We consider a domain optimization problem of an unbounded domain, which models scattering of a time-harmonic acoustic wave at the junction of two closed waveguides. Solutions of our problem fulfill the Helmholtz equation with a real wavenumber, a modal radiation condition and homogeneous Dirichlet boundary conditions. We derive an a-priori bound for the solution on a certain class of domains (which is compact in the Hausdorff metric topology) and show that within this class the solution depends -continuously on the domain. Furthermore, we show some numerical examples to illustrate our results, which were calculated using the domain (or shape) derivative of our problem.