Find fourier series of wave function

[leroyjenkens]

Homework Statement

Find Fourier series of f(x) = Acos($\pi$x/L)
I know how to do this, I just don't know the value of L. If it's equal to $\lambda$/2, then I know the solution. But the question does not specify the value of L. L is just the length of the entire wave that I'm working with, right? If it's $\lambda$/2, then that's only half the wave, but the answer to the question works out to be what the solution manual says. How was I supposed to know that L = $\lambda$/2? The only information given is what I wrote down.
Thanks

 

[BvU]

before a question like "Find Fourier series of $f(x) = A\; \cos( \pi x/L)$ "" can be answered, one would need to agree on the range of x. From the word "series" in the question, a periodicity is implied. It makes a big differerence if the range is $[0,L]$ or $[0,2L]$. The first leads to an odd function, the second to an even function (other , more pathological ranges can be imagined as well..). This second is almost trivial: to me f(x) = A cos(π x/L) is already a Fourier series of one contributing component.

So it depends on the context of the preceding expose in your textbook whether you can take the easy route (second) or whether you have to do some real work ( range is $[0,L]$ and the function is periodically continued).

Apparently they want you to do some real work. There's no need for them to leave you guessing, so in that respect I'm with you in feeling a bit misguided. Doesn't do much good, but there it is :)

 

[leroyjenkens]

before a question like "Find Fourier series of $f(x) = A\; \cos( \pi x/L)$ "" can be answered, one would need to agree on the range of x. From the word "series" in the question, a periodicity is implied. It makes a big differerence if the range is $[0,L]$ or $[0,2L]$. The first leads to an odd function, the second to an even function (other , more pathological ranges can be imagined as well..). This second is almost trivial: to me f(x) = A cos(π x/L) is already a Fourier series of one contributing component.

So it depends on the context of the preceding expose in your textbook whether you can take the easy route (second) or whether you have to do some real work ( range is $[0,L]$ and the function is periodically continued).

Apparently they want you to do some real work. There's no need for them to leave you guessing, so in that respect I'm with you in feeling a bit misguided. Doesn't do much good, but there it is :)

Thanks for the response. Going by the solution manual, it just looks like it was supposed to be assumed that L = λ/2. I don't know why, it doesn't say anything about it in the chapter. I thought maybe there was a typical L that was used in this sort of problem, but I guess not.
Unless I'm given L, I have no idea how to find out what L is from that information I was given in the question (just the function itself that includes L).

 

[BvU]

You're right it sucks, but now it's time to do the integrals !

 

[HalfLostAndSearching]

for your question. The value of L in Fourier series is typically determined by the boundary conditions of the system in question. In this case, it seems like the system is a periodic function with a wavelength of \lambda. Therefore, the length L of the wave would be equal to the wavelength \lambda, as the periodicity repeats itself every \lambda units. However, if the system has different boundary conditions, the value of L may be different and would need to be specified in the question. It is important to carefully read the question and understand the given information to determine the appropriate value of L for the Fourier series.