Various types of waves exist in the atmosphere over a wide range of time and spatial scales. Some familiar examples are vortex waves within hurricanes, mountain waves that generate the turbulence often experienced by air travellers, and the polar vortex which sometimes affects our winter weather here in Canada. Atmospheric waves affect weather and climate and so it is important to understand their properties and interactions for accurate weather prediction and climate modeling. Applied mathematicians study atmospheric fluid flows and waves using mathematical equations based on the laws of physics. In this presentation I will describe how mathematical and computational methods can be used to solve and analyze these equations. I will discuss some recent progress from my research in this area. My results describe the effects of wave interactions on the background atmospheric flow. The goal is to advance our understanding of waves and their effects on the atmosphere.