There are indeed two nodes in the rubber band example. They are on the end of a rubber band and are held with your fingers. Although a standing wave has only half a wavelength, there is certainly a relationship between the length of the rubber band and the size of the wavelength.
guitar strings
Putting all these ideas together, let’s take a look at guitar strings. When you strike that string, it generates a standing wave with an antinode in the center and two nodes of he at the ends. This is called the first harmonic.
It is also possible to generate second harmonics (with a node in the middle) and even higher harmonics. However, due to the drag forces on the string, these higher frequencies disappear rather quickly, leaving only a standing wave with a wavelength equal to twice the length of the string.
But you don’t strum a guitar string to see standing waves. No, you strum your guitar because you want to make a sound. Maybe it’s music. What we really care about is the frequency of the vibrating guitar string. Let’s use some realistic values. If you use the highest frequency string, it can oscillate at 330 Hz. In terms of musical notes, this is E. Let’s also assume that the length of the string is 76.5 centimeters (30 inches). The length of this string gives us a wavelength of 1.53 meters.Currently in use v = λfwe find that the speed of the wave is 504.9 meters per second.
What if I want to play a G note, or 391 Hz, on the same string? You can do that by using your fingers to press down on the string on the fingerboard. This effectively changes the length of the string, changing the wavelength. The calculations show that an effective length of 64.6 centimeters (25.4 inches) reduces the wavelength enough to increase the frequency to 391 Hz. If you want to produce even higher frequency sounds, shorten the string even further.
How do you make a guitar sound lower than 330 Hz? You can’t do it with the same string. However, you can also get another string of the same length that has a higher linear density, or mass per unit length. This is why guitar strings have different thicknesses. Recall that you can change the speed of waves on a string by changing its properties. The higher the density, the slower the wave speed, and hence the lower the frequency. The rest is just music.
What should you do if your guitar doesn’t sound right, such as when an E note is being played at 325 Hz instead of 330 Hz? You can fix this problem by tuning your guitar. There is a tuning peg at the end of each string on the guitar. Turning this will increase or decrease the tension on the string. As you increase the tension, the speed of the waves on that string also increases, increasing their frequency. Now you’re not just playing guitar, you’re a guitar hero. Wait, it’s a video game. I don’t care.