Vibroacoustic Simulation. Alexander PeifferЧитать онлайн книгу.

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Figure 2.3 One-dimensional harmonic waves travelling in the positive x-direction (c0=2m/s, T=2.2s). Source: Alexander Peiffer.

z0=ρ0c0 is called the characteristic acoustic impedance of the fluid. The specific acoustic impedance z is complex, because for waves that are not plane the velocity may be out of phase with the pressure. However, for plane waves the specific acoustic impedance is real and an important fluid property.

The above description of plane waves can be extended to three-dimensional space by introducing a wavenumber vector k.

bold-italic p equals bold-italic upper A e Superscript j left-parenthesis omega t minus bold k bold x right-parenthesis (2.40)

2.3.2 Helmholtz equation

Entering (2.40) into the wave Equation (2.27) provides

(2.41)

The ejωt term is often omitted and with k=ω/c0 we get the homogeneous.

left-parenthesis k squared plus normal upper Delta right-parenthesis bold-italic p left-parenthesis bold x comma omega right-parenthesis equals 0 (2.42)

2.3.3 Field Quantities: Sound Intensity, Energy Density and Sound Power

A sound wave carries a certain amount of energy that is moving with the speed of sound. We start with the instantaneous acoustic power Π:

normal upper Pi left-parenthesis t right-parenthesis equals bold upper F bold v (2.43)

F is the force acting on a fluid particle and v the associated velocity. The acoustic intensity I is defined as the power per unit area A=An in the direction of the unit vector n and with F=pAn we get:

bold upper I left-parenthesis t right-parenthesis equals p bold v bold n (2.44)

As in Equation (1.48) the time average is given by:

mathematical left-angle bold upper I mathematical right-angle Subscript upper T Baseline equals StartFraction 1 Over upper T EndFraction integral Subscript 0 Superscript upper T Baseline p bold v d t right double arrow one-half upper R e left-bracket bold-italic p bold-italic v Subscript x Superscript asterisk Baseline right-bracket (2.45)

Using the harmonic plane wave solutions for pressure (2.34) and velocity (2.36)

StartLayout 1st Row 1st Column bold-italic p left-parenthesis x comma t right-parenthesis 2nd Column equals 3rd Column bold-italic upper A e Superscript j left-parenthesis omega t minus k x right-parenthesis 2nd Row 1st Column bold-italic v Subscript x Superscript asterisk Baseline left-parenthesis x comma t right-parenthesis 2nd Column equals 3rd Column StartFraction bold-italic upper A Superscript asterisk Baseline Over rho 0 c 0 EndFraction e Superscript j left-parenthesis omega t minus k x right-parenthesis EndLayout

the time averaged mean intensity yields:

(2.46)

and finally:

mathematical left-angle upper I mathematical right-angle Subscript upper T Baseline equals StartFraction ModifyingAbove p With caret squared Over 2 rho 0 c 0 EndFraction equals StartFraction p Subscript normal r normal m normal s Superscript 2 Baseline Over rho 0 c 0 EndFraction equals one-half rho 0 c 0 ModifyingAbove v With caret squared (2.47)

We see that the specific impedance z0=ρ0c0 relates the intensity to the squared pressure.

The kinetic energy density ekin in a control volume V0 is written as

e Subscript normal k normal i normal n Baseline equals StartFraction upper E Subscript normal k normal i normal n Baseline Over upper V 0 EndFraction equals one-half rho 0 v Subscript normal x Superscript 2 Baseline equals StartFraction p squared Over 2 rho 0 c 0 squared EndFraction (2.48)

The potential energy density epot follows from the adiabatic work integral as in equation (2.9)

e Subscript normal p normal o normal t Baseline equals StartFraction upper E Subscript normal p normal o normal t Baseline Over upper V 0 EndFraction equals minus StartFraction 1 Over upper V 0 EndFraction integral Subscript upper V Baseline 0 Superscript upper V Baseline upper P d upper V (2.49)

If we use Equation (2.15) we get the change in density as a start for the change in volume

d rho equals StartFraction 1 Over c 0 squared EndFraction d upper P

With unit mass M in the control volume V0 it follows from ρ=M/V0 that

d upper V equals minus StartFraction upper V Over rho 0 EndFraction d rho equals minus StartFraction upper V Over rho 0 c 0 squared EndFraction d upper P

Finally we get:

e Subscript normal p normal o normal t Baseline equals StartFraction upper E Subscript normal p normal o normal t Baseline Over upper V 0 EndFraction equals integral Subscript upper P 0 Superscript upper P 0 plus p Baseline StartFraction upper P Over rho 0 c 0 squared EndFraction d upper P equals StartFraction p squared Over 2 rho 0 c 0 squared EndFraction (2.50)

Pressure and velocity are in phase for plane waves; the same is true for the potential and kinetic energy density, so the total energy density is given by:

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