Vibroacoustic Simulation. Alexander Peiffer

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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 of vibrating structures and thus, how they exchange acoustic energy.

      2.4.1 Monopoles – Spherical Sources

The most simple geometry we might think of is a point in space. For simple derivation of the sound field of a point source the spherical coordinate system is introduced as shown in Figure 2.4 and defined by the following coordinate transformation

       (2.62a)

       (2.62b)

       (2.62c)

Figure 2.4 Definition of a spherical coordinate system. Source: Alexander Peiffer.

      Using this coordinate system and neglecting the angular components the Laplace operator Δ reads as

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