a particle of such a medium from its "resting" position due elastic vibrations, understood to be a function of the rest position and time . The deformation of the medium at that point can be described by the strain tensor , the 3×3 matrix whose elements are where denotes partial derivative with respect to position coordinate . The strain tensor is related to the 3×3 stress tensor by the equation Here is the Kronecker delta (1 if , 0 otherwise) and and are the Lamé parameters ( being the material's shear modulus). It follows that From Newton's law of inertia, one also gets where is the density (mass per unit volume) of the medium at that point, and denotes partial derivative with respect to time. Combining the last two equations one gets the seismic wave equation in homogeneous media Using the nabla operator notation of vector calculus, , with some approximations, this equation can be written as Taking the curl of this equation and applying vector identities, one gets This formula is the wave equation applied to the vector quantity , which is the material's shear strain. Its solutions, the S waves, are linear combinations of sinusoidal plane waves of various wavelengths and directions of propagation, but all with the same speed Taking the divergence of seismic wave equation in homogeneous media, instead of the curl, yields a wave equation describing propagation of the quantity , which is the material's compression strain. The solutions of this equation, the P waves, travel at the speed that is more than twice the speed of S waves. The steady state SH waves are defined by the Helmholtz equation where is the wave number. See also Earthquake Early Warning (Japan) Lamb waves Longitudinal wave Love wave P wave Rayleigh wave Seismic wave Shear wave splitting References Further reading Waves Seismology