Vibroacoustic Simulation - Alexander Peiffer

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

       k 1 sine theta 1 equals k 2 sine theta 2 (2.112)

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

       StartFraction sine theta 1 Over c 1 EndFraction equals StartFraction sine theta 2 Over c 2 EndFraction (2.113)

      With these conditions we can factor out the exponential function

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

       bold-italic upper R equals StartFraction normal upper Phi 1 Superscript left-parenthesis upper R right-parenthesis Baseline Over normal upper Phi 1 EndFraction equals StartStartFraction StartFraction z 2 Over cosine theta 2 EndFraction minus StartFraction z 1 Over cosine theta 1 EndFraction OverOver StartFraction z 2 Over cosine theta 2 EndFraction plus StartFraction z 1 Over cosine theta 1 EndFraction EndEndFraction (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,

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

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

StartLayout 1st Row 1st Column normal upper Pi Subscript in 2nd Column equals StartFraction upper A Over 2 EndFraction upper R e left-parenthesis StartFraction ModifyingAbove p With caret Subscript 1 Superscript 2 Baseline Over bold-italic z 1 EndFraction right-parenthesis 3rd Column normal upper Pi Subscript trans 4th Column StartFraction upper A Over 2 EndFraction upper R e left-parenthesis StartFraction ModifyingAbove p With caret Subscript 2 Superscript 2 Baseline Over bold-italic z 2 EndFraction right-parenthesis equals StartFraction upper A Over 2 EndFraction upper R e left-parenthesis StartFraction bold-italic upper T squared ModifyingAbove p With caret Subscript 1 Superscript 2 Baseline Over bold-italic z 2 EndFraction right-parenthesis EndLayout

      With (2.117) this reads as:

       tau equals StartFraction 4 upper R e left-parenthesis bold-italic z 1 right-parenthesis upper R e left-parenthesis bold-italic z 2 right-parenthesis Over StartAbsoluteValue bold-italic z 1 plus bold-italic z 2 EndAbsoluteValue squared EndFraction (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.

      2.7 Inhomogeneous Wave Equation

      In the considerations in this chapter so far, we neglected the source terms related to the conservation of mass and momentum. All sources discussed until now are caused by vibrating surfaces. For establishing a physical link between the source term and the specific mass flow q˙s in Equation (2.3) and force density term f in Equation (2.8) we keep the terms this time. The source terms are not influenced by the linearization procedure; thus, the inhomogeneous and linear equations of momentum (2.24) and continuity (2.23) read as

       StartLayout 1st Row 1st Column StartFraction partial-differential rho prime Over partial-differential t EndFraction plus rho 0 nabla bold v prime 2nd Column equals ModifyingAbove rho With dot Subscript normal s Baseline 3rd Column rho 0 StartFraction partial-differential bold v prime Over partial-differential t EndFraction plus nabla p 4th Column equals bold f EndLayout (2.121)

      Repeating the steps of section 2.2.5 we finally get the inhomogeneous wave equation

       StartFraction 1 Over c 0 squared EndFraction StartFraction partial-differential squared p Over partial-differential t squared EndFraction minus nabla squared p equals ModifyingAbove rho With two-dots Subscript s Baseline minus nabla bold f (2.122)

      The density source is converted into a volume source strength density by ρ˙s=ρ0qs(t). The above equation can also be converted into the frequency domain and hence to the inhomogeneous Helmholtz equation

       left-parenthesis k squared plus normal upper Delta right-parenthesis bold-italic p left-parenthesis bold x comma omega right-parenthesis equals minus j omega rho 0 bold-italic q Subscript s Baseline plus nabla bold f equals minus bold-italic f Subscript q Baseline left-parenthesis bold r right-parenthesis (2.123)

      2.7.1 Acoustic Green’s Functions

      This section presents the concept of the Green’s function that uses a formalism to calculate the sound field for arbitrary source and boundary configurations as shown for example by Morse and Ingard (1968).

      The Green’s function is defined as the solution of the following inhomogeneous wave equation.