Chemical waves can travel in well-defined packets. Two types of phase wave packets, distinguished by whether their component waves move toward or away from an initiating perturbation, are found in a reaction-diffusion model with a finite wave instability. Their propagation, reflection, and spreading are studied numerically and analytically. Reflection from a boundary or collision of two identical packets can result in standing waves. When two wave packets collide, they can interact briefly and then pass through each other without modification. The phase velocity, group velocity, and spreading velocity calculated by linear stability analysis with the inclusion of quadratic dispersion agree well with the results of numerical simulations.