This paper deals with the coupling between one-dimensional heat and wave equations in unbounded subdomains, as a simplified prototype of fluid-structure interaction problems. First we apply appropriate artificial boundary conditions that yield an equivalent problem, but with bounded subdomains, and we carry out the stability analysis for this coupled problem in truncated domains. Then we devise an optimized Schwarz-in-time (or Schwarz Waveform Relaxation) method for the numerical solving of the coupled equations. Particular emphasis is made on the design of optimized transmission conditions. Notably, for this setting, the optimal transmission conditions can be expressed analytically in a very simple manner. This result is illustrated by some numerical experiments.