As an illustration of the results on the propagation of singularities (or rather the propagation of regularity) proved in Chapter 8 , we shall solve wave-type equations \(P u=f\) on suitable spacetime domains (Theorem 9.33). This substantially generalizes Example 7.5 . An important feature of the approach used here, which distinguishes it from that of Chapter 7 , is that neither the operator nor the function spaces are defined with respect to any splitting into time and space variables. Moreover, we deduce the existence of solutions of \(P u=f\) by duality from an a priori (energy) estimate for \(P^*\) . (Duality methods are powerful tools for solving PDEs and will feature again in Chapter 11 .) Energy methods are often used in the study of wave equations; we shall only develop the basics here.