Slinky Wave Lab

The slinky as a chain of masses

The slinky is modelled as N equal coils of mass m = M/N joined by springs. Whole-slinky spring constant K gives per-spring constant k_L = N K for compression, and the tension T = K (L - L0) of the stretched slinky gives k_T = T / a for sideways motion, where a = L/(N+1) is the coil spacing.

Equation of motion

m u_i'' = k (u_(i+1) - 2 u_i + u_(i-1)) - 2 gamma m u_i', with gamma = omega_0 / (2Q) and omega_0 = sqrt(k/m). It is integrated with the velocity-Verlet method, which conserves energy over long runs.

Wave speed

For long waves v = a sqrt(k/m). This gives v_L = L sqrt(K/M) for compression waves and v_T = sqrt(K (L - L0) L / M) for sideways waves. Both increase with stretch and fall with mass.

Dispersion and cutoff

A real chain of discrete coils obeys omega(k) = 2 omega_0 sin(k a / 2). It follows omega = v k only for long waves; the highest frequency that can propagate is omega_max = 2 omega_0. Above it the hand only makes a local, decaying disturbance. Short pulses therefore spread out as they travel (try the few-coil preset).

Reflection

A fixed end reflects a pulse with inverted displacement; a free end reflects it unchanged; a dashpot with c = sqrt(k m) matches the chain impedance and absorbs the pulse almost completely.

Standing waves

With both ends fixed the normal modes of the discrete chain are u_i ~ sin(n pi i / (N+1)) with omega_n = 2 omega_0 sin(n pi / (2 (N+1))). For small n this is the string result f_n = n v / (2L). Drive the left end at one of these frequencies and the amplitude builds up.

Simplifications

Linear springs (small displacements), no coil mass offsets, no friction with the table except viscous damping set by Q, and the transverse and longitudinal motions are kept separate. Real slinkies also show a slight low-frequency dispersion from their helical structure.

1. Transverse pulse with a fixed end: the reflected pulse comes back upside down. Switch the right end to Free to keep it upright, or to Matched absorber to make it disappear.
2. Longitudinal preset: the compression front moves at about the same speed as the sideways pulse, but the coils move along the slinky.
3. Standing wave: excite mode 3, then use the time-space graph to see two fixed nodes along the slinky.
4. Driven at the fundamental: the amplitude builds over many hand cycles. Detune the hand frequency by 10 percent and it never builds.
5. Few coils and a short pulse: it spreads into a wave train. In the dispersion graph, the hand frequency above the cutoff line gives no propagating wave.