Click here to purchase the entire book in PDF format. Standing waves (aka Resonant Frequencies) Take a look at a guitar string from the side as is shown in the top diagram in Figure 3.38. It's a long, thin piece of metal that's pulled tight and, on each end it's held up by a piece of plastic called the bridge at the bottom and the nut at the top. Since the nut and the bridge are firmly attached to heavy guitar bits, they're harder to move than the string, therefore we can say that the string is terminated at both ends with a high impedance. Figure 3.38: A frame-by-frame diagram showing what happens to a guitar string when you pluck it. Let's look at what happens when you pluck a perfect string from the the centre point in slow motion. Before you pluck the string, it's sitting at the equilibrium position as shown in the top diagram in Figure 3.38. You grab a point on the string, and pull it up so that it looks like the second diagram in Figure 3.38. Then you let go... One way to think of this is as follows: The string wants to get back to its equilibrium position, so it tries to flatten out. When it does get to the equilibrium position, however, it has some momentum, so it passes that point and keeps going in the opposite direction until it winds up on the other side as far as it was pulled in the first place. If we think of a single molecule at the centre of the string where you plucked, then the behaviour is exactly like a simple pendulum. At any other point on the string, the molecules were moved by the adjacent molecules, so they also behave like pendulums that weren't pulled as far away from equilibrium as the centre one. So, in total, the string can be seen as an infinite number of pendulums, all connected together in a line, just as is explained in Huygens theory. If we were to pluck the string and wait a bit, then it would settle down into a pretty predictable and regular motion, swinging back and forth looking a bit like a skipping rope being turned by a couple of kids. If we were to make a move of this movement, and look at a bunch of frames of the film all at the same time, they might look something like Figure 3.39. Figure 3.39: The first mode of vibration of a string anchored on both ends. This will vibrate at a frequency $f$. In Figure 3.39, we can see that the string swings back and forth, with the point of largest displacement being in the centre of the string, halfway between the two anchored points at either end. Depending on the length, the tension and the mass of the string, it will swing back and forth at some speed (we'll look at how to calculate this a little later...) which will determine the number of times per second it oscillates. That frequency is called the fundamental resonant frequency of the string. If it's in the right range (between 20 Hz and 20,000 Hz) then you'll hear this frequency as a musical pitch. In reality, this is a bit of an oversimplification. The string actually resonates at other frequencies. For example, if you look at Figure 3.40, you'll see a different mode of oscillation. Notice that the string still can't move at the two anchored end points, but it now also does not move in the centre. In fact, if you get a skipping rope or a telephone cord and wiggle it back and forth regularly at the right speed, you can get it to do exactly this pattern. I would highly recommend trying. A short word here about technical terms. The point on the string that doesn't move is called a node. You can have more than one node on a string as we'll see below. The point on the string that has the highest amplitude of movement is called the antinode, because it's the evil opposite of the node, I suppose... Figure 3.40: The second mode of vibration of a string anchored on both ends. This will vibrate at a frequency $2 f$. One of the interesting things about the mode shown in Figure 3.40 is that its wavelength on the string is exactly half the wavelength of the mode shown in Figure 3.39. As a result, it vibrates back and forth twice as fast and therefore has twice the frequency. Consequently, the pitch of this vibration is exactly one octave higher than the first. This pattern continues upwards. For example, we have seen modes of vibration with one and with two ``bumps'' on the string, but we could also have three as is shown in Figure 3.41. The frequency of this mode would be three times the first. This trend continues upwards with an integer number of bumps on the string until you get to infinity. Figure 3.41: The third mode of vibration of a string anchored on both ends. This will vibrate at a frequency $3 f$. Since the string is actually vibrating with all of these modes at the same time, with some relative balance between them, we wind up hearing a fundamental frequency with a number of harmonics. The combined timbre (or sound colour) of the sound of the string is determined by the relative levels of each of these harmonics as they evolve over time. For example, if you listen to the sound of a guitar string, you might notice that it has a very bright sound immediately after it has been plucked, and that the sound gets darker over time. This is because at the start of the sound, there is a relatively high amount of energy in the upper harmonics, but these decay more quickly than the lower ones and the fundamental. Therefore, at the end of the sound, you get only the lowest harmonics and fundamental of the string, and therefore a darker sound quality. It might be a little difficult to think that the string is moving at a maximum in the middle for some modes of vibration in exactly the same place as it's not moving at all for other modes. If this is confusing, don't worry, you're completely normal. Take a look at Figure 3.42 which might help to alleviate the confusion. Each mode is an independent component that can be considered on its own, but the total movement of the string is the result of the sum of all of them. Figure 3.42: The sum of the three first modes of vibration of a string. The top three plots show the first, second and third modes of vibration with relative amplitudes 1, 0.5 and 0.25 respectively. The bottom plot is the sum of the top three. One of the neat tricks that you can do on a stringed instrument such as a violin or guitar is to play these modes selectively. For example, the normal way to play a violin is to clamp the string down to the fingerboard of the instrument with your finger to effectively shorten it. This will produce a higher note if the string is plucked or bowed. However, you could gently touch the string at exactly the halfway point and play. In this case, the string still has the same length as when you're not touching it. However, your finger is preventing the string from moving at one particular point. That point (if your finger is halfway up the string) is supposed to be the point of maximum movement for all of the odd harmonics (which include the fundamental - the 1st harmonic). Since your finger is there, these harmonics can't vibrate at all, so the only modes of vibration that work are the even-numbered harmonics. This means that the second harmonic of the string is the lowest one that's vibrating, and therefore you hear a note an octave higher than normal. If you know any string players, get them to show you this effect. It's particularly good on a cell or double bass because you can actually see the harmonics as a shape of the string.