Vibroacoustic Simulation. Alexander Peiffer

(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

       (2.28)

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

(2.29)

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

(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:

(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):

       (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

       (2.33)

      and get

(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:

(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:

       (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

(2.38)

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

       (2.39)