How Do Nodes Function in Standing Waves with Fixed and Open Ends?

[kathyt.25]

Homework Statement

I don't have an actual problem to solve, it's more of a conceptual problem.

I'm trying to understand harmonics and standing waves, and how the nodes work out, but it's very confusing.

(1) If a wave is FIXED at both ends, for the fundamental or 1st harmonic, does it have 0 nodes, 1 node, or 2 nodes? Do you count both "end nodes" where the wave is fixed?

(2) If a wave is FIXED at one end, and OPEN at the other end, I know that the equation for frequency is f(n) = n*v / 4L, where n=1,3,5,7...
Does "n" represent the # nodes, or the harmonic level?
This is super confusing for me, because for f(1), there is 1 node at the fixed end, which makes sense, because this is the first harmonic.
However, for f(2), there are 2 nodes, but "n" can only equal odd numbers. So there is no second harmonic... Why is that? So to find values of n, we have to go by harmonic number, and not the number of nodes, for all situations (ie. open, closed, open-closed waves)?

Homework Equations

The Attempt at a Solution

 

[diazona]

(1) If a wave is FIXED at both ends, for the fundamental or 1st harmonic, does it have 0 nodes, 1 node, or 2 nodes? Do you count both "end nodes" where the wave is fixed?

Yes you do.

(2) If a wave is FIXED at one end, and OPEN at the other end, I know that the equation for frequency is f(n) = n*v / 4L, where n=1,3,5,7...
Does "n" represent the # nodes, or the harmonic level?
This is super confusing for me, because for f(1), there is 1 node at the fixed end, which makes sense, because this is the first harmonic.
However, for f(2), there are 2 nodes, but "n" can only equal odd numbers. So there is no second harmonic... Why is that? So to find values of n, we have to go by harmonic number, and not the number of nodes, for all situations (ie. open, closed, open-closed waves)?

Think of it this way: n represents the number of half-wavelengths that fit into the length L. A wave that consists of an odd number of half-wavelengths has a node at one end and an antinode at the other end, which fits perfectly into a pipe that's open at one end and closed at the other. Alternatively, for a wave that has an even number of half-wavelengths, there are two possibilities: either it has a node at both ends, which fits a pipe that's closed at both ends, or it has an antinode at both ends, which fits a pipe that's open at both ends. That's why for pipes that are open at both ends or closed at both ends, n only takes even values.

 

[nlightner]

In response to your questions:

1) For a wave that is fixed at both ends, the fundamental or 1st harmonic will have 0 nodes. This is because the nodes are points along the wave where there is no displacement, and since the wave is fixed at both ends, there is no room for any nodes to form. So, you do not count the "end nodes" in this case.

2) The "n" in the equation f(n) = n*v / 4L represents the harmonic level, not the number of nodes. The harmonic level refers to the number of times the wave repeats within the length of the medium (in this case, the length L). So for f(1), there is 1 node at the fixed end, and for f(2), there are 2 nodes, but this does not mean there is a second harmonic. The second harmonic would actually be f(3), where there are 3 nodes. This is because the equation only applies to odd values of n, as you mentioned. This is because the nodes must always be located at the ends of the medium, and in between the nodes, the wave must have a half-integer number of wavelengths. This is why the equation only works for odd values of n.

I hope this helps clarify the concept of nodes in standing waves. If you have any further questions, please feel free to ask.