Vibroacoustic Simulation. Alexander PeifferЧитать онлайн книгу.

href="#fb3_img_img_71ff5497-f0af-5691-a451-7b6bbead1f7f.png" alt="StartFraction 1 Over c 0 squared EndFraction StartFraction partial-differential squared p Over partial-differential t squared EndFraction minus nabla squared p equals 0"/> (2.27)

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

p equals minus rho 0 StartFraction partial-differential normal upper Phi Over partial-differential t EndFraction (2.29)

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

StartFraction 1 Over c 0 squared EndFraction StartFraction partial-differential squared normal upper Phi Over partial-differential t squared EndFraction minus nabla squared normal upper Phi equals 0 (2.30)

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:

bold v Superscript prime Baseline equals nabla normal upper Phi equals minus StartFraction 1 Over rho 0 EndFraction integral nabla p d t (2.31)

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 of basic concepts the wave equation is used in one dimensional form.

2.3.1 Harmonic Waves

According to D’Alambert every function of the form p(x,t)=Af(x−c0t)+Bg(x+c0t) is a solution of the one-dimensional wave equation. In the following we will consider harmonic motion or waves so we replace the functions f and g by the exponential function with

f left-parenthesis x right-parenthesis comma g left-parenthesis x right-parenthesis equals e Superscript j omega x slash c 0 Baseline equals e Superscript j k x Baseline with k equals StartFraction omega Over c 0 EndFraction (2.33)

and get

bold-italic p left-parenthesis x comma t right-parenthesis equals bold-italic upper A e Superscript j left-parenthesis omega t minus k x right-parenthesis Baseline plus bold-italic upper B e Superscript j left-parenthesis omega t plus k x right-parenthesis (2.34)

The first term of the right hand side of this equation is travelling in positive directions, the second in negative directions². Harmonic waves are characterized by two quantities, the angular frequency ω and the wavenumber k. The first is the frequency (in time) as for the harmonic oscillator, and the second is a frequency in space. A similar relationship can be found between the time period T and the wavelength λ. Space and time domains are coupled by the sound velocity c0 as shown in Table 2.1.

Table 2.1 Quantities of wave propagation in time and space domains.

Name	Time	Space
	Symbol	Unit	Symbol	Unit
Period	T	s	λ=c0T	m
Frequency	f=1T	s −1(Hz)	(⋅)=1λ	m −1
Angular frequency	ω=2πf=2πT	s −1	k=2πλ=ω/c0	m −1

The time integration in Equation (2.31) corresponds to the factor 1/(jω) and reads in the frequency domain:

bold v equals minus StartFraction 1 Over j omega rho 0 EndFraction nabla bold-italic p (2.35)

For one-dimensional waves in the x-direction this leads to:

(2.36)

Depending on the wave orientation the ratio between pressure and velocity is given by:

bold-italic v Subscript x Baseline equals plus-or-minus StartFraction 1 Over rho 0 c 0 EndFraction bold-italic p (2.37)

In accordance with the impedance concept from section 1.2.3 we define the ratio of complex pressure and velocity as specific acoustic impedance z

bold-italic z equals StartFraction bold-italic p Over bold-italic v EndFraction (2.38)

also called acoustic impedance. For plane waves this leads to:

z 0 equals plus-or-minus rho 0 c 0 (2.39)

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