Wavelengths and Superposition Principle?

Question

Two questions

1) What is the Superposition Principle?

Can you walk me through this problem?

2) A guitar string is 0.75 m long. Find the wavelengths of the fundamental and the second, third, and fourth harmonics of vibration.

Answer 1

1. In most general terms, superposition says that the response of a linear system to two or more signals added together equals the sum of its responses to each signal alone. This means, for instance, that if air were a totally linear transmitter of sound, two sound waves would add together and we would get effects such as reinforcement and cancellation. In fact we do get these phenomena, so air is generally thought of as linear. (In extreme cases it isn't; pressure can increase to very high levels, but can only decrease to zero.)

2. This is important to understand. Since the string is contrained at both ends, it can only resonate to a wave that has displacement nulls, or nodes, at the ends. If you look at a sine wave, you see it has nodes spaced 1/2 wavelength apart. This means the longest wavelength the string can accommodate is twice the length of the string, 2L. That's the string's fundamental wavelength.

All the harmonics sustainable by the string are similarly required to have nodes at either end, so the next highest has those two plus one in the middle; so the 2nd harmonic's wavelength equals L. The 3rd, 4th etc. harmonics have wavelengths of 2L/3, 2L/4, etc. So all the string's resonant wavelengths are integer divisors of 2L. For this reason, the string's resonant frequencies are integer multiples of the fundamental frequencies.  

These same rules determine the resonant wavelengths and frequencies of pipes that are open at both ends, except that you replace displacement nodes with displacement maxima (antinodes) and pressure nodes.