Needing explanation of Kittel Intro to Solid States chapter 4.

[lylos]

In chapter 4, Kittel is setting up the differential equations for the forces involved in lattice vibrations. I understand the concept, the only problem I'm running into is why he chose the solution he did.

Here goes:
He states that the differential equation is
$$M\ddot{u_{s}}=C(u_{s+1}+u_{s-1}-2u_{s})$$

I understand this. He then goes to say that all displacements must have some time dependence of
$$exp(-Iwt)$$

Again, I understand why he says this. Then if you put this into the differential equation you get the following
$$-w^2 u_{s} M=C(u_{s+1}+u_{s-1}-2u_{s})$$

He then states "This is a difference equation in the displacements u and has traveling wave solutions of the form:"
$$u_{s\pm1}=uexp(IsKa)exp(\pm IKa)$$

This is where I'm running into the issue. Why does he have uexp(IsKa)exp(IKa) to the solution of u(s+-1)?

 

[marcusl]

The difference equation is a second order differential equation (a wave equation, to be exact) like the one for the time dependence, but written instead in the form of discrete differences. You can verify this starting with the definition of the derivative

$$\frac{du}{dx}=\lim_{\Delta x \rightarrow 0} \frac{u(s)-u(s-1)}{\Delta x}$$

Using the difference form is valid since the spacing a between planes is very small. Now take the second derivative to get the same second-order finite difference equation that you wrote above. The solutions to this discretized wave equation are traveling waves

$$u(s+p)=u(0)e^{i[(s+p)Ka-\omega t]}$$

for any p. (This is close to but not quite what you wrote...)