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When you push on something it moves forward, even if only a very tiny bit, and when you let go it relaxes and moves back. If you do this very quickly over and over again, the object moves back and forth continuously. We say the object is vibrating, and that's what the glass is doing as you rub back and forth- it's vibrating. You don't see it move, however, because only the rim of the glass is moving a very tiny distance, and because it's actually moving much, much faster than your finger is (to understand why this is so you can read the more detailed explanation below).
Now you have surely made water waves in the bath tub by moving your hand back and forth (you were vibrating your hand). If the walls of your bath tub were thin and flexible enough to move easily, you could even make waves in the water by pushing or tapping on the sides of the tub instead. When the wine glass vibrates while it's full of water it also makes waves (you can also tap on the sides of the glass to see them), and if these waves are big enough they can even splash water droplets right out of the glass!
more detailed explanation
Objects can be made to vibrate by forcing them to move back and forth quickly. A dry finger will stick to the rim of a wine glass pretty well, but when you wet your finger a little, it will begin to slide. Either way, when you rub the rim of the wine glass in just the right way, your finger will stick for a short time, then slide a little, then stick again, then slide again, etc. (the same thing happens as you rub the handles of the spouting bowl with your wet hands). This is called "stick-slip" friction, and in a sense it's like tapping on the rim of the glass (or bowl) very quickly. This makes the glass begin to move back and forth or vibrate, literally bulging in and out at various places around the rim. The number of times the rim vibrates back and forth per second is called the frequency, and many different vibration frequencies are excited as you rub the rim of the glass. These movements are much to small in magnitude and much too fast to see with your eyes, but we can easily see the waves that are created by the rim pushing on the water in the glass.
Since these vibrations are so small, most of them lose their energy and die out very quickly. Any solid object, however, has a set of special frequencies (and shapes) at which it prefers to vibrate, called its natural or resonant frequencies (and modal shapes). At these resonant frequencies it takes only a very small amount of input motion or energy to produce very large vibrations and large output energies. As you start the wine glass or spouting bowl vibrating with your stick-slip motion, these resonant frequencies are also excited, but since they require only a small input energy to produce large output vibrations, they quickly dominate the motion and last much longer. This is called resonance, and we say the object is resonating. The sound you hear is produced by the resonant vibration of the glass or bowl.
Now back to the waves in the water. The large vibrations along the rim of the glass push on the water, sending waves traveling across the surface. When the waves hit the other side of the glass they bounce back (reflections) and run into other waves traveling in the opposite direction. All of these waves, which are being launched at precise time intervals, begin to combine. In places where two or more wave crests or high points meet, the combined wave will be even higher. Similarly, in places where two or more wave troughs or low points meet, the combined wave will be even lower. In other places crests and troughs from different waves will meet and cancel each other. This creates what are called standing wave patterns (i.e. the combined wave pattern appears to stand still) on the surface of the water. FInally, because the standing wave patterns are created by large resonant vibrations of the glass (or bowl), the standing water waves become very large also, eventually splashing water high into the air. The locations where the water splashes highest corresponds to locations where the glass (or bowl) is moving the most, called anti-nodes. Halfway between each anti-node is a node, a location where the rim of the glass (or bowl) is not moving at all. Near these nodes the standing waves are very small and no splashing occurs. Since the position of the handles are fixed on the spouting bowl, the positions of the nodes and anti-nodes are also fixed, and the water always splashes in the same locations, For the wine glass, however, your finger is moving around the rim, thus the vibrations of the rim as well as the standing wave patterns in the water also move with your finger.
variations and related activities
Just as the moving glass strikes water molecules inside the glass producing water waves, it also strikes air molecules to produce similar waves that travel through the air. We can't see those, but we do hear them as sound waves, and since the glass vibrates most at its resonant frequencies, those are the frequencies of sound (or musical notes) that we hear when it rings. These resonant or natural frequencies of your glass depend on its dimensions and the type of glass from which its made [you might experiment with wine glasses of different sizes and shapes], but as you may have noticed as you performed the experiment, it's actually very easy to change the resonant frequency of any glass- just add water! To demonstrate this, listen carefully to the note as you make the empty glass ring. Next fill the glass about half full of water and ring it again. The note you hear now should be much lower in pitch, because the glass is vibrating with a lower frequency. The mass of the water in the glass makes it heavier and causes it to vibrate more slowly. Experiment with different amounts of water to see how the note it makes as it rings changes. If you have several wine glasses, each with a different amount of water, you can even make a simple musical instrument (sometimes called a glass harp) to play a song (see the video link below). Ben Franklin actually invented a musical instrument based on this that he called the Glass Armonica.
You may not be able to directly see the wine glass move as it vibrates, but there are other ways to prove that it's moving. Sprinkle a few drops of water on the outside of the glass near the top then wet your finger and rub the rim of the glass until it rings as you did before. You should observe the droplets vibrating, showing that the glass is moving. Another way is to place a drinking straw or even a pencil inside the (empty) glass, leaning against the rim. Again wet your finger and rub the rim of the glass until it rings and the straw will begin to bounce around as the vibrating glass strikes it. As this continues the straw might actually stop moving for a moment and remain in the same location even though the glass is still vibrating (which you know because it is still ringing). This is because the rim of the glass moves more in some places and less in others, and the straw has happened to land in a spot on the rim where it is not moving enough to bounce he straw. In fact, if you could rub the glass without moving your finger around (of course that's not really possible, but bear with us), you would find that there are at least 4 locations along the rim where it is not moving at all (called nodes), 4 other locations halfway between each pair of nodes where the rim is moving the maximum amount (called anti-nodes), and at all other locations the amount of movement (or displacement) falls somewhere between the maximum and minimum. The shapes or patterns that the glass makes as it vibrates are called modes, and each frequency your hear corresponds to a different mode of vibration. One of the video links below shows the vibration mode of a wine glass driven by sound waves very nicely. This also demonstrates that just as a vibrating glass generates sound waves in the air, sound waves in the air from another source (in this case a loudspeaker) can actually strike an initially motionless glass and cause it to begin vibrating at the same frequency. If the sound waves are loud enough the glass may even vibrate too much and shatter!
Singing Bowls (coming soon).
references and links to more information
Resonance with wine glasses and singing bowls:
Vibrating a wine glass with sound waves: