The rubber band example does indeed have two nodes—they are at the ends of the rubber band where your fingers hold it. We only have half a wavelength in the standing wave, but there is indeed a relationship between the length of the rubber band and the size of the wavelength.
Guitar Strings
It’s time to put all these ideas together and look at a guitar string. Once you hit that string, it’s going to create a standing wave with an antinode in the middle and two nodes on the ends. This is called the first harmonic wave.
It’s possible to also produce a second harmonic wave (with a node in the middle) and even higher harmonics. However, because of drag forces on the string, these higher frequencies die out fairly quickly so that you are just left with a standing wave that has a wavelength equal to twice the length of the string.
But you don’t strum a guitar string to see a standing wave. No, you strum the guitar because you want to make a sound—maybe even some music. What we really care about is the frequency of that oscillating guitar string. Let’s use some realistic values. If you use the highest-frequency string, it could oscillate at 330 Hz. In terms of musical notes, that’s an E. Let’s also assume that the length of the string is 76.5 centimeters (30 inches). From this string length we can get a wavelength of 1.53 meters. Now using v=λf, we find a wave speed of 504.9 meters per second.
What if I want to play a G note, or 391 Hz, on the same string? I can do that by using my finger to push the string down on the fretboard. This effectively changes the length of the string and changes the wavelength. We can do the math and find that with an effective length of 64.6 centimeters (25.4 inches), the wavelength will decrease enough to cause the frequency to increase to 391 Hz. If you want an even higher-frequency note, just make the string even shorter.
How do you make a guitar note that’s lower than 330 Hz? You can’t do it with that same string. But you can get another string that has the same length but a higher linear density, or mass per unit length—which is why the strings on a guitar have different thicknesses. Remember that we can change the speed of the waves on the string by changing the properties of the string. With a higher density you get a lower wave speed, which means a lower frequency. The rest is just music.
What if your guitar doesn’t sound right, like if your E note is playing at 325 Hz instead of 330 Hz? You can solve this problem by tuning your guitar. At the end of each guitar string is a tuning peg. If you turn this, you will either increase or decrease the string’s tension. Increasing the tension will also increase the wave speed on that string, which increases the frequency. Now you aren’t just playing a guitar, you are a guitar hero. Wait, that’s a video game. Never mind.
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