Vibroacoustic Simulation. Alexander Peiffer
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Название: Vibroacoustic Simulation

Автор: Alexander Peiffer

Издательство: John Wiley & Sons Limited

Жанр: Отраслевые издания

Серия:

isbn: 9781119849865

isbn:

СКАЧАТЬ 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.

       StartLayout 1st Row 1st Column upper K Subscript normal s 2nd Column rho left-parenthesis StartFraction partial-differential upper P Over partial-differential rho EndFraction right-parenthesis Subscript normal s Baseline equals kappa upper P 3rd Column upper K Subscript upper T 4th Column rho left-parenthesis StartFraction partial-differential upper P Over partial-differential rho EndFraction right-parenthesis Subscript upper T Baseline equals upper P EndLayout (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 upper P 2nd Column equals 3rd Column upper P 0 plus p EndLayout (2.21)

      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:

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

      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.

      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

      Inserting the velocity v′=∇Φ derived from the potential Φ into the linear equation of motion (2.24) provides the required relation between pressure and the velocity potential

       rho 0 StartFraction partial-differential Over partial-differential t EndFraction nabla normal upper Phi plus nabla p equals nabla left-parenthesis rho 0 StartFraction partial-differential normal upper Phi Over partial-differential t EndFraction plus p right-parenthesis equals 0 (2.28)

      Thus, the relationship between pressure p and the velocity potential Φ is

      Entering this into the wave equation (2.27) and eliminating one time derivative gives:

      The definition of the velocity potential (2.25) and equation (2.29) can be applied for the derivation of a relationship between acoustic velocity and pressure:

      2.3 Solutions of the Wave Equation

      In acoustics we stay in most cases in the linear domain, so we change the notations from equations (2.20)–(2.22):

       bold v Superscript prime Baseline right-arrow bold v rho Superscript prime Baseline right-arrow rho (2.32)

      Equations (2.27) and (2.30) define the mathematical law for the propagation of waves. For the explanation СКАЧАТЬ