator. One can generalise this equation in a number of ways, for instance by replacing the spatial domain $latex {{bf R}^d}&fg=000000$ with some other manifold and replacing the Laplacian $latex {Delta}&fg=000000$ with the Laplace-Beltrami operator or adding lower order terms (such as a potential, or a coupling with a magnetic field). But for sake of discussion let us work with the classical wave equation on $latex {{bf R}^d}&fg=000000$. We will work formally in this post, being unconcerned with issues of convergence, justifying interchange of integrals, derivatives, or limits, etc.. One then has a conserved energy


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3 thoughts on “Discretised wave equations

  1. In short, no. To understand this, you would need to understand the concept of a limit. This in turn requires you to understand what an infinite sequence of numbers is. As very basic first step, you need to acknowledge that one can have an infinite sequence of numbers. Since you don’t acknowledge this, we can’t explain it to you.

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