Out-of-phase normal modes show how coupling stiffness shifts the squared frequency. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
Zero coupling leaves the wall spring alone
Out-of-phase motion stretches the coupling spring. With mass 2 kg and wall stiffness 2 newtons per metre, coupling stiffness 0 gives omega squared 1 per second squared.
ω2=22+2×0=1
One coupling spring doubles the squared frequency
Out-of-phase motion stretches the coupling spring. With mass 2 kg and wall stiffness 2 newtons per metre, coupling stiffness 1 gives omega squared 2 per second squared.
ω2=22+2×1=2
Three coupling units make four squared-frequency units
Out-of-phase motion stretches the coupling spring. With mass 2 kg and wall stiffness 2 newtons per metre, coupling stiffness 3 gives omega squared 4 per second squared.