  Wave propagation of heat in solids, which is often called `second sound', has
attracted increasing attention to its various potential applications such as in
thermal diodes and waveguides. It has been suggested that ferroelectrics are
strong candidates for thermal-wave media$^1$, and several reports on an
incipient ferroelectric, strontium titanate (SrTiO$_3$), have shown a novel
wave-like excitation in light-scattering spectra$^{2,3}$. Second sound has been
proposed as the most plausible origin for this excitation based on rather
indirect demonstrations$^{3,4}$. However, the lack of a direct demonstration
still leaves controversy$^5$ on the existence of second sound in SrTiO$_3$.
Here, we present a direct demonstration of an optically generated coherent heat
wave, i.e, coherent second sound, in bulk single crystals of SrTiO$_3$. The
generated wave-excitation has a linear dispersion relation with the predicted
velocity of second sound in SrTiO$_3$, and is overdamped at high temperatures
or low frequencies, exhibiting apparently diffusive behaviour with the decay
rates defined by the known values of the thermal diffusivity of SrTiO$_3$. This
work offers not only convincing evidence for the existence of second sound in
SrTiO$_3$, but also a first demonstration of coherent generation and
wave-control of gigahertz temperature fields in a 3D bulk material. As
high-quality SrTiO$_3$ is widely available, and it is also known as a mother
material of superconductors, the coherent excitation of temperature wave in
this material offers a pathway for potential novel wave-based applications in
heat management such as construction of thermal waveguides and directional
devices or heat-control of superconductivity.
