Question

Consider the normal modes of a linear diatomic chain in which the force constants between nearest neighbour atoms are alternately C and 5C. Let the masses be equal, and let the nearest neighbour separation be a/2. If the equations of motion for such a system is m d^2u_s/dt^2 = C(v_s + 5v_{s-1} - 6u_s) m d^2v_s/dt^2 = C(u_s + 5u_{s+1} - 6v_s) and we choose travelling wave solutions u_s = u e^{j(ska-?t)} and v_s = v e^{j(ska-?t)}, find ?(k) at k = 0 and k = ?/a.

          Consider the normal modes of a linear diatomic chain in which the force constants between nearest neighbour atoms are alternately C and 5C. Let the masses be equal, and let the nearest neighbour separation be a/2.

If the equations of motion for such a system is

m d^2u_s/dt^2 = C(v_s + 5v_{s-1} - 6u_s)

m d^2v_s/dt^2 = C(u_s + 5u_{s+1} - 6v_s)

and we choose travelling wave solutions u_s = u e^{j(ska-?t)} and v_s = v e^{j(ska-?t)},

find ?(k) at k = 0 and k = ?/a.

Consider the normal modes of a linear diatomic chain in which the force constants between nearest neighbour atoms are alternately C and 5C. Let the masses be equal, and let the nearest neighbour separation be a/2.

If the equations of motion for such a system is

m d^2us/dt^2 = C(vs + 5vs-1 - 6us)

m d^2vs/dt^2 = C(us + 5us+1 - 6vs)

and we choose travelling wave solutions us = u e^j(ska-?t) and vs = v e^j(ska-?t),

find ?(k) at k = 0 and k = ?/a.

Added by Brian L.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Sri K

11/13/2022

Step 1

This gives us: d^2ue^(i(ska-wt))/dt^2 = C(ve^(i(ska-wt)) + 5ve^(i((s-1)ka-wt)) - 6ue^(i(ska-wt))) and d^2ve^(i(ska-wt))/dt^2 = C(ue^(i(ska-wt)) + 5ue^(i((s+1)ka-wt)) - 6ve^(i(ska-wt))) Simplifying these equations, we get: -w^2u = C(v + 5ve^(-ika) - Show more…

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Consider the normal modes of a linear diatomic chain in which the force constants between nearest neighbor atoms are alternately C and 5C. Let the masses be equal, and let the nearest neighbor separation be a/2. If the equations of motion for such a system are d^2u_s/dt^2 = C(v_s + 5v_{s-1} - 6u_s) and d^2v_s/dt^2 = C(u_s + 5u_{s+1} - 6v_s), and we choose traveling wave solutions u_s = u e^{j(ska-ωt)} and v_s = v e^{j(ska-ωt)}, find ω(k) at k = 0 and k = i/a.