I'm gonna link to Fuck Yeah Fluid Dynamics a bunch, most likely, because it's fantastic.
In this case...what a wonderful opportunity to express how neat resonance is!
So, the sum of that paper is that the stride we use while walking (a quick 'step' followed by a slow lurching over your balanced and stiff leg) resonates slightly with the 'natural frequency' of your average coffee cup.
Now, you have this walking motion (quick-then-slow), a sort of simple sinusoidal motion, matched with a fluid circulating around a small cup (try it at home, but not on carpet). What resonance means here is that - considering both actions as simple waves - these waves hit their peaks at the same time. So the liquid sloshing around may do so twice as quickly the step as the person walking with it; if it is under the influence of this resonance, it is not 2.5 times as quickly, it is two...or it could be three, but not 3.12...These fractional values do not resonate. Resonance means an exacting relation between distinct waves.
Meaning the wave of our step-slow-step oscillation matches, in some integer multiple, the oscillation of our coffee cups. And then it spills. Like a jerk.
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