Phonons Are Finite Standing Modes

Third Standing Mode Fits the Chain

A finite standing mode closes an exact boundary ledger. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

A standing mode must fit both ends of the chain

The chain length is 12 meters and the mode index is 3.

L=12 m,n=3L=12\ \mathrm{m},\quad n=3
Third modeThe mode index counts half-waves.L=12 mn=3lambda=8 mfit=1nodes=4

The fit condition closes exactly

Wavelength 8 meters gives 24 meters on the left and 24 meters on the right.

nλ=3⋅8=24 m=2L,nodes=4n\lambda=3\cdot 8=24\ \mathrm{m}=2L,\quad nodes=4
Fit ledgerNode count is tied to the mode index.L=12 mn=3lambda=8 mfit=1nodes=4