For the reflection coefficients, rₛ = rₑ = 1. Equation (4.4) establishes a relationship between the distance travelled and the magnitude of the pressure. What is still missing is a relationship to the quantities used to excite the transducers harmonically and to measure their response, namely the voltages Uₛ and Uₑ. We therefore seek a transfer function T(f):

The magnitude of this function is calculated as:

Here T(f) denotes the complex transfer function of the system; Uₑ and Uₛ are the input and output voltages; rₑ and rₛ are the reflection coefficients at receiver and transmitter; α is the attenuation exponent of the liquid; γ = α + iβ is the complex propagation constant; k = 2πf/cₛ is the wave number; cₛ is the speed of sound; l is the distance between the transducer plates; and P₀ is the acoustic-pressure amplitude at z = 0.

The solution describes a Lorentzian curve, which occurs whenever the sine term in Eq. (4.6) vanishes. For rₛ = rₑ = 1, the following relation holds:

An explicit relationship for α(f) is still needed. It is obtained by considering the half-width Δf of the Lorentzian curve. After mathematical rearrangement, one obtains:

Original mathematical notation is preserved in the extracted image assets.