The Inverted Glass Harp

Sliding a finger around the edge of a glass causes it to resonate; any 5-year-old can tell you that. When you line up a bunch of glasses with water at different heights, it's often called a "glass harp" and is the instrument of choice for many professional musicians. But you can also use a single empty glass and control the pitch by raising and lowering it into a basin of water:

  

Let's call that second kind the "inverted glass harp". The inverted harp has less range and can only play one note at a time, but it also has some key advantages: it's quicker to set up, it's much cheaper, and it can play "slides" like a trombone. Both the normal and inverted harp basically work the same way. The glass has to accelerate the neighboring water and therefore behaves as if it has more mass; scientifically speaking, you'd say the glass some additional "virtual mass" when it's full (or submerged). The virtual mass lowers the resonant frequency of the glass, so it produces a lower pitched sound.

You can estimate the virtual mass of the wine glass by solving for the motion of the water inside or around the vibrating glass. If you assume that viscous effects and vertical motions in the water are small, you can plot exact solutions for the flow. Here are some birds-eye views of the solutions for water speed, both radially (toward/away from the center) and azimuthally (around the center):

The accelerations you see are what give the glass its virtual mass, because the vibrating glass is pushing and pulling this water back and forth. As it turns out, the extra virtual mass is the same for both the normal and inverted harps (see the details here). In other words, the pitch is lowered about the same amount whether you fill the glass or submerge the glass to some height. Even though the inverted harp has to push against more water, it doesn't have to push as hard, and the effect ends up being the same (in practice, the inverted harp seems to have a similar but slightly lower frequency, probably because of 3D effects).

One more interesting property of the two flow solutions is that they're members of a family of solutions to a more general problem: a glass with radius a vibrating in the presence of a cylindrical boundary with some radius ao. The normal glass harp is the special case where ao = 0, and the inverted glass harp is the special case where ao = ∞. The in-between cases offer yet more ways you can play around with the pitch, either by inserting something into the glass (0 < ao< a) or by putting the glass in a larger but comparable container (a < ao< ∞). Here are the four types of solution along with photos of their application:

Just remember to practice before your next dinner party; these instruments are not crowd-pleasers as you're first learning, trust me.

Special thanks to Brian Rosenberg, co-discoverer of the inverted glass harp and the underlying mathematics. Want more? Follow @danbquinn for updates, submit a post idea, or explore on your own by downloading the Mathematica code (.nb, .pdf) used to make this post .