e Superscript minus j k sine theta x Baseline plus normal upper Phi Superscript left-parenthesis upper R right-parenthesis Baseline e Superscript minus j k sine theta Super Superscript left-parenthesis upper R right-parenthesis Superscript x Baseline right-parenthesis"/> (2.100)
and the velocity in z-direction reads
We certainly shall not be able to match the impedance z2=p/vz at every surface position unless the arguments of the exponential functions are equal, hence
So, we get from the surface impedance condition
With z0=ρ0c0 and rearranging the above equation, the reflection factor is given by
The ratio between irradiated power to reflector power is the squared reflection factor called the.
Note that those coefficients are exclusively described by the impedance of fluid and surface and not density or speed of sound. Thus, the impedance is the relevant quantity here.
2.6 Reflection and Transmission of Plane Waves
A plane wave passing a flat interface between two infinite fluid volumes with different density and sound velocity as shown in Figure 2.8 is a first example of continuous systems exchanging acoustic energy. Applications of such a system could be for example the interface between a liquid (water) and a gas (air) or just different gases.
Figure 2.8 Transmission and reflection of a plane wave at the interface of two fluids. Source: Alexander Peiffer.
Region 1 of the incoming wave has two wave components, the incoming and the reflected wave, and region 2 the transmitted wave. Thus, both velocity potentials read
Using the given angles as sketched in Figure 2.8 the wavenumber space vector products are given by
with k1=ω/c1 and k2=ω/c2. The contact face between the fluid requires the continuity of pressure and velocity in the z-direction. We start with the pressure p1/2=jωρ1/2Φ1/2. Entering equations (2.107)–(2.109) into (2.106) and determining the pressure relation
gives
A solution for any x is only possible if the arguments of the exponential functions are equal.
So
