Energy sloshes between the two masses — a beat phenomenon from normal-mode interference.
Two Springs, One Connection
Two identical mass-spring systems are joined by a third, weaker coupling spring. Pull one mass to the side and let go, and something elegant happens: the energy doesn't stay in that mass — it sloshes back and forth between the two, transferring completely from one to the other and back again.
This is not an exponential decay. It's a beat — two frequencies close together producing a slow modulation of amplitude. The energy alternates at a frequency equal to half the difference between the two normal-mode frequencies.
Normal Modes
The system has two natural "shapes" of oscillation where both masses move at the same frequency indefinitely:
- Mode 1 (ω₁ = √(k/m)): both masses move in phase — the coupling spring is never stretched, so k_c doesn't matter.
- Mode 2 (ω₂ = √((k + 2k_c)/m)): both masses move out of phase — the coupling spring stretches maximally, adding its stiffness.
Select "Normal mode 1" or "Normal mode 2" from the dropdown to see these pure modes. Now select "Left displaced" — watch how the amplitude beats between m₁ and m₂. That pattern is the sum of the two normal modes interfering.
The Beat Frequency
When you start with only m₁ displaced, you're exciting both normal modes equally. The displacement of each mass over time is:
x₁(t) = A cos(ω_beat t) cos(ω_avg t)
x₂(t) = A sin(ω_beat t) sin(ω_avg t)
where ω_beat = (ω₂ − ω₁) / 2 and ω_avg = (ω₂ + ω₁) / 2. The beat period gets shorter as the coupling spring gets stronger — increase k_c and watch the energy slosh faster.