# Mathematical expressions

In the case of multiple layers, Miki noticed that the real part of the surface impedance when computed with the original Delany-Bazley model sometimes becomes negative at low frequencies denoting a non-physical result. From Delany and Bazley measurements data realized 20 years before on fibrous materials with porosities close to 1.00, Miki proposes to use the following expressions for the wave number and characteristic impedance (a $+j\omega t$ time convention is assumed):