Question

1. Vibrations of square lattice. We consider transverse vibrations of a planar square lattice of rows and columns of identical atoms, and let $u_{l,m}$ denote the displacement normal to the plane of the lattice of the atom in the $l$th column and $m$th row (Fig. 13). The mass of each atom is $M$, and $C$ is the force constant for nearest neighbor atoms. (a) Show that the equation of motion is $M(d^2u_{l,m}/dt^2) = C[(u_{l+1,m} + u_{l-1,m} - 2u_{l,m}) + (u_{l,m+1} + u_{l,m-1} - 2u_{l,m})]$. Figure 13 Square array of lattice constant a. The displacements considered are normal to the plane of the lattice. (b) Assume solutions of the form $u_{l,m} = u(0) \exp[i(lka + mka - \omega t)]$ where a is the spacing between nearest-neighbor atoms. Show that the equation of

          1. Vibrations of square lattice. We consider transverse vibrations of a planar square
lattice of rows and columns of identical atoms, and let $u_{l,m}$ denote the displacement
normal to the plane of the lattice of the atom in the $l$th column and $m$th row (Fig. 13).
The mass of each atom is $M$, and $C$ is the force constant for nearest neighbor atoms.
(a) Show that the equation of motion is
$M(d^2u_{l,m}/dt^2) = C[(u_{l+1,m} + u_{l-1,m} - 2u_{l,m}) + (u_{l,m+1} + u_{l,m-1} - 2u_{l,m})]$.
Figure 13 Square array of lattice constant a. The
displacements considered are normal to the plane
of the lattice.
(b) Assume solutions of the form
$u_{l,m} = u(0) \exp[i(lka + mka - \omega t)]$
where a is the spacing between nearest-neighbor atoms. Show that the equation of

1. Vibrations of square lattice. We consider transverse vibrations of a planar square
lattice of rows and columns of identical atoms, and let ul,m denote the displacement
normal to the plane of the lattice of the atom in the lth column and mth row (Fig. 13).
The mass of each atom is M, and C is the force constant for nearest neighbor atoms.
(a) Show that the equation of motion is
M(d^2ul,m/dt^2) = C[(ul+1,m + ul-1,m - 2ul,m) + (ul,m+1 + ul,m-1 - 2ul,m)].
Figure 13 Square array of lattice constant a. The
displacements considered are normal to the plane
of the lattice.
(b) Assume solutions of the form
ul,m = u(0) [i(lka + mka - ω t)]
where a is the spacing between nearest-neighbor atoms. Show that the equation of

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