To any mathematician out there… Help!
I am writting some code that measures the effects a curved surface has on wave motion. Here is my question:
On a straight surface, a wave traveling along it will have the normal opposing peaks with the same (but opposing) amplitude on each peak. If the surface was to be made circular (bringing the two ends together), how does this affect each amplitude of the peaks? And how does the ‘symetry’ of the wavelegth change? 
Well, I suppose nothing will change.
I’m trying to understand your question. In a normal one-cycle per unit of time representation of a sin wave. represented in 2D cartesian, the x-axis would be time. Do you mean to wrap the x-axis around in a circle? That’s chewy.
If you mean that the x-axis is irregular, do you desire some capability to correct that irregularity to a linear function?
If you don’t ask time to repeat itself, your frequency values increase. If you do ask for multiple time values, someone besides me will have to answer.
If you’re not trying to bend the time axis around to repeat time values, I think all you’re looking for is the derivative.
I’m going to bed.
Well, not quite yet. Would this circular surface be something that, like the curvature of the Earth, can be treated locally as a planar surface?
Wrapping the x-axis is what I want. What I have so far is that the ‘area’ under each of the curves (on a sine wave) would be the same (???). But is the ‘shape’ of the curve on the inner surface (of the x-axis) different than the shape of the outer surface? And if so, is the inner curve ‘thinner and longer’ (more of a peak) than the outer (which would be ‘shallower and flatter’)?
Is it possible to place a picture withought having it on the net?
Thats it… But imagine a huge tidle wave, even bigger than Hollywood could ever picture, that would leave most of the ocean beds dry. How does that curvature affect both the peak and trough?
i.e Is there a place to upload it in the SDMB
No, but there’s lots of image hosting sites, like Village Photos.
Say you have either a cylinder or a sphere, with circumference c.
Unless the wavelegth, λ, is such that c = n λ, where n is an integer, you’ll get destructive interference.
Are you talking about water waves? Or about something like vibrations of a wide metal sheet?
In either case, if the wave amplitude is very small, and if the wave length is much smaller than the radius of curvature of the ocean (or the metal sheet), then the sine wave remains “pure” and symmetrical. I think your question only involves large amplitudes where the peak of the wave is a good fraction of the radius of curvature of the medium the wave travels across.
In other words, I don’t know. 
Thanx Q.E.D
I hope this works.
http://www.villagephotos.com/usergallery-viewimage.asp?id_=4963026
I didn’t include any formulas or anything but you can see on the curved surface, it is like 2 waves being added together. Is this right? Or is the shape spread over the whole wave, changing the x-intersection?
…And how would this wave retain its energy over many revolutions? Do I need some constant (to keep it together)? If so, what would it be? (I’m assuming its the wavelength, but even then, how does the wavelegth change on the ‘sharp’ curves illustrated in the pic)
I have seen some complex formulas, but I need to approach this from a more abstract level. Ie Understand what they mean.
I’m not much of a mathemetician, but I can put two and two together.
[sub]fixed link - DrMatrix[/sub]
Desmostylus, I realize that in the case of c = n ë, we can have a kind of ‘natural-vibration’, but does the wavelength change in this case?
Alright.
For your plane wave, you can express the amplitude at any point either as a function of x or t.
A = A[sub]max[/sub] sin([symbol]a[/symbol][sub]1[/sub] + k[sub]1[/sub]x), or
A = A[sub]max[/sub] sin([symbol]a[/symbol][sub]2[/sub] + k[sub]2[/sub]t).
On your cylinder or sphere, A with respect to t could remain unchanged. A with respect to x changes to:
A = A[sub]max[/sub] sin([symbol]a[/symbol][sub]1[/sub] + k[sub]1[/sub]θr).
However, in extreme cases, when the wavelength becomes comparable to r, or A[sub]max[sub] is comparable to r, all sorts of other things could come into play that cause asymmetry (i.e., so that a sinusoid is not applicable).
What causes there to be a wave in the first place? Is it the action of gravity on a liquid? Is it surface tension on a liquid? Sound waves on a solid?
If it’s gravity, and A[sub]max[/sub] is comparable to r, you can no longer assume that the difference between gravitaional forces at peaks and troughs is negligible. You also have to formulate the equation so that the total enclosed volume remains constant.
Tricky stuff.
Crap. Try again:
Alright.
For your plane wave, you can express the amplitude at any point either as a function of x or t.
A = A[sub]max[/sub] sin([symbol]a[/symbol][sub]1[/sub] + k[sub]1[/sub]x), or
A = A[sub]max[/sub] sin([symbol]a[/symbol][sub]2[/sub] + k[sub]2[/sub]t).
On your cylinder or sphere, A with respect to t could remain unchanged. A with respect to x changes to:
A = A[sub]max[/sub] sin([symbol]a[/symbol][sub]1[/sub] + k[sub]1[/sub]θr).
However, in extreme cases, when the wavelength becomes comparable to r, or A[sub]max[/sub] is comparable to r, all sorts of other things could come into play that cause asymmetry (i.e., so that a sinusoid is not applicable).
What causes there to be a wave in the first place? Is it the action of gravity on a liquid? Is it surface tension on a liquid? Sound waves on a solid?
If it’s gravity, and A[sub]max[/sub] is comparable to r, you can no longer assume that the difference between gravitaional forces at peaks and troughs is negligible. You also have to formulate the equation so that the total enclosed volume remains constant.
Tricky stuff.
Tell me about it!!!
Thanx. I’ll have a think about it and get back to you. I have heaps of other work right now so it may be a couple of days.
okay, I thought about it, and i,ll start again, if I may.
The way I imagined it is that the vave motion describes an oscillation (so time will come into it… Eventually). However, there are two ‘oscillations’ working independently, each is in its own ‘dimension’. ie. I can express the surface as a wave (wavelength = 1, which describes the path of some other motion with time). I have then introduced another ‘wave’ on its ‘surface’, and now am wondering how the two will interact with each other as the wavelength of one approaches the other.
The way I see it is like 2 waves being added together. I could be wrong, but the above formula, if applied to discribing the motion of the moon around the earth (small wave) and the orbit of the earth around the sun (surface), then the ‘shape’ of the moons orbit would yield no difference in orbit as the ‘time of orbit’ increases (or distance from earth increases).
It is this kind of ‘interaction’ which I am after - and probaly less in the ‘weak’ gravitational form, and more on the ‘strong’ sub-atomic form.