  Magnetohydrodynamic (MHD) waves have been considered as energy sources for
heating the solar chromosphere and the corona. Although MHD waves have been
observed in the solar atmosphere, there are a lack of quantitative estimates on
the energy transfer and dissipation in the atmosphere. We performed
simultaneous Hinode and IRIS observations of a sunspot umbra to derive the
upward energy fluxes at two different atmospheric layers (photosphere and lower
transition region) and estimate the energy dissipation. The observations
revealed some properties of the observed periodic oscillations in physical
quantities, such as their phase relations, temporal behaviors, and power
spectra, making a conclusion that standing slow-mode waves are dominant at the
photosphere with their high-frequency leakage, which is observed as upward
waves at the chromosphere and the lower transition region. Our estimates of
upward energy fluxes are $2.0\times10^7$ erg cm$^{-2}$ s$^{-1}$ at the
photospheric level and $8.3\times10^4$ erg cm$^{-2}$ s$^{-1}$ at the lower
transition region level. The difference between the energy fluxes is larger
than the energy required to maintain the chromosphere in the sunspot umbrae,
suggesting that the observed waves can make a crucial contribution to the
heating of the chromosphere in the sunspot umbrae. In contrast, the upward
energy flux derived at the lower transition region level is smaller than the
energy flux required for heating the corona, implying that we may need another
heating mechanism. We should, however, note a possibility that the energy
dissipated at the chromosphere might be overestimated because of the opacity
effect.
