# Matrix methods for wave equations

## TL;DR

This paper establishes conditions under which 2x2 operator matrices generate cosine operator functions, with applications to wave equations, boundary value problems, and overdamped systems, advancing the mathematical understanding of wave dynamics.

## Contribution

It provides new criteria for operator matrices to generate cosine functions, extending previous work on semigroups to second-order wave systems.

## Key findings

- Derived conditions for operator matrices to generate cosine operator functions.
- Applied criteria to systems of wave equations and boundary value problems.
- Extended analysis to overdamped wave equations.

## Abstract

In analogy to a characterisation of operator matrices generating $C_0$-semigroups due to R. Nagel (\cite{[Na89]}), we give conditions on its entries in order that a $2\times 2$ operator matrix generates a cosine operator function. We apply this to systems of wave equations, to second order initial-boundary value problems, and to overdamped wave equations.