In the past two decades, significant progress has been made in understanding random dispersive PDEs with polynomial nonlinearities. However, non-polynomial cases remain far less understood. This talk presents recent progress on the well-posedness of the two-dimensional damped wave equation, one with a sine nonlinearity and the other with an exponential nonlinearity, both driven by additive space-time white noise. Our analysis leads to a novel physical space approach to address the challenges of random wave equations in this setting. This talk is partially based on joint work with Justin Forlano (Monash University, Australia).