  We study a simple one-dimensional coupled wave-heat system and obtain a sharp
estimate for the rate of energy decay of classical solutions. Our approach is
based on the asymptotic theory of $C_0$-semigroups and in particular on a
result due to Borichev and Tomilov (Math. Ann., 2010), which reduces the
problem of estimating the rate of energy decay to finding a growth bound for
the resolvent of the semigroup generator. This technique not only leads to an
optimal result, it is also simpler than the methods used by other authors in
similar situations.
