Mode shapes of global stable (top) and unstable (bottom) density waves of the particle phase in a protoplanetary disk with μ = 0.04, M _p/ M _g = 1/8, and . Contour plots show the positive parts of the perturbed density Σ ₁ at t = 0. In all cases, density waves rotate counterclockwise, and their corresponding pattern speeds are given in Table 2. The outer density bump of unstable mode A ₁ is a trailing spiral. The inner and outer wave packets of unstable mode A ₂ lead and trail their stable counterparts, respectively. Solid circle shows the location of the central star.