The heating of the solar atmosphere is a fundamental problem of modern
solar and astrophysics. A review of the seismological aspects of
magnetohydrodynamic (MHD) waves with an emphasis on standing
longitudinal waves in the context of coronal heating is presented.
Efforts made recently may be split into two categories: forward
modelling and data inversion. Forward modelling can be applied to
predict the observational footprints of various heating scenarios. A new
diagnostic method based on the analysis of Doppler shift time series is
outlined with specific application to solar coronal conditions. The
power of the method is demonstrated and tested using synthetic data and
comparing them with actual high-resolution (e.g. SoHO/SUMER)
observations. Further, related recent examples of standing longitudinal
oscillations in coronal loop structures observed with the new Hinode/EIS
instrument are also presented. These latter observations provide an
advanced ground for MHD seismology as a tool for plasma heating
diagnostics in the atmosphere of the Sun.