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ANNEX U.K.

Appendix E The finite segment correction

E2 ESTIMATION OF THE ENERGY FRACTION U.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.

(E-2)

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,

(E-3)

where p ² and p_p ² are the observed mean-square sound pressures produced by the aircraft as it passes points P and P_p .

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 d_p 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

and

the mean-square pressure can be expressed as a function of time (again disregarding sound propagation time):

(E-4)

Putting this into equation (E-2) and performing the substitution

(E-5)

the sound exposure at the observer from the flypast between the time interval [τ₁ ,τ₂ ] can be expressed as

(E-6)

The solution of this integral is:

(E-7)

Integration over the interval [–∞,+∞] (i.e. over the whole infinite flight path) yields the following expression for the total exposure E_∞ :

(E-8)

and hence the energy fraction according to equation E-1 is

(E-9)