Can plantar fasciitis cause pain under the heel and a rhythmic throb?

Yes. Yes it can, get it evaluated. I recommend a course of physical therapy, stretching, icing (10 minutes on). See your local Podiatrist as you might benefit from custom inserts.
Foot pain. Plantar fasciitis typically causes a stabbing pain in the bottom of your foot near the heel. The pain is usually worst with the first few steps after awakening, although it can also be triggered by long periods of standing or getting up from a seated position.