Solving the (Abstract) Wave Equation using the Helmholtz equation

The talk by Prof. Mike Meylan addresses the solution of the abstract wave equation using the Helmholtz equation. He begins by defining the abstract wave equation with a positive self-adjoint operator and emphasizes the importance of inner products. He illustrates the solution via simple harmonic motion and matrix exponentials, showing that the solution operator is a one-parameter group. He then generalizes to infinite-dimensional operators, discussing the spectral theory and how to compute the cosine of an operator. He introduces the Helmholtz equation and shows how to construct the time-domain solution using spectral expansions, reducing it to matrix multiplication. He presents numerical simulations of wave scattering by cylinders, demonstrating the method. He then moves to a specific application: a simple model for an ice shelf, which is a coupled elastic-fluid system. He defines the abstract operator and shows that it is self-adjoint in a suitable Hilbert space. He introduces a transform that maps the problem to a space where the evolution is a simple shift, simplifying the solution. He discusses the structure of the reflection coefficient and its poles and zeros, leading to a complete description of the ice shelf dynamics in terms of complex resonances. He concludes by discussing the biorthogonal system and the potential for a complete theory for non-self-adjoint operators.