A matrizant relates field variables at one depth in a stratified material to variables at some other depth. That is, we will solve the problem for arbitrary depth variations in density and in compressibility. Then we will come back and develop analytical solutions for the special case of constant material properties. For simplicity we may take and use subscripts to indicate the z coordinate. The matrix is also called an integral matrix.
Physical problems present themselves in different ways with different boundary conditions.
For the acoustic problem discussed earlier is a two-component vector involving pressure and of the stratified medium. The solution to the problem comes only when two boundary conditions are introduced. From these, pressure and displacement could be determined everywhere. If portions of the material have constant material properties and contain no sources, then it is possible to find an analytical expression for the matrizant. A matrizant which takes one across such a layer of constant properties is called, appropriately enough, a layer matrix .