| d²ξ | = b² ( | d²ξ | + | d²ξ | + | d²ξ | ) + (a² − b²) | d² | ( | d²ξ | + | d²η | + | d²ζ | ), |
| dt² | dx² | dy² | dz² | dx | dx² | dy² | dz² |
where a² and b² denote the two arbitrary constants. Put for shortness
| d²ξ | + | d²η | + | d²ζ | ≈ δ (1), |
| dx² | dy² | dz² |
and represent by Δ²χ the quantity multiplied by b². According to this notation, the three equations of motion are
| d²ξ | b²Δ²ξ + (a² − b²) | dδ | } (2). |
| dt² | dx | ||
| d²η | b²Δ²η + (a² − b²) | dδ | |
| dt² | dy | ||
| d²ζ | b²Δ²ζ + (a² − b²) | dδ | |
| dt² | dz |
It is to be observed that S denotes the dilatation of volume of the element situated at (x, y, z). In the limiting case in which the medium is regarded as absolutely incompressible δ vanishes; but, in order that equations (2) may preserve their generality, we must suppose a at the same time to become infinite, and replace a²δ by a new function of the co-ordinates.
These equations simplify very much in their application to plane waves. If the ray be parallel to OX, and the direction of vibration parallel to OZ, we have ξ = 0, η = 0, while ζ is a function of x and t only. Equation (1) and the first pair of equations (2) are thus satisfied identically. The third equation gives
| d²ζ | = b² | d²ζ | (3), |
| dt² | dx² |
of which the solution is
ζ = ƒ(bt − x) (4),