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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

[tex]M\ddot{u_{s}}=C(u_{s+1}+u_{s-1}-2u_{s})[/tex]

I understand this. He then goes to say that all displacements must have some time dependence of

[tex]exp(-Iwt)[/tex]

Again, I understand why he says this. Then if you put this into the differential equation you get the following

[tex]-w^2 u_{s} M=C(u_{s+1}+u_{s-1}-2u_{s})[/tex]

He then states "This is a difference equation in the displacements u and has traveling wave solutions of the form:"

[tex]u_{s\pm1}=uexp(IsKa)exp(\pm IKa)[/tex]

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

Here goes:

He states that the differential equation is

[tex]M\ddot{u_{s}}=C(u_{s+1}+u_{s-1}-2u_{s})[/tex]

I understand this. He then goes to say that all displacements must have some time dependence of

[tex]exp(-Iwt)[/tex]

Again, I understand why he says this. Then if you put this into the differential equation you get the following

[tex]-w^2 u_{s} M=C(u_{s+1}+u_{s-1}-2u_{s})[/tex]

He then states "This is a difference equation in the displacements u and has traveling wave solutions of the form:"

[tex]u_{s\pm1}=uexp(IsKa)exp(\pm IKa)[/tex]

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

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