PHY.5:5.2 - Decide when a chain can be treated as a continuous medium

Consider an infinite ideal one-dimensional chain with identical masses m, spacing a and linear springs of stiffness kappa. Each mass moves a small distance u_j from its reference position. The balance is

m*u_j'' = kappa*(u_(j+1)-2*u_j+u_(j-1)).

For waves with wavenumber q in 0<q*a<pi, substitution of a sinusoidal wave gives

omega^2 = (4*kappa/m)*sin^2(q*a/2).

For wavelengths long compared with a, expanding the neighboring displacements gives the continuum equation u_tt=c^2*u_xx with c=a*sqrt(kappa/m). It predicts both phase and group speed c. The chain’s phase speed divided by c is sin(q*a/2)/(q*a/2); its group speed divided by c is cos(q*a/2). Expanding those ratios gives v_phase/c = 1-(q*a)^2/24+O((q*a)^4) and v_group/c = 1-(q*a)^2/8+O((q*a)^4), exposing their different first corrections.

At q*a=0.2 these ratios are about 0.99833 and 0.99500. A half-percent allowance for these speeds accommodates this ideal comparison, subject to the question’s waveform and other physical premises. The group-speed difference is already larger than the phase-speed difference.

Changed work. A disturbance containing wavelengths near the shortest traveling waves of the chain probes q*a near pi. The chain’s group speed tends to zero, while the continuum account keeps c. A question about that disturbance’s propagation requires retaining the discrete dispersion or an adequate extension. Making the computation of the uncorrected continuum equation more accurate cannot recover the omitted physical dependence.

This case concerns an ideal linear chain. It demonstrates choosing spatial resolution from the wave that matters. It supplies no claim that every material, boundary or large deformation obeys the same chain law.