Vibroacoustic Simulation. Alexander Peiffer

Baseline Over upper T EndFraction d upper T 2nd Column plus 3rd Column StartFraction upper P Over upper T EndFraction d left-parenthesis StartFraction 1 Over rho EndFraction right-parenthesis 4th Row 1st Column equals StartFraction c Subscript normal v Baseline Over upper T EndFraction d upper T 2nd Column plus 3rd Column StartFraction upper P Over upper T rho squared EndFraction d rho EndLayout"/> (2.10)

      The relation dv=d(1/ρ) comes from the fact that v is a mass specific value and therefore the reciprocal of the density ρ=1/v. For an ideal gas we have

(2.11)

      cp and cv are the specific thermal heat capacities for constant pressure and volume, respectively. That is the ratio of temperature change ∂T per increase of heat ∂q. From the total differential

(2.12)

      we can derive

       (2.13)

      Using all above relations the change in density dρ is:

       (2.14)

      with κ=cv/cp. In most acoustic cases the process is isotropic: i.e. time scales are too short for heat exchange in a free gas; thus ds=0, and the change of pressure per density is

(2.15)

      In case of constant temperature (isothermal) dT=0 we get with (2.12) and the ideal gas law (2.11):

(2.16)

As we will later see, c0 is the . Newton calculated the wrong speed of sound based on the assumption of constant temperature that was later corrected by Laplace by the conclusion that the process is adiabatic. For fluids and liquids like water a different quantity is used because there is no such expression as the ideal gas law. The bulk modulus is defined as:

       (2.17)

      Due to (2.15) and (2.16) the relationship between the bulk modulus K and c0 is:

       (2.18)

      The bulk modulus can be defined for gases too, but we must distinguish between isothermal or adiabatic processes.

       (2.19)

      2.2.4 Linearized Equations

      Equations (2.3) and (2.8) can be linearized if small changes around a certain equilibrium are considered:

(2.20)

       (2.21)

(2.22)

      Inserting (2.22) into the equation of continuity (2.3), neglecting all second order terms as far as source terms, and setting¹ v0=0 the linear equation of continuity is:

(2.23)

      Doing the same for the equation of motion (2.8) leads to:

(2.24)

      Using the curl(∇×) of this equation it can be shown that the acoustic velocity v′ can be expressed using a so-called velocity potential which will be useful for the calculation of some wave propagation phenomena.

(2.25)

2.2.5 Acoustic Wave Equation

      From the following operation

      follows

       (2.26)

      With the equation of state (2.15) for the density we get the linear wave equation for the acoustic pressure p