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.
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