Graph based physical models for sound synthesis

We focus on physical models in which multiple strings are connected via junctions to form graphs. Starting with the case of the 1D wave equation, we show how to extend it to a string branching into two other strings, and from there how to build complex cyclic and acyclic graphs. We introduce the concept of dense models and show that a discretization of the 2D wave equation can be built using our methods, and that there are more efficient ways of modelling 2D wave propagation than a rectangular grid. We discuss how to apply Dirichlet and Neumann boundary conditions to a graph model, and show how to compute the frequency content of a graph using common methods. We then prove general lower and upper bounds computational complexity. Lastly, we show how to extend our results to other kinds of acoustical objects, such as linear bars, and how to add dampening to a graph model. A reference implementation in MATLAB and an interactive JUCE/C++ application is available online.