Quasi-continuous spatial motion of a mass-spring chain

The three-dimensional motion of mass-spring chain in close to continuum conditions (a large number of mass-springs per unit length) is shown to be governed by r_tt= T(A) Ar_5S+ h² 12r_sstt, r=(x,y,z), A≡|r_S|, where T is the tension function of the continuum (arbitrary but known) and A is the stretch, r is the spatial displacement vector, s is the reference coordinate along the chain and h is equilibrium discreteness length. Some exact solutions for the string (h≡0) and the chain are derived. While the string supports only trivial travelling waves (the stretch must be constant) the chain admits a travelling wave confined to a plane with the stretch propagating as a solitary wave. In general the dispersion born out of the discreteness counteracts the steepening of waves caused by the nonlinearity and leads to the formation of nonlinear structures.