  We derive analytic forms for spin waves in pyrochlore magnets with
dipolar-octupolar interactions, such as ${\rm Nd}_2{\rm Zr}_2{\rm O}_7$. We
obtain full knowledge of the diagonalized magnonic Hamiltonian within the
linear spin wave approximation. We also consider the effect of a "breathing
mode" as a perturbation of this system. The breathing mode lifts the degeneracy
of the upper band of the spin wave dispersion along the direction $X\to W$ in
$k$-space.
