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ThomasMagnus
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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 [itex]\frac{n\lambda}{2}[/itex]. 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:
[itex]\int \sqrt{1+cos(x)^{2}}dx[/itex] (with limits 0 and a). So one wavelength would be equal to a distance of 2[itex]\pi[/itex]. The integral between 0 and [itex]\pi[/itex] [itex]\approx 3.82 [/itex] while the integral between 0 and 2[itex]\pi[/itex] [itex]\approx 7.64 [/itex]
So, that means if we had a wave that was a sine function defined between 0 and [itex]\pi[/itex], the wavelength would be 1/2 that of a sin function between 0 and 2[itex]\pi[/itex] and it would have a length of 1/2 that of the function defined between 0 and 2[itex]\pi[/itex].
What I am confused about is that I keep just getting:
First Harmonic [itex]\frac{\lambda}{2}[/itex]=[itex]\frac{L}{2}[/itex] ([itex]\lambda[/itex]=L)
Apart from it being the wrong answer, this would mean the wavelength, [itex]\pi[/itex], 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 [itex]\pi[/itex] wouldn't I be able to say that the wavelength= [itex]\pi[/itex] and by definition L=wavelength/2 for first harmonic. Then I would get [itex]\pi[/itex]/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.
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