When you push someone on a swing, a series of small pushes makes the swing move through a large amplitude. To accomplish this, you time your pushes to match the swing’s natural frequency, the rate at which the swing tends to move back and forth.
The same principle is at work here. When you shake the two-by-four assembly at just the right frequency, a series of small shakes adds up to a large vibration of a particular dowel. The shaking beam sets the dowel vibrating. If the next shake is timed just right to reinforce the next vibration of the dowel, the vibration in the dowel builds up. This process of using a series of small inputs to create a large motion is known as resonance.
The longer the dowel, the more slowly it tends to vibrate, and the lower its natural frequency. Thus, the long dowel will resonate at lower frequencies than the short dowel.
Stiffer dowels have higher resonant frequencies. The 3/8-in (9.5-mm) dowel is much stiffer than the 1/4-in (6-mm) dowels, and so it tends to resonate at higher frequencies than the thinner dowels. Note also that each dowel may have more than one resonant frequency.
Not all objects resonate. Any object that dissipates energy faster than the energy is added will not resonate. Try, for example, shaking the dowels under water. The friction of the dowel moving through water will dissipate the energy faster than you add it. Because the motion of the dowel will not build up at any frequency, there is no resonance.