# Standing Waves with One Fixed End: Relationship Explained

• theneedtoknow
In summary, standing waves produced in a rope or musical instrument with both ends fixed have a relationship between length and possible wavelengths characterized by wavelength = 2xlength / n, where n is an integer. When only one end is fixed, the wave will have an antinode at the unfixed end, but the relationship between length and wavelength remains the same.
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?

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.

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.

## 1. What is a standing wave with one fixed end?

A standing wave with one fixed end is a type of wave that occurs when one end of a medium is fixed while the other end is free to move. This creates a reflected wave that interferes with the incoming wave, causing stationary points of maximum and minimum amplitude.

## 2. How is the relationship between wavelength and frequency explained in standing waves with one fixed end?

The relationship between wavelength and frequency in standing waves with one fixed end is that the wavelength is equal to twice the length of the vibrating medium, while the frequency is determined by the speed of the wave and the length of the medium.

## 3. What is the significance of nodes and antinodes in standing waves with one fixed end?

Nodes are points of zero amplitude in a standing wave, while antinodes are points of maximum amplitude. In a standing wave with one fixed end, there will always be a node at the fixed end and an antinode at the free end. The number and location of nodes and antinodes can help determine the wavelength and frequency of the wave.

## 4. How do standing waves with one fixed end differ from those with two fixed ends?

The main difference between standing waves with one fixed end and two fixed ends is that in the latter, both ends of the medium are fixed and there is no free end. This results in a different pattern of nodes and antinodes and a different relationship between wavelength and frequency.

## 5. What are some real-life examples of standing waves with one fixed end?

Standing waves with one fixed end can be observed in a variety of real-life situations, such as in musical instruments like flutes and clarinets, where the air column inside acts as the vibrating medium. They can also be seen in strings of musical instruments, such as guitars and violins, or in ropes and cables that are fixed at one end and free to vibrate at the other. Additionally, standing waves with one fixed end can be created in water by partially submerging a rod or stick and vibrating it at one end.

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