Sympathetic Resonance
Drive one oscillator at a chosen frequency and watch energy flow selectively into those with matching natural frequencies. Non-resonant oscillators barely stir. The resonance curve below shows amplitude versus frequency — the sharper the peak, the lower the damping.
About this lab
Sympathetic resonance occurs when an oscillating body causes another body with the same or a harmonically related natural frequency to vibrate. The phenomenon is ubiquitous in music — when you press the sustain pedal on a piano and sing a note, the corresponding piano strings begin to ring "in sympathy." It happens because energy transfers efficiently between systems when the driving frequency matches the natural frequency of the receiver, a condition called resonance. Off-resonance, the driving force and the oscillator's restoring force work against each other, preventing significant energy buildup.
The physics of a driven damped oscillator is governed by the equation m*x'' + b*x' + k*x = F_0*cos(omega_d*t), where m is mass, b is the damping coefficient, k is the spring constant, and omega_d is the driving angular frequency. The steady-state amplitude is A = F_0 / sqrt((k - m*omega_d^2)^2 + (b*omega_d)^2). This peaks when omega_d equals the natural frequency omega_0 = sqrt(k/m) (approximately, for light damping). The sharpness of the resonance peak is quantified by the quality factor Q = sqrt(m*k) / b — high Q means narrow, tall resonance peaks and slow energy transfer, while low Q means broad peaks and quick but less selective response.
Coupling between oscillators adds another layer of richness. When two identical oscillators are coupled (for example, two pendulums connected by a spring, or two tuning forks on a shared resonant board), energy sloshes back and forth between them at a "beat frequency" that depends on the coupling strength. This produces the phenomenon of normal modes: the system oscillates as a whole in symmetric and antisymmetric patterns. In this simulation, the coupling allows the driven oscillator to transfer energy to its neighbors through a shared medium — just as a vibrating tuning fork on a wooden table can set a nearby matching fork ringing.