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 build artificial boundary conditions for each subproblem so as to solve it numerically in a bounded subdomain. Then we devise an optimized Schwarz-in-time (or Schwarz Waveform Relaxation) method for the numerical solving of the coupled equations , which allows possibly different solvers and different time steps for each separated problem. 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 .