1

This may seem backwards since a fourier series isn't typically used this way but I'm trying to prove whether or not the sum of sin and cos waves could produce a sin wave with a wave length that is not in any of the summed waves.

I don't intend the use of the fourier series as a restriction. It just seemed an obvious place to start thinking about this problem.

The restriction is, make a sin of finite wavelength L by summing any sin's and cos's so long as they do not have that same wavelength L.

candied_orange
  • 672

1 Answers1

1

If you are working with a finite interval, you have available to you the sine and cosine waves with frequencies that are multiples of $\frac {2\pi}L$ These waves are all orthogonal, so you cannot approximate any one as a sum of the others.

Ross Millikan
  • 374,822
  • If I'm understanding this correctly it's saying that many shorter and shorter wavelength sins could NOT be added up to make a sin with a wavelength that is bigger than any of the others? Even if they were allowed to be phase shifted? – candied_orange Mar 16 '15 at 04:53
  • That is correct. The phase shift doesn't help because you already have the sines and cosines of that wavelength. The combination can represent any phase shift. – Ross Millikan Mar 16 '15 at 04:56
  • I'm not 100% sure I understand the finite interval concern. I intend that the desired sin wave not have an infinite wavelength. I do not intend to restrict the fourier transformation to a particular method. – candied_orange Mar 16 '15 at 05:13
  • The finite interval is needed to make sure the function is periodic, so that it can be expressed as a Fourier series. The point then is that you need all the terms of the Fourier series available because you cannot express one in terms of the others. Using an infinite interval and the Fourier integral raises convergence concerns, but I think comes to the same conclusion as long as you demand that the sine wave continue for all time. I do't understand what you mean by "not restrict the Fourier transformation to a particular method". What new freedom do you want to give? – Ross Millikan Mar 16 '15 at 05:19
  • So if I allowed the wave length to decay there might still be a way? – candied_orange Mar 16 '15 at 05:22
  • What are you expressing it in, if not sine and cosine waves that are multiples of a given frequency? They can only express things that are periodic with the same period. If you use the integral, you will still be missing the piece that comes from the integral with the sine wave you have removed. You can see what the result is-take your function and subtract the component that you have removed. The expression will match the new function. – Ross Millikan Mar 16 '15 at 05:28
  • I didn't want to restrict your methods to a fourier transform. I gave an answer to a question about how multiple moons would affect the tides. After I did I started thinking about how a fourier transform allows you to build up any shape. However, the point of the multiple moons was to fill in for a single one that would have produced the same tides. I didn't think it could ever work out perfectly but I wanted to be sure. You've pretty much confirmed that multiple moons can't be a perfect substitute. http://worldbuilding.stackexchange.com/a/11847/6866 – candied_orange Mar 16 '15 at 05:34
  • You can make an analogy with vectors in $\Bbb R^n$, because the Fourier functions form a vector space. If you have the standard basis, ${e_1,e_2,\dots,e_n}$, you can't express $e_1$ or anything like it in terms of the rest. The same thing goes on in Fourier space, but our intuition is not as good. – Ross Millikan Mar 16 '15 at 14:38