The building blocks

The field variable transformation

$$\pmatrix{ v \cr \sigma } = \pmatrix{ 1 & 1 \cr \surd(\rho/s) & - \surd(\rho/s) } \pmatrix{ v^{+} \cr v^{-} }$$ (1)

and its inverse

$$\pmatrix{ v^{+} \cr v^{-} } = \frac{1}{2} \pmatrix... .../\rho) \cr 1 & -\surd(s/\rho) } \pmatrix{ v \cr \sigma }$$ (2)

and the translation equation

$$\pmatrix{ v^{+}(z) \cr v^{-}(z) \cr } = \pmatrix{ ... ...ega z \surd(\rho s)} } \pmatrix{ v^{+}(0) \cr v^{-}(0) }$$ (3)

concatenate to form a basic building block  

$$\pmatrix{ v(z) \cr \sigma(z) \cr } = \pmatrix{ \cos(\o... ...s(\omega z \surd(\rho s)) } \pmatrix{ v(0) \cr \sigma(0) }$$ (4)

which may also be written

$$\pmatrix{ v(z) \cr \sigma(z) \cr } = \pmatrix{ \cos(\o... ...& \cos(\omega z \lambda) } \pmatrix{ v(0) \cr \sigma(0) }$$ (5)

and in case slowness is a constant, then we can re-write this equation in a $\cal Z$-transform notation

$$\pmatrix{ v(z) \cr \sigma(z) \cr } = \frac{1}{2 {\cal Z... ...\cal Z}) & (1 + {\cal Z}) } \pmatrix{ v(0) \cr \sigma(0) }$$ (6)

If we denote the RHS operator above by ${\cal P}({\cal Z})$, note the property

$${\cal P}^{2}({\cal Z}) = {\cal P}({\cal Z}^{2})$$ (7)