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e Subscript normal p normal o normal t Baseline equals StartFraction p left-parenthesis x comma t right-parenthesis squared Over rho 0 c 0 squared EndFraction"/> (2.51)

      Using Equation (2.34) the time average over one period leads to:

(2.52)

      Finally, we can see that the speed of sound relates energy density to the sound intensity.

       (2.53)

      All those above expressions are useful for the description and evaluation of sound fields. Especially in case of statistical methods that are based on the energy density of acoustic subsystems they link the wave fields to the energy in the systems and the power irradiating at the system boundaries.

      If the intensity can be determined over a certain surface the source power is calculated by integrating the intensity component perpendicular to the surface

(2.54)

      2.3.4 Damping in Waves

      There is no motion without damping, and a sound wave propagating over a long distance will vanish. This is considered by adding a damping component to the one-dimensional solution of the wave equation similar to the decay rate in (1.22)

       (2.55)

      Here α is the damping constant. There are several reasons for the attenuation of acoustic waves:

       Viscous damping due to inner viscosity.

       Thermal damping due to irreversible heat flow during wave propagation.

       Molecular damping due to excitation of degrees of freedom of molecules (for additional content of the gas, e.g. humidity in air).

      The damping loss η as defined in (1.68) is based on the amount of energy dissipated during one cycle of wave motion. The harmonic pressure wave performs one cycle of oscillation in one period in time T or space λ. So we get for η:

       (2.56)

      For small damping the exponential function can be approximated by ex≈1+x−… providing the relationship between damping loss and fluid wave attenuation.

       (2.57)

      Hence, the attenuation can be given by:

       (2.58)

      An appropriate way to consider this relationship in the solution of the wave equation is to include this into a complex wavenumber k:

       (2.59)

      This complex wavenumber naturally impacts the speed of sound

       (2.60)

      and the acoustic impedance

       (2.61)

The shown quantities of the plane wave field can also be applied in three-dimensional space and they are summarized in Table 2.2.

Table 2.2 Field and energy properties of acoustic waves.

Quantity	Symbol	Formula	Units	Plane wave	Equation
Acoustic velocity	v	1jωc0∇p	m/s	pρ0c0	(2.35)
Acoustic impedance	z	p/v	Pa s/m	z0=ρ0c0	(2.38)
Intensity	I	12Re(pv*)	Pa m/s	⟨I⟩T=p^22ρ0c0	(2.47)
Energy density	e		J/m 3	⟨e⟩T=p^22ρ0c02	(2.52)
Acoustic power	Π	Π=IA	W	⟨Π⟩T=Ap^22ρ0c0	(2.43)

      2.4 Fundamental Acoustic Sources

      The radiation of sound is key to understanding how energy is introduced into wave fields. Depending on the wavelength, geometry, and dimension of the source the behavior varies. A detailed understanding of fundamental sources is helpful for the radiation

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