Monopole, Dipole, and Quadrupole Models
In principle, the propagating sound waves caused by a fluctuating source, when there is no reflection from another wall, is not very different from the noise in a tube and the propagation of the sound wave for the pressure fluctuation, except that the bou
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Monopole, Dipole, and Quadrupole Models
In principle, the propagating sound waves caused by a fluctuating source, when there is no reflection from another wall, is not very different from the noise in a tube and the propagation of the sound wave for the pressure fluctuation, except that the boundary conditions must be matched for a given fluctuating source strength. Let a disturbance be made at r = 0 at time t1 due to a fluctuating source, which reaches the observer at a distance ro at time t2 . For the propagating medium (air) at rest, the disturbance propagates at sonic speed co . Thus, t1 = t − ro /co . It is evident that in the case of a wave train we may define the disturbance by a characteristic value in the wave, for example, by density, which may be given in terms of a Fourier series. For a fluctuating variable in the wave, it can be shown that for a simple harmonic wave the density fluctuation is approximately given proportional to cos[ω (t − r/co )], and thus 1 ∂ ∂ =− . ∂r co ∂ t
(2.1)
For noise generated due to fluctuations in mass, force, and turbulence, Lighthill [61, 62] has shown that the acoustic equation can be derived directly from the standard flow equations of conservation and momentum, and some of the terms may be considered to be due to a fluctuating monopole, dipole, or quadrupole. We will now discuss these in a fairly heuristic manner.
2.1 Fluctuating Monopole 2.1.1 Fluctuating Point Mass Source The fluctuating mass source being considered here initially has an infinitely small volume. In practice, however, this is not true, but one can always consider a case in which the observer hearing the sound is at a sufficiently large distance in relation to T. Bose, Aerodynamic Noise: An Introduction for Physicists and Engineers, Springer Aerospace Technology 7, DOI 10.1007/978-1-4614-5019-1 2, © Springer Science+Business Media New York 2013
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2 Monopole, Dipole, and Quadrupole Models
Fig. 2.1 Fluctuating mass source
the characteristic dimension of the source and the wavelength of the radiated sound. Physically, a fluctuating mass source can be generated by several methods (Fig. 2.1). On the left in Fig. 2.1 is a nozzle through which a mass flow rate q˙m = q˙m (t)[kg s−1 ] is flowing. A slightly different arrangement is a rotating perforated disk in front of a nozzle, technically known as a siren. On the right is an example of a fluctuating balloon, which will also be analyzed later and has the same effect. Note that q˙m with the dimension of the mass flow rate per unit time is independent of the volume of the actual source. Thus, let t1 = the time at which a sound wave originates at source (= 0), t2 = the time at which the sound wave reaches the observer (= ro /co ), and t = r/co be the time it takes to travel from the source to the destination. We consider now a mass flow rate q˙ [kg s−1 ] originating at a point. Let the mass flow rate have a time-averaged part and a time-dependent part, and let the disturbance be fluctuating at a point r = 0 at time t1 and reach the observer at dista
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