Coupled Resonators Have Normal Modes

In-Phase Mode Keeps the Coupling Spring Quiet

Equal displacements leave the coupling spring unstretched. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

In phase means both masses move together

The sign pattern is plus plus, so the coupling spring is not stretched by the mode.

signs=++\mathrm{signs}=++
In-phase modeThe checked sign pattern is plus plus.mode=in-phasem=2 kgk_wall=2 N/mk_couple=1 N/msigns=+,+omega2=1 1/s^2

Only the wall stiffness sets this frequency squared

Wall stiffness 2 newtons per metre divided by mass 2 kilograms gives frequency squared 1.

ω+2=22=1\omega_+^{2}={2\over 2}=1
In-phase frequency ledgerThe coupling stiffness is not included for this sign pattern.mode=in-phasem=2 kgk_wall=2 N/mk_couple=1 N/msigns=+,+omega2=1 1/s^2