Back

Transverse vs Longitudinal Modes on a 1D Chain

What you are seeing: a 1D chain of masses connected by springs supports both transverse waves (particles bob perpendicular to the chain) and longitudinal waves (particles compress along the chain). Both share the dispersion relation ω(k)=2K/m∣sin⁡(ka/2)∣\omega(k) = 2\sqrt{K/m}|\sin(ka/2)|.

Figure 1. Both modes shown with the same dispersion; the difference is in the polarization direction.
k (1/a)1.0
amplitude0.20
view

WHAT TO TRY

  • Compare the two rows: transverse particles bob across the chain, longitudinal ones compress along it, yet both obey the same dispersion omega(k) of the spring chain.
  • Sweep the wavenumber k toward the zone boundary: the wavelength shrinks to two lattice spacings and the group velocity falls to zero, the same cutoff for both polarisations.
  • Longitudinal waves carry sound through a solid, transverse waves carry light and shear, the distinction that lets seismologists map the Earth liquid core from the shadow of S-waves.