Compact Sources in Motion

For both numerical implementation issues and deeper insight into the general solution, the possibility of reducing the extended source region to a single point source or to a discretized set of sources involves the property of source compactness. A source region is said acoustically compact when retarded time variations over it are negligible in comparison with a charac­teristic period Tn. This condition is simple for stationary sources but more subtle for moving sources. It reads

where L is the extent of the source region and 1 — Mr an average value of the Doppler factor.

A simple example is detailed below to illustrate this key notion. Consider an emitting segment of length L moving at some subsonic speed V0 = M0 c0, as depicted in Fig. 15. At time t the segment is between two points (A, B) but the sound received at observer comes from retarded locations between the points (A’, B’) that are the retarded locations of (A, B) and that define the ’acoustic image’ of the segment. Because the propagation distances from

all retarded locations are different, the length of the acoustic image differs from L. It is longer if the segment is approaching the observer and shorter if it is leaving. Motion artificially induces spatial stretching of any source distribution. Because compactness must be evaluated on the acoustic image, a given source region is more compact when leaving the observer and less compact when approaching. Assuming dipoles on the segment, the general expression of the acoustic pressure is provided for instance by the second term of Ffowcs Williams & Hawkings’ solution specified as the line integral

For simplicity consider a uniform distribution of identical dipoles with axis aligned along the segment direction ei. The strength of the dipoles is P = P (t) e1 = Ae гші e1. At the origin of time the center of the segment will be at observer’s location, so that the times of interest are negative. The observer is assumed in the acoustical and geometrical far field, in front of the segment and on its trajectory. In these conditions the general formula reduces to

where the retarded time is a function of (x, n, t). At any time t’ the center of the segment is distant from the observer of —M0 c01’ and the element of the segment of coordinate n is at distance — (n + M0 c0t’) . Therefore te(n) = (t + n/co)/(1 — Mo). Furthermore in the far field 1/Re(t) — 1/R0(t) where R0(t) is the retarded distance of the center of the segment. It is worth noting that the parameters involved in the far-field assumptions must be evaluated at the retarded time and position. Therefore the condition of geometrical far field reads Re(t) ^ L/(1 — M0) and involves the length of the acoustic image of the segment. The acoustic pressure is finally written as

_ —ikAe-iut/(1-M°) L sin [kL/(2 (1 — M0))]

= 4n (1 — M0) R0(t) 1 — M0 kL/(2(1 — M0)) ‘

The result for a stationary segment with the same source distribution would be obtained by putting M0 = 0. The expression involves the sine – cardinal function which approaches 1 at very low frequencies such that kL/(1 — M0) ^ 1. Under this condition the acoustic pressure is pro­portional to the length of the segment, because the distributed sources are identical and perfectly in phase in the example, and the whole segment is said compact. When the argument is a multiple of n, the function is ex­actly zero and no sound can be heard in the conditions of the example: all sources interfere in such a way to produce a complete cancellation. The crucial point associated with source motion is that the Mach number now enters the definition of the argument. Typically at kL = n assuming source compactness for the stationary segment remains acceptable, whereas for the approaching segment at Mach number 0.5 the same frequency corresponds to a total extinction of the sound.

It is worth noting that the length of the acoustic image of the segment extends to infinity if the segment approaches the observer at exactly the speed of sound. In such conditions the solving procedure has been recog­nized as no longer valid. More generally the sonic singularity also means that any discretization cell of the source domain will not fulfill the compact­ness condition, leading to possible issues for numerical implementation.

Going back to the general result the assumed dominant loading-noise term for a compact surface in Ffowcs Williams & Hawkings’ solution sim­plifies to

±_ [ Ri д f Pi

4 n JS |_c0 R2(1 — Mr) dt’ V1 — Mr)

where the total aerodynamic force induced by the fluid on the surface F is introduced and where the over-bar, omitted in the following, stands for an average value over the surface. This simplified expression is the basis of the dimensional analysis proposed in the next section.