Fundamentals of General Linear Acoustics
Finn Jacobsen, Peter Moller Juhl
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Acoustics deals with the production, control, transmission, reception, and effects of sound. Owing to acoustics being an interdisciplinary field, this book is intended to be equally accessible to readers from a range of backgrounds including electrical engineering, physics and mechanical engineering.
This book introduces the fundamentals of acoustic wave motion. It addresses in a clear and systematic way some of the most difficult parts of acoustics for beginners, such as the widely different approximations due to the wide frequency range, the apparently arbitrary choice between the use of analytical solutions to the wave equation with boundary conditions, and the fundamentally different energy-based considerations used in noise control. As a result, it provides readers with a self-contained source of information on acoustics which can be used for self-study or as a graduate course text.
- Places an emphasis on detailed derivations based on the fundamental laws of physics and interpretations of the resulting formulas.
- Avoids, where possible, electrical and mechanical equivalent circuits, so as to make it accessible to readers with different backgrounds.
- Introduces duct acoustics, sound in enclosures, and sound radiation and scattering.
- Contains a set of appendices which includes material on signal analysis and processing as these tools are essential for the modern acoustician.
In practice we are often concerned with the time-averaged sound intensity in stationary sound fields. For simplicity we shall use the symbol I for this quantity (rather than I t ), that is, (6.9) I = p(t)u(t) t . Examination of Equation (6.6) leads to the conclusion that the divergence of the time averaged sound intensity is identically zero unless we are at a source point, ∇ · I = 0, (6.10) from which it follows that in a stationary sound field the time-average of the instantaneous net flow of
eigenfunctions. Spinning modes in ducts with circular cross section are degenerate. Duct Acoustics 117 0.8 m 0.6 m 1.05 m Figure 7.34 Cross section of a uniform duct with rigid walls However, this equation can be written 2 − kn2 )ψm ψn∗ = 0. ∇ · (ψn∗ ∇ψm − ψm ∇ψn∗ ) + (km (7.148) Integrating over the cross section and applying Gauss’s theorem5 to the ﬁrst term gives s 2 (ψn∗ ∇ψm − ψm ∇ψn∗ ) · nds + (km − kn2 ) S ψm ψn∗ dS = 0, (7.149) where s is the perimeter of the cross section
right-hand side of Equation (8.24)) can also be expanded into a sum of modes, −δ(r − r0 ) = Bm ψm (r). (8.28) m Multiplying with ψn and integrating over the volume of the room gives, if we make use of the fact that the modes are orthogonal (cf. Equation (8.22)), − V δ(r − r0 )ψn (r)dV = − ψn (r0 ) = which shows that Bm = − V m Bm ψm (r)ψn (r)dV = Bn V , ψm (r0 ) , V (8.29) (8.30) Sound in Enclosures 135 and thus −δ(r − r0 ) = − 1 V ψm (r)ψm (r0 ). (8.31) m It now follows that (∇
Reverberation Rooms Reverberation rooms are used for a number of standardised measurements, some of the most important of which are brieﬂy described in the following. 8.5.1 Sound Power Determination Reverberation rooms are useful for measurement of sound power, in particular the sound power of machines that operate in long cycles, since the alternatives, the sound intensity method and the free ﬁeld method, are not very suitable for such sources. The considerations presented in Sections 8.3.3
rigid surface, which acts as an image source. Assuming that the distance to the plane is h we can calculate the resulting sound power in the same way as we used in deriving Equation (9.14): Pa = 1 lim Re 2 r1 → 0 jωρQe−jkr1 jωρQe−jkr2 + 4πr1 4πr2 Q∗ r2 → 2h = ρck 2 |Q|2 8π 1+ sin 2kh 2kh . The factor in parentheses is shown in Figure 9.2. It is apparent that the sound power is doubled if kh 1. However, the rigid surface is seen to have an insigniﬁcant inﬂuence on the sound power output