Water waves are ubiquitous in nature, occurring everywhere from oceans and lakes to swimming pools and bathtubs. Their study has led to important engineering applications, such as tsunami detection, wave energy harvesting, and coastal protection using breakwaters. Water waves are typically studied under the assumption that they are periodic (i.e., possessing a spatially repeating pattern) or that they decay to zero at infinity (i.e., no waves are coming from the far away). However, these assumptions are often too restrictive for exploring more complex and realistic phenomena. For instance, large bodies of water such as oceans are often covered with waves of different wavelengths, propagating in all directions over vast distances. Furthermore, these wavelengths can be incommensurate. To accurately model the interactions of water waves, the investigator studies them in a spatially quasi-periodic setting. This project will advance the modeling and simulation of water waves, with broad applications in ocean wave forecasting and shoreline protection. The project will also offer research and training opportunities for students. One of the main goals of the project is to develop accurate and stable numerical methods for studying spatially quasi-periodic water waves in both two and three dimensions. These methods will then be used to simulate such waves over flat or variable bottom topography. Traveling wave solutions will be computed, and their stability will be analyzed. In addi