As an example of how the balance law of momentum is used to obtain an important equation of mathematical physics, suppose S = kF where k is a constant and F is the deformation gradient and let u ≡ y − x. Thus u is the displacement. Then from (17.13) you can verify the following holds.
In the case where ρ0 is a constant and b0 = 0, this yields
The wave equation is utt − cΔu = 0 and so the above gives three wave equations, one for each component.