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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)

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.

(2.40)