Vibroacoustic Simulation. Alexander Peiffer

and levels

      The dynamic range of acoustic quantities can be very high; thus, a logarithmic scale is well established for the quantification of acoustic signals. For convenience a certain time averaged quantity, for example the mean square pressure ⟨p⟩T2=prms2, is compared to a mean square reference value pref2. The pressure level in decibels is defined as follows:

       (2.166)

      The factor of 10 is introduced to spread the scale. Linear quantities such as pressure, velocity, or displacement use the mean square values. As level and decibel are used on time signals too, one should not apply the decibel scale to amplitudes. This may lead to confusion, as it is not clear if the mean square values of the amplitude is meant. This makes even more sense when the energy and power levels are defined. The energy must be averaged, as there is no constant value over the period – see Equation (2.46). Energy quantities such as energy, intensity, or power are compared with mean values and not mean square values; hence:

       (2.167)

Table 2.3 Field and energy properties of acoustic waves

Source	Source strength	Impedance Velocity	Pressure Radiated power
Mono pole	Q,jωV	Zrad=k2ρ0c04π	p=jkρ0c0Q(ω)4πre−jkr
		vr=Q(ω)4πrjk(1+1jkr)e−jkr	Qrms2k2ρ0c04π
Breath. sphere	Q=4πR2vr	zR=jρ0c0kR1+jkR	Q4πr[jkρ0c01+jkR]e−jk(r−R)
		Q4πr2[1+jkr1+jkR]e−jk(r−R)	Qrms2k2ρ0c04π(1+k2R2)
Piston	Q=πR2vz	zR=ρ0c0(1−J1(2kR)kR+jH1(2kR)kR)	jωρ02πrvz[2J1(kRsin⁡ϑ)kRsin⁡ϑ]
			ρ0c0(1−J1(2kR)kR+jH1(2kR)kR)Qrms2

      In addition, the decibel scale is used for ratios of similar quantities. A typical example is the transmission loss that is the decibel scale of the transmission factor from Equation (2.118) that relates the transmitted to the radiated power. The definition of the transmission loss (TL) is:

       (2.168)

      The reciprocal definition was chosen in order to get positive values for losses. When linear quantities are compared, for example the pressure at two locations, the mean square values are related. When the squared pressure is compared to the situation with and without a specific equipment or installation, this is called insertion loss (IL)

       (2.169)

      The reference quantities for power and pressure are chosen conveniently to simplify the calculations with levels. Taking the equation for the spherical source (2.82) and dividing it by the squared reference value for the pressure yields

       (2.170)

      and taking the decibel of this

      yields

       (2.171)

Entering typical values for air with ρ0=1.23 kg/m 3 and c0=343 m/s using Aref=1 m 2 we get

      matching well to the reference value of acoustic power.

      Bibliography

      1 W.R. Graham. BOUNDARY LAYER INDUCED NOISE IN AIRCRAFT, PART I: THE FLAT PLATE MODEL. Journal of Sound and Vibration, 192(1): 101–120, April 1996. ISSN 0022-460X.

      2 Finn Jacobsen. PROPAGATION OF SOUND WAVES IN DUCTS. Technical Note 31260, Technical University of Denmark, Lynby, Denmark, September 2011.

      3 Reinhard Lerch and H. Landes. Grundlagen der Technischen Akustik, September 2012.

      4 P.M.C. Morse and K.U. Ingard Theoretical Acoustics. International Series in Pure and Applied Physics. Princeton University Press, 1968. ISBN 978-0-691-02401-1.

      Notes

      1 1 Keeping v0≠0 would lead to the convective wave equation that is used in the context of flow related acoustic problems, which is more a topic of aero acoustics

      2 2