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July 18th, 2014, 04:52 AM   #1
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A challenge, to find the equation for a function

So after really considering this album cover (titled 'AM'), I decided that I really wanted to come up with a function that resembled it. Even with my relatively limited understanding of functions, I decided to give it a shot. However, the best thing I could come up with is this:

f(x) = sin(x) * (35-x) * (x+35)

Its definitely not much and quite possibly useless, but from x=-35 to x=35 it is quite close to the waves on the left of the cover and has the same number of crests/troughs. My initial thought would be to use cosine so that the highest crest would be in the centre, however having looked at the image further, I noticed the highest crest is slightly off-centre and there is also a single lowest crest, so sine is actually more accurate. On another note, the image I linked to is technically not the album cover, it has been extended to be flat on either end. I imagine it would be quite difficult to do that, but I thought it might be worth mentioning as I think it would look good. Otherwise just a gradual decrease in amplitude would look fine.

This may seem quite pedantic (and pointless...), but its all part of the challenge (and fun?) I suppose. Anyway, I've become quite frustrated with this problem and I feel somewhat obliged to see it solved. So, does anyone have any suggestions (or answers?!), as to what I can do?

Also, something tells me such a graph might not be possible, so someone please say so if it isn't :P

PS. Oh, and the waves in the centre of the image actually say 'AM', so I can't overlook that...

Last edited by opaque; July 18th, 2014 at 04:56 AM.
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July 18th, 2014, 05:08 AM   #2
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I suspect wave-packets might achieve what you're after. They are of the form

$\displaystyle u(x,t) = Ae^{-(x-ct)^2 + ik(x-ct)}$

where $\displaystyle c$ is the speed of the wave, $\displaystyle k$ is the wave number and $\displaystyle i = \sqrt{-1}$.

I imagine you can take two wavepackets, one which is displaced relative to the first one and add them together carefully to make the function that looks like the one on the cover.

Wave packet - Wikipedia, the free encyclopedia
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