Wave propagation at oblique shocks: How did Tycho get its stripes?

We describe a new model for the "stripes" of synchrotron radiation seen in the remnant of Tycho's supernova. In our picture, cosmic rays streaming ahead of the forward shock generate parallel propagating (with respect to the local magnetic field direction) circularly polarized Alfvén waves that are almost free of dissipation, and, due to being circularly polarized, exhibit no spatial variation of magnetic field strength. Following the interaction with the supernova remnant (SNR) shock with nonzero obliquity, these parallel propagating waves become obliquely propagating, due the the wave refraction (which is different, in principle, for the different plane wave components), and dissipation sets in. The magnetosonic polarization decays faster, due to transit time damping, leaving only the Alfvén mode. This surviving mode now exhibits a spatial variation of the magnetic field, leading to local maxima and minima in the synchrotron emission, i.e., the stripes. We attribute the initial wave generation to the Bell instability, which in contrast to the resonant generation of upstream Alfvén waves, gives rise to a preferred wavelength, and hence the single wave period at which the stripes are seen. Based on estimates for damping rates due to turbulent cascade and transit time damping, we estimate the dependence of the visibility of the stripes on the shock obliquity and determine a maximum cosmic ray energy in Tycho's SNR in the range $6\times {10}^{14}-1\times {10}^{15}$ eV.