Green’s functions

So, what do these wave equations represent? Well, simply stated: they describe propagative wave-like fluctuations of the density or pressure in a

quiescent fluid medium[6]. Such wave-like motion will only be sustained by the medium for space-time scales that satisfy the balance expressed by the equation. A Fourier transform of the wave equation can help illustrate this:

w p = c0k p,

wp’ = Cokp’. (13)

This is known as the dispersion relation for the wave equation, and what it states is that for propagation to be supported in the quiescent, homoge­neous fluid medium considered, the time scales of the motion, w-1, must be matched with the space scales, к-1, by the speed of sound, co. When such a system is excited by a disturbance that does not satisfy this criterion, the associated motions will not be supported as a propagating wave, and will tend, rather, to evanescence (very rapid decay). This concept is central to understanding the mechanisms by which a given source structure [7] gener­ates a propagative energy flux, and these mechanisms can be most clearly seen by looking at integral solutions of the wave equation, which can be obtained by means of an appropriate Green’s function.

The Green’s function, G(x, ty, r), describes the wave-like response (as described by the wave equation) of the quiescent fluid medium to an im­pulse localised at x = y and at time t = т. Where the free-field Green’s function is concerned, a single clap of your hands in a large open space is an approximate equivalent of this. Mathematically, this can be expressed as:

rf-G

-д£Г – c2ag = s(x – y)s(t – т). (14)

Once we have found the Green’s function we are equipped with a filter which, when convolved with a given source, will extract the space-time scales of the source structure that match the balance expressed by the propagation operator (-дрг = c2o Ap’), and which are therefore capable of producing a propagating wave. For example, consider the physical problem described

by ‘

^2 – clAp’ = q(x, t), (15)

where q(x, t) is some (known) source (this could be an unsteady, spatially – distributed force field, or an unsteady, spatially-distributed, addition of
mass), that drives sound waves in a quiescent medium. Multiplying equa­tion 14 by p’, equation 15 by G, integrating in both space and time (ne­glecting the effect of initial conditions), and subtracting the former from the latter, we get, provided there are no solid boundaries, and after a little manipulation

(16)

The right hand side of this equation describes the filtering of q(y, t) by G(x, ty, r): G(x, ty, t) allows us to extract, from the heart of what might be an extremely complex, and largely (acoustically) ineffective, source struc­ture, q(y, T), only those scales that are acoustically-matched.

This is the key to analysing and understanding aeroacoustic systems, experimentally, numerically or theoretically. It is necessary to identify the space-time scales (or flow behaviour that leads to the generation of such scales) that are actually efficient in the generation of sound waves—the vast majority are not. In the context of Lighthill’s acoustic analogy the problem is exactly that described here, insofar as the wave equation used has the same form as 15. For the more sophisticated acoustic analogies, while the wave equations and source descriptions change, conceptually we are dealing with the same scenario: the dispersion-relations and Green’s functions will change, and this will modify the criterion by which we identify the pertinent space-time scales of the ‘source’ quantity (which it is then necessary to relate to the turbulence characteristics of the jet). Further discussion on this point is provided in the next section.