*2.1. Attenuation Due to Reflection*

Attenuation of the ultrasound signal due to reflection at phase boundaries occurs if two adjacent media have differing wave impedances. The larger the wave impedance difference, the more reflection occurs. The wave impedance can be calculated for many materials with: [26]

$$\mathbf{Z} = \mathfrak{p} \times \mathfrak{c},\tag{1}$$

With Z being the wave impedance (kg/m<sup>2</sup> s), ρ the material's density (kg/m<sup>3</sup> ), and c the speed of sound in the material (m/s). For the case of perpendicular incidence of an ultrasonic compressional wave the reflection coefficient R at the boundary is: [26]

$$\mathbf{R} = \frac{\mathbf{Z}\_1 - \mathbf{Z}\_2}{\mathbf{Z}\_1 + \mathbf{Z}\_2} \tag{2}$$

Z<sup>1</sup> and Z<sup>2</sup> being the wave impedances of the two media. Equation (2) shows that less reflection occurs if the wave impedances of the two media are similar. Table 1 shows the wave impedances for air, water, and pulp fibers in paper calculated from density and speed of sound. During the penetration of a paper sample with water, the ultrasound signal can be reflected at the boundaries between air and water, air and fibers, and water and fibers. From the wave impedances, the reflection coefficient at each type of boundary can be calculated (see Table 2).

**Medium Speed of Sound (m**/**s) Density (kg**/**m**<sup>3</sup> **) Wave Impedance (kg**/**m**<sup>2</sup> **s)** air <sup>1</sup> 343 1.29 442.47 water <sup>1</sup> 1490 998 1,487,020 fiber <sup>2</sup> 1493 1500 2,239,548

**Table 1.** Wave impedances for air, water, and pulp fibers.

<sup>1</sup> values at 20 ◦C taken from [27]; <sup>2</sup> values based on [28].

**Table 2.** Reflection coefficients for air, water, and fiber boundaries.


The reflection coefficients show that at an air-fiber boundary most of the signal was reflected. The same is true for air–water boundaries. If the air within a paper sample is completely replaced by water, water–fiber boundaries are replacing the air-fiber boundaries. The reflection coefficient for water-fiber boundaries was much lower, yielding a better signal transmission. This means that in general the received ultrasound signal should increase when the air in the sample is replaced by the test liquid during penetration. Please note that the wave impedance value for fibers should not be understood to be valid for all paper fibers, since the speed of sound in the fibers largely depends on the manufacturing process (compare [29]).

The system of air, water, and pulp fibers is used as an example for a liquid penetrating into a porous material. Since solids and liquids generally have more similar wave impedances compared to gases [27], the considerations made above are also valid for other porous solids being penetrated by any kind of liquid. In conclusion, displacement of air by liquid in the substrate pores thus can only be responsible for an increase in ultrasound intensity over time, the curves however often show a (pronounced) decrease.
