Relating Arc Length and Standing Wave Patterns

[ThomasMagnus]

Homework Statement

I am currently reviewing the physics of 'standing waves on a string'. I know that for the nth harmonic, the length of the 'string' is $\frac{n\lambda}{2}$. Instead of just memorizing these, I have been trying to apply my knowledge of Calculus to figure out why these numbers are what they are. However, I am having a bit of trouble. Here's what I have tried to do thus far:

The Attempt at a Solution

If you take the graph of Sin(x), the arc length of the curve between 0 and a can be calculated with the following integral:

$\int \sqrt{1+cos(x)^{2}}dx$ (with limits 0 and a). So one wavelength would be equal to a distance of 2$\pi$. The integral between 0 and $\pi$ $\approx 3.82$ while the integral between 0 and 2$\pi$ $\approx 7.64$

So, that means if we had a wave that was a sine function defined between 0 and $\pi$, the wavelength would be 1/2 that of a sin function between 0 and 2$\pi$ and it would have a length of 1/2 that of the function defined between 0 and 2$\pi$.

What I am confused about is that I keep just getting:

First Harmonic $\frac{\lambda}{2}$=$\frac{L}{2}$ ($\lambda$=L)

Apart from it being the wrong answer, this would mean the wavelength, $\pi$, is equal to L which by the arc length formula is 3.82. Furthermore, would the wavelength have to be the same as the arc length (i.e. length of the string?)

Also if I said I wanted to find the length of the sine wave between 0 and $\pi$ wouldn't I be able to say that the wavelength= $\pi$ and by definition L=wavelength/2 for first harmonic. Then I would get $\pi$/2 = 3.82. 1.57≠3.82

Maybe by Length of the string they don't mean arc length?

Am I just completely missing something here, or am I using Calculus in the wrong place.

Thanks!(EDIT) I think I figure it out. I was making it WAY more complicated then it has to be. Since there are two nodes on each end, it is like fixing a string in two positions and pulling it up, so the length of the string can't change then.

 

[Simon Bridge]

(EDIT) I think I figure it out. I was making it WAY more complicated then it has to be. Since there are two nodes on each end, it is like fixing a string in two positions and pulling it up, so the length of the string can't change then.

Well done :)

Basically the model is for those situations where the overall length of the string can change while the tension remains constant and the separation of the endpoints is a constant ... where there are not a lot of losses. If there are no losses, the standing wave does not even have to be a sine wave. (i.e. pluck it in the middle and you have a triangular standing wave.)

The equation you start with for calculus is the wave equation in one dimension:
http://en.wikipedia.org/wiki/Wave_equation
... the solution is generally not trivial (from scratch) but the upshot is that the solutions can be expanded in a trigonometric series, and you are back to counting nodes and antinodes between 0 and L.

Aren't you glad you asked ?!

 

[CWatters]

Maybe by Length of the string they don't mean arc length?

Correct. The wave length λ is the straight line distance between two nodes (or antinodes). The length of string refers to the length with no wave on it.

Edit: oops my post crossed with Simons.