Particles on a String: Max Frequency & Effects of Oscillation

[aaaa202]

Consider N atoms separated by springs of force constant C. If one writes up the N linear differential equations for the displacements of the string, you find that the solutions are traveling wavse of the form exp(ikx) and on finds a dispersional relation of the form:
sin(...), which means that there is a maximum frequency which the oscillations can support. I want to understand this last phenomenon that there is an upper bound on the frequence. Why is that so? What happens at this maximal frequency and what happens if we try to oscillate the atoms with a higher frequency that the maximal?

 

[Simon Bridge]

Why is that so?

It tells you:
"Because there is a dispersional relation of form..."

Why is that so? What happens at this maximal frequency and what happens if we try to oscillate the atoms with a higher frequency that the maximal?

Why not set up an equation and put some vales in and find out?

Also consider: what does "dispersion" mean?

A high frequency means a small wavelength - if there are N small masses joined by springs, so the separation of the masses is $\small \Delta x$ what does a wave with wavelength $\small 2\Delta x$ look like?

 

[aaaa202]

"A high frequency means a small wavelength". This is exactly where I have trouble. Because I am used to continuous strings where frequency is proportional to 1/wavelength. But in this case, and any other case, is it always true that small wavelength equals small frequency?

 

[Simon Bridge]

is it always true that small wavelength equals small frequency?

It's never true.
Small wavelength always means high frequency ... all other things remaining equal.
You meant to say that right?

 

[aaaa202]

oops yes.

 

[aaaa202]

Okay so now I have drawn the problem for the minimal wavelength and I could see what you meant. That trying to make the wavelength shorter only resulted in an actually longer wavelength due to the discreteness of the particles on the string. Now it all makes sense but one stupid thing: Why does small wave length always imply high frequency and vice versa? I can see it from the math but what is the physical interpretation?

 

[Simon Bridge]

Why does small wave length always imply high frequency and vice versa? I can see it from the math but what is the physical interpretation?

Why is everything the other way up when you stand on your head?

You can work out the math for what each word means, separately, and then derive the relationship.

... "frequency" and "wavelength over speed" are different ways of looking at the same thing
... noticing that wavelength and frequency have an inverse relationship is like noticing that everything turns the other way up when you stand on your head.