Julius Smith

Digital waveguide synthesis is the synthesis of audio using a digital waveguide. Digital waveguides are efficient computational models for physical media through which acoustic waves propagate. For this reason, digital waveguides constitute a major part of most modern physical modeling synthesizers.

A lossless digital waveguide realizes the discrete form of d'Alembert's solution of the one-dimensional wave equation as the superposition of a right-going wave and a left-going wave,

where ${\displaystyle y^{+}}$ is the right-going wave and ${\displaystyle y^{-}}$ is the left-going wave. It can be seen from this representation that sampling the function ${\displaystyle y}$ at a given point ${\displaystyle m}$ and time ${\displaystyle n}$ merely involves summing two delayed copies of its traveling waves. These traveling waves will reflect at boundaries such as the suspension points of vibrating strings or the open or closed ends of tubes. Hence the waves travel along closed loops.

...
Wikipedia