ANNEXU.K.
Appendix E The finite segment correction
E2ESTIMATION OF THE ENERGY FRACTIONU.K.
The basic concept of the energy fraction is to express the noise exposure E produced at the observer position from a flight path segment P1P2 (with a start-point P1 and an end-point P2 ) by multiplying the exposure E∞ from the whole infinite path flyby by a simple factor — the energy fraction factor F:
| E = F · E∞ | (E-1) |
Since the exposure can be expressed in terms of the time-integral of the mean-square (weighted) sound pressure level, i.e.
to calculate E, the mean-square pressure has to be expressed as a function of the known geometric and operational parameters. For a 90° dipole source,
where p 2 and pp 2 are the observed mean-square sound pressures produced by the aircraft as it passes points P and Pp .
This relatively simple relationship has been found to provide a good simulation of jet aircraft noise, even though the real mechanisms involved are extremely complex. The term dp 2/d2 in equation E-3 describes just the mechanism of spherical spreading appropriate to a point source, an infinite sound speed and a uniform, non-dissipative atmosphere. All other physical effects — source directivity, finite sound speed, atmospheric absorption, Doppler-shift etc. — are implicitly covered by the sin2ψ term. This factor causes the mean square pressure to decrease inversely as d4 ; whence the expression ‘fourth power’ source.
Introducing the substitutions
the mean-square pressure can be expressed as a function of time (again disregarding sound propagation time):
Putting this into equation (E-2) and performing the substitution
the sound exposure at the observer from the flypast between the time interval [τ1 ,τ2 ] can be expressed as
The solution of this integral is:
Integration over the interval [–∞,+∞] (i.e. over the whole infinite flight path) yields the following expression for the total exposure E∞ :
and hence the energy fraction according to equation E-1 is
