Vibroacoustic Simulation. Alexander PeifferЧитать онлайн книгу.

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

upper P equals rho upper R upper T with upper R equals c Subscript normal p Baseline minus c Subscript normal v Baseline (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

d upper P left-parenthesis upper T comma rho right-parenthesis equals left-parenthesis StartFraction partial-differential p Over partial-differential upper T EndFraction right-parenthesis Subscript rho Baseline d upper T plus left-parenthesis StartFraction partial-differential p Over partial-differential rho EndFraction right-parenthesis Subscript upper T Baseline d rho (2.12)

we can derive

StartFraction d upper T Over upper T EndFraction equals StartFraction d upper P Over upper P EndFraction minus StartFraction d rho Over rho EndFraction (2.13)

Using all above relations the change in density dρ is:

d rho equals StartFraction rho Over kappa upper P EndFraction d upper P minus StartFraction rho Over c Subscript normal p Baseline EndFraction d s (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

left-parenthesis StartFraction d upper P Over d rho EndFraction right-parenthesis Subscript normal s Baseline equals kappa StartFraction upper P Over rho EndFraction equals c 0 squared (2.15)

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

left-parenthesis StartFraction d upper P Over d rho EndFraction right-parenthesis Subscript upper T Baseline equals StartFraction upper P Over rho EndFraction equals c Subscript 0 upper T Superscript 2 (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:

upper K equals rho left-parenthesis StartFraction partial-differential upper P Over partial-differential rho EndFraction right-parenthesis (2.17)

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

c 0 squared equals StartFraction upper K Over rho EndFraction (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:

StartLayout 1st Row 1st Column rho 2nd Column equals 3rd Column rho 0 plus rho prime EndLayout (2.20)

StartLayout 1st Row 1st Column upper P 2nd Column equals 3rd Column upper P 0 plus p EndLayout (2.21)

StartLayout 1st Row 1st Column bold v 2nd Column equals 3rd Column bold v 0 plus bold v prime EndLayout (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:

StartFraction partial-differential rho prime Over partial-differential t EndFraction plus rho 0 nabla bold v Superscript prime Baseline equals 0 (2.23)

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

rho 0 StartFraction partial-differential bold v prime Over partial-differential t EndFraction plus nabla p equals 0 (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.

bold v prime equals nabla normal upper Phi (2.25)

2.2.5 Acoustic Wave Equation

From the following operation

StartFraction partial-differential Over partial-differential t EndFraction left-parenthesis 2.23 right-parenthesis minus nabla left-parenthesis 2.24 right-parenthesis

follows

StartFraction partial-differential squared rho prime Over partial-differential t squared EndFraction minus nabla squared p equals 0 (2.26)

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

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