Division Finite segment lateral attenuation

Equations (2.7.41) to (2.7.44) describe the lateral attenuation Λ(β,ℓ) of sound arriving at the observer from an aeroplane in steady flight along an infinite, level flight path. When applying them to finite path segments that are not level, the attenuation has to be calculated for an equivalent level path — as the closest point on a simple extension of the inclined segment (that passes through the ground surface at some point) generally does not yield an appropriate elevation angle β.

The determination of lateral attenuation for finite segments differs markedly for L_max and L_E metrics. Segment maximum levels L_max are determined from NPD data as a function of propagation distance d from the nearest point on the segment; no corrections are required to account for the dimensions of the segment. Likewise, lateral attenuation of L_max is assumed to depend only on the elevation angle of, and ground distance to, the same point. Thus only the coordinates of that point are required. But for L_E , the process is more complicated.

The baseline event level L_E(P,d) that is determined from the NPD data, even though for finite segment parameters, applies nevertheless to an infinite flight path. The event exposure level from a segment, L_E,seg , is of course less than the baseline level — by the amount of the finite segment correction defined later in Section 2.7.19. That correction, a function of the geometry of triangles OS₁S₂ in Figures 2.7.j to 2.7.l, defines what proportion of the total infinite path noise energy received at O comes from the segment; the same correction applies, whether or not there is any lateral attenuation. But any lateral attenuation shall be calculated for the infinite flight path, i.e. as a function of its displacement and elevation, not those of the finite segment.

Adding the corrections Δ _V and Δ _I , and subtracting lateral attenuation Λ(β,ℓ) from the NPD baseline level gives the adjusted event noise level for equivalent steady level flight on an adjacent, infinite straight path. But the actual flight path segments being modelled, those that affect the noise contours, are rarely level; aircraft are usually climbing or descending.

Figure 2.7.p illustrates a departure segment S₁S₂ — the aircraft is climbing at an angle γ — but the considerations remain very similar for an arrival. The remainder of the ‘real’ flight path is not shown; suffice it to state that S₁S₂ represents just a part of the whole path (which in general will be curved). In this case, the observer O is alongside, and to the left of, the segment. The aircraft is banked (anti-clockwise about the flight path) at an angle ε to the lateral horizontal axis. The depression angle φ from the wing plane, of which the installation effect Δ _I is a function (equation (2.7.39)), lies in the plane normal to the flight path in which ε is defined. Thus φ = β – ε where β = tan^–1(h/ℓ) and ℓ is the perpendicular distance OR from the observer to the ground track; i.e. the lateral displacement of the observer^f1. The aeroplane's closest point of approach to the observer, S, is defined by the perpendicular OS, of length (slant distance) d_p . The triangle OS₁S₂ accords with Figure 2.7.k, the geometry for calculating the segment correction Δ _F .

To calculate the lateral attenuation using equation (2.7.40) (where β is measured in a vertical plane), an equivalent level flight path is defined in the vertical plane through S₁S₂ and with the same perpendicular slant distance d_p from the observer. This is visualised by rotating the triangle ORS, and its attached flight path about OR (see Figure 2.7.p) through angle γ thus forming the triangle ORS′. The elevation angle of this equivalent level path (now in a vertical plane) is β = tan^–1(h/ℓ) (ℓ remains unchanged). In this case, observer alongside, the lateral attenuation Λ(β,ℓ) is the same for L_E and L_max metrics.

Figure 2.7.q illustrates the situation when the observer point O lies behind the finite segment, not alongside. Here the segment is observed as a more distant part of an infinite path; a perpendicular can only be drawn to point S_p on its extension. The triangle OS₁S₂ accords with Figure 2.7.j which defines the segment correction Δ _F . But in this case the parameters for lateral directivity and attenuation are less obvious.

Remembering that, as conceived for modelling purposes, lateral directivity (installation effect) is two-dimensional, the defining depression angle φ is still measured laterally from the aircraft wing plane. (The baseline event level is still that generated by the aircraft traversing the infinite flight path represented by the extended segment.) Thus the depression angle is determined at the closest point of approach, i.e. φ = β_p - ε where β_p is angle S_pOC.

For maximum level metrics, the NPD distance parameter is taken as the shortest distance to the segment, i.e. d = d ₁. For exposure level metrics, it is the shortest distance d_p from O to S_p on the extended flight path; i.e. the level interpolated from the NPD table is L_E∞ (P ₁, d_p ).

The geometrical parameters for lateral attenuation also differ for maximum and exposure level calculations. For maximum level metrics the adjustment L(β,ℓ) is given by equation (2.7.40) with β = β ₁ = sin^–1(z₁/d₁ ) and where β₁ and d₁ are defined by the triangle OC₁S₁ in the vertical plane through O and S₁ .

When calculating the lateral attenuation for airborne segments only and exposure level metrics, ℓ remains the shortest lateral displacement from the segment extension (OC). But to define an appropriate value of β it is again necessary to visualise an (infinite) equivalent level flight path of which the segment can be considered part. This is drawn through S₁′, height h above the surface, where h is equal to the length of RS₁ the perpendicular from the ground track to the segment. This is equivalent to rotating the actual extended flight path through angle γ about point R (see Figure 2.7.q). Insofar as R is on the perpendicular to S₁ , the point on the segment that is closest to O, the construction of the equivalent level path is the same as when O is alongside the segment.

The closest point of approach of the equivalent level path to the observer O is at S′, slant distance d, so that the triangle OCS′ so formed in the vertical plane then defines the elevation angle β = cos^–1(ℓ/d). Although this transformation might seem rather convoluted, it should be noted that the basic source geometry (defined by d₁ , d₂ and φ) remains untouched, the sound travelling from the segment towards the observer is simply what it would be if the entire flight along the infinitely extended inclined segment (of which for modelling purposes the segment forms part) were at constant speed V and power P₁ . The lateral attenuation of sound from the segment received by the observer, on the other hand, is related not to β_p , the elevation angle of the extended path, but to β, that of the equivalent level path.

The case of an observer ahead of the segment is not described separately; it is evident that this is essentially the same as the case of the observer behind.

However, for exposure level metrics where observer locations are behind ground segments during the takeoff roll and locations ahead of ground segments during the landing roll, the value of β becomes the same as that for maximum level metrics, i.e. β = β₁ = sin^–1(z₁/d₁ ) and