Standing Waves with One Fixed End: Relationship Explained

[theneedtoknow]

I know that typically, standing waves (liek those produced in a musical instrument sloed at both ends, or by a rope tied to a point) have a relationship between the length (of the rope/instrument) and the possible wavelengths characterized by wavelength = 2xlength / n
where n is an integer number, and that produces 1st harmonic, 2nd harmonic, so on.

But what about if only one end is fixed and the other is not? (like jiggling a ripe tied with a loop to a pole such that the loop can move up and down)?
How is their relationship defined then?
I know that the closed end still remains as a node, but the other end will now be an antinode instead of a node
so would it be
wavelength = 2xlength / n-0.5?
since there's 0.5 less "loops" produced?

 

[Valerie Park]

No, the relationship between the length of the rope and the possible wavelengths remains the same: wavelength = 2xlength / n, where n is an integer number. The only difference is that when one end is not fixed, the wave will be an antinode rather than a node at the other end.

 

[sir-pinski]

Yes, you are correct in your understanding of the relationship between length and wavelength for standing waves with one fixed end. The formula is still the same, but the difference is that the number of nodes (and therefore, the number of loops) will be one less than the number of antinodes. This is because the fixed end acts as a node, but the other end can now move up and down, producing an antinode.

So the formula would be: wavelength = 2xlength / (n-0.5), where n is an integer number representing the number of antinodes. This means that the possible wavelengths will be slightly shorter than those produced in a standing wave with both ends fixed.

One interesting thing to note is that the fundamental frequency (1st harmonic) for a standing wave with one fixed end will have the same wavelength as the first overtone (2nd harmonic) for a standing wave with both ends fixed. This is because in both cases, there is one antinode and one node.

Overall, the relationship between length and wavelength for standing waves with one fixed end is still based on the fundamental equation of wavelength = 2xlength / n, but the interpretation and application of this formula may be slightly different due to the presence of a fixed end.