By M. S. Howe
ISBN-10: 052181281X
ISBN-13: 9780521812818
This is often an advent to the department of fluid mechanics considering the construction of sound by means of hydrodynamic flows. it's designed for a one semester introductory direction on the complex undergraduate or graduate point. nice care is taken to provide an explanation for underlying fluid mechanical and acoustic techniques, and to explain totally the stairs in a sophisticated derivation. The dialogue offers in particular with low Mach quantity flows, which allows the sound produced via `vortex-surface' interactions to be analyzed utilizing the `compact Green's function'. this gives a regimen process for estimating the sound, and a straightforward id of these elements of a constitution which are more likely to be vital assets of sound.
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Example text
9) Now, xj ∂ . 10) Applying this formula for j = 1, we find (putting r = |x| and x1 = r cos θ) ϕ=− a 3 cos θ r U t− 2r 2 c0 near field − a 3 cos θ ∂U r t− 2c0r ∂t c0 far field . The near-field term is dominant at sufficiently small distances r from the origin such that 1 r 1 1 ∂U f ∼ , c0 U ∂t c0 where f is the characteristic frequency of the oscillations of the sphere. But, sound of frequency f travels a distance c0 / f = λ ≡ one acoustic wavelength in one period of oscillation 1/ f . Hence, the near-field term is dominant when r λ.
The result is usually expressed as the simple reciprocal relation ˆ ˆ G(x, y, ω) = G(y, x, ω). 4 Time-Harmonic Compact Green’s Function ˆ We are now ready to derive the compact Green’s function G(x, y, ω) for the problem depicted in Fig. 1. 1) where the rigid body S is assumed to be acoustically compact. The influence of Fig. 1.
The value of the integral over the eddy then may be estimated to be of order ρ0 v 2 3 . 2 Lighthill’s v 8 Law 31 Fig. 1. The order of magnitude of the time derivative for changes in the source region is ∂ v ∼ . 3) that, for one eddy, the far-field acoustic pressure satisfies p∼ ρ0 v 4 ρ0 v 2 M 2 . 4) taken over a large sphere centered on the eddy. Thus, in order of magnitude, acoustic power radiated by one eddy ∼4π|x|2 p2 ∼ ρ0 c0 2 ρ0 v 8 = c05 2 ρ0 v 3 M 5 . 5) This is Lighthill’s ‘eighth power’ law.
Theory of vortex sound by M. S. Howe
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