  Recent observations of protoplanetary disks, as well as simulations of
planet-disk interaction, have suggested that a single planet may excite
multiple spiral arms in the disk, in contrast to the previous expectations
based on linear theory (predicting a one-armed density wave). We re-assess the
origin of multiple arms in the framework of linear theory, by solving for the
global two-dimensional response of a non-barotropic disk to an orbiting planet.
We show that the formation of a secondary arm in the inner disk, at about half
of the orbital radius of the planet, is a robust prediction of linear theory.
This arm becomes stronger than the primary spiral at several tenths of the
orbital radius of the planet. Several additional, weaker spiral arms may also
form in the inner disk. On the contrary, a secondary spiral arm is unlikely to
form in the outer disk. Our linear calculations, fully accounting for the
global behavior of both the phases and amplitudes of perturbations, generally
support the recently proposed WKB phase argument for the secondary arm origin
(as caused by the intricacy of constructive interference of azimuthal harmonics
of the perturbation at different radii). We provide analytical arguments
showing that the process of a single spiral wake splitting up into multiple
arms is a generic linear outcome of wave propagation in differentially rotating
disks. It is not unique to planet-driven waves and occurs also in linear
calculations of spiral wakes freely propagating with no external torques. These
results are relevant for understanding formation of multiple rings and gaps in
protoplanetary disks.
