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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

(2.106)

Using the given angles as sketched in Figure 2.8 the wavenumber space vector products are given by

(2.107)

(2.108)

(2.109)

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

(2.110)

A solution for any x is only possible if the arguments of the exponential functions are equal.

(2.111)

So also in the transmission case the incident angle equals the angle of the reflected wave. Additionally we have

(2.112)

This represents the acoustic equivalent of Snell’s law of transmission:

(2.113)

With these conditions we can factor out the exponential function

(2.114)

After clarifying the angles of reflection and transmission the next point is to assess the fraction of transmitted and reflected wave. From the continuity condition for the velocity in the z-direction we get with vz=−∂Φ/∂z when going through the algebra

(2.115)

Rearranging equations (2.114) and (2.115) the reflection factor is

(2.116)

This expression is similar to (2.103) except the angle factor that represents the physics of the wave propagation in the second medium. The transmission factor is defined by the ratio of pressure amplitudes p1=jωρ1Φ1 and p2=jωρ2Φ2. Hence,

(2.117)

The transmitted acoustic power follows from the square of the amplitudes. We introduce a transmission coefficient τ by

(2.118)

Without loss of generality we assume ϑ=0 so each power is given by

With (2.117) this reads as:

(2.120)

It should be noted that the transmission coefficient of the flat interface between two fluids is determined by the impedance of each half space ‘seen’ from the other side. This is the first indication for the coupling of subsystems determined by the radiation impedance into the free fields of each subsystem. Similar expressions will be found in Sec. 8.2.4.1 when transmission is dealt with in the context of coupled random subsystems.