Improving AEDT Noise Modeling of Mixed Ground Surfaces (2017)

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3-1    CHAPTER 3. THEORY The calculation of propagation of sound through the atmosphere consists of solving the wave equation, which for a uniform medium is shown in equation (1)   ׏ଶ݌ ൌ 1ܿଶ ߲ଶ݌ ߲ݐଶ   (1)  where p is the acoustic pressure, t is time and c is the speed of sound. Solutions of the wave equation detail the pressure field as a function of time and space in the air around the noise source. It is customary to separate the time dependence out of the solution by assuming that it is sinusoidal. This is represented by a factor of ݁ି௜ఠ௧. (The minus sign convention is used throughout this report except where noted.) By separating out the time dependence from the solution to the wave equation all that is left is the pressure distribution as a function of space; thus, all mention of the pressure function p is understood to have the sinusoidal time factor. If the noise source is above a boundary or behind a barrier, the pressure field must detail their effects on the sound as it travels from source to receiver. Ground effects describe how interactions between a propagating acoustic wave and a boundary affect the acoustic field at a receiver location. These interactions can be caused by reflections and diffractions when treating the propagating sound as a ray traveling in a straight line. The ground effect can also be thought of in terms of the interference pattern of the reflected sound combining with the direct sound when considering the wave aspect of the sound field spherically radiating from all points on the boundary and arriving at the receiver. 3.1. Ray Theory In ray acoustics, sound energy travels along ray paths. Consider an acoustic point source above an infinite plane radiating sound to a receiver also above the same surface as shown in Figure 1. The sound is considered to propagate along the lines that join the source and receiver directly. The reflection of sound from the surface is treated as sound traveling on the line from the source’s image to the receiver.   FIGURE 1. Basic ray geometry with reflection. Direct Ray Reflected Ray Receiver Source Air Ground Refracted Ray Image Source hs hr x1 x2 Z1 Z2 θφ1 ra r1 φ φ2 θ1 θ2 rb

3-2    Assuming that the acoustic waves are planar, the acoustic field, p, at the receiver can be described by:   ݌ ݌଴ ൌ ݁௜௞భ௥భ ݎଵ ൅ ܴ௉ ݁௜௞భ௥మ ݎଶ   (2) where p0 is the pressure a unit distance from the source, k1 is the wave number in air, r1 is the length of the direct path, r2 is the length of the reflected path (where r2 = ra + rb), and RP is the plane wave reflection coefficient, given by Rudnick (1947) and Chessell (1977) as:   ܴ௉ ൌ ܼଶ cosሺߠଵ ሻ െ ܼଵcosሺߠଶሻ ܼଶ cosሺߠଵሻ ൅ ܼଵcosሺߠଶሻ  (3) and   ܴ௉ ൌ ܼଶ sinሺϕଵ ሻ െ ܼଵsinሺϕଶሻ ܼଶ sinሺϕଵሻ ൅ ܼଵsinሺϕଶሻ  (4) where Z1 is the specific acoustic impedance of the air, Z2 is the specific acoustic impedance of the ground,  and are shown in Figure 1, and and are not necessarily equal nor are  and necessarily equal. For rigid surfaces, with no wave transmission normal to the surface, and locally reacting surfaces, with  and are equal and  and are equal (Salomons The time dependence, ݁ି௜ఠ௧, is implied. Depending on the assumptions made about the boundary, RP can be simplified to various forms. For example, the ground can be treated as a locally reacting surface, an isotropic fluid capable of transmitting dilatational waves, or it can be treated as an elastic solid capable of transmitting both dilatational and shear waves (Piercy, 1977). For a locally reacting surface, where  = the plane wave reflection coefficient is given by Morse (1958) as:   ܴ௉ ൌ sinሺϕሻ െ ܼଵ/ܼଶsinሺϕሻ ൅ ܼଵ/ܼଶ  (5) For an extended reacting surface, the plane wave reflection coefficient is given by Embleton et al. (1983) as:   ܴ௉ ൌ Zଶݏ݅݊ϕ െ ܼଵሺ1 െ ݇ଵ ଶ/݇ଶଶ cosଶ ϕሻ ଵ ଶ Zଶݏ݅݊ϕ ൅ ܼଵሺ1 െ ݇ଵଶ/݇ଶଶ cosଶ ϕሻ ଵ ଶ   (6) where k2 is the wave number for the ground. For a grazing angle equal to zero ( = 0), Rp = -1 and r1 = r2. Thus the direct and reflected waves will completely cancel. This is not consistent with measurements and indicates that when the grazing angle is small, the plane wave assumption is not appropriate. This conclusion is not dependent on the boundary conditions as the results can be derived from the general expression for the plane wave reflection coefficient. In order to address this issue, Rudnick (1947) borrowed from electromagnetic theory developed by Sommerfeld (1909, 1926), Van Der Pol (1935), and Norton (1936, 1937). The development of this theory is beyond the scope of this present review, but the results can be described as follows. Equation (2) is replaced by the Weyl-Van Der Pol solution of the wave equation for spherical waves:   ݌ ݌଴ ൌ ቆ ݁௜௞భ௥భ ݇ଵݎଵ ቇ ൅ ܴ௣ ቆ ݁௜௞భ௥మ ݇ଵݎଶ ቇ ൅ ሺ1 െ ܴ௉ሻܨሺݓሻ݁௜௞భ௥మ ݇ଵݎଶ   (7)

3-3    where the “ground wave” (also called the Boundary Loss Factor) function F(w) is given by:   ܨሺݓሻ ൌ 1 ൅ ݅√ߨ ⋅ ݓ ⋅ ݁ି௪erfc൫െ݅√ݓ൯  (8) with w being the “numerical distance”. This treatment of ray theory with a spherical reflection at the boundary will continue to be referred to as straight ray theory for the remainder of this report. At this point a digression is required for clarity. There are numerous estimates of the numerical distance in the literature. Each is based on a set of assumptions and/or approximations as well as the form of the ground wave function. Another form of Eq. (8) defines the numerical distance squared (Pirinchieva, 1991). Three forms are listed here for normal impedance (Lawhead and Rudnick, 1951), Equation (9), locally reacting (Chessell, 1977), Equation (10), and extended reaction (Embleton et al., 1983), Equation (11):   ݓ ൌ ݅ 2݇ଵܴଶ൫1 െ ܴ௣൯ଶ ൬ߩଵܿଵܼଶ ൰ ଶ   (9)   ݓ ൌ 12 ݅݇ଵܴଶ ቆݏ݅݊߶ ൅ ൬ ܼଵ ܼଶ൰ቇ ଶ /ሺ1 ൅ ൬ܼଵܼଶ൰ ݏ݅݊߶ሻ  (10)   ݓ ൌ ݅ 2݇ଵܴଶ൫1 െ ܴ௣൯ଶ ܼଵଶ ܼଶଶ ቆ1 െ ݇ଵଶ ݇ଶଶ cos ଶ ϕቇ  (11) Note equations (9) through (11) all use the form of the ground wave function given in Equation (8). Equation (9) has the specific acoustic impedance of the air shown as ߩଵܿଵ emphasizing the resistance – only impedance of the air. For consistency with the literature, it should also be mentioned that Equation (7) can also be expressed as:   ݌ ݌଴ ൌ ቆ ݁௜௞భ௥భ ݇ଵݎଵ ቇ ൅ ܳ ቆ ݁௜௞భ௥మ ݇ଵݎଶ ቇ  (12) where, the “image strength” or “spherical wave reflection coefficient”, Q, is given by (Chien and Soroka,1975):   ܳ ൌ ܴ௉ ൅ ሺ1 െ ܴ௉ሻܨሺݓሻ (13) Equation (13) provides a succinct form to express the contribution of the reflected wave to the acoustic field as being a portion that can be modeled by a plane wave and by an additional component that accounts for the differences between a plane wave reflection and a spherical wave reflection. This difference can be referred to as the ground wave. The above formulation is for a pure-tone sound field (i.e. a single frequency); whereas, jet aircraft noise is comprised of a broad spectrum. Equation (14) can be used to calculate the excess ground attenuation in decibels for a point source of one-third octave band noise (Wyle, 1985) and has been shown to agree with jet noise propagation over ground (Howes, 1958).   Aୣ୥ୟ ൌ 10 logଵ଴ሾ1 ൅ |୕| మ ୰ᇲమ ൅ ଶ|୕| ୰ᇲ ୱ୧୬ሺ୤ ୲ሻ ୡ୭ୱሺ୤୲ା ሻ ୤୲ expሺെ0.5௣ଶ ሻ]  (14) where f = is the one-third octave band center frequency, r’ = r2/r1

3-4    t = difference in transit time between the reflected ray path and the direct ray path  = the phase angle of Q  = 2f/(2f)  = 2[1 + (f/2f)2]0.5 f = bandwidth of the one-third octave band centered at frequency f p = standard deviation of the path difference fluctuation caused by the atmospheric turbulence Numerical approximation of the ground wave function is required because the complementary error function does not yield a readily accessible closed form solution for complex arguments. Various approximations have been proposed and are a function of the numerical distance, w. Example approximations can be found in (Rudnick, 1947), (Ingard, 1957), (Chien and Soroka, 1975), (Chessell, 1977), (Daigle et al., 1985), (Nobile and Hayek, 1985), (Pirinchieva, 1991), (Salomons, pg. 134, 2001), (Plovsing et al., Appendix D, 2001), and (Attenborough et al., pg. 36-37, 2007). Empirical equations for the specific acoustic impedance, Z2 of acoustically absorptive surfaces, such as a typical ground surface have been developed by Delany and Bazley (1970). They created a one- parameter model of the ground using an effective flow resistivity as the dependent variable. The ground in this case is considered as a semi-infinite layer whose properties do not change with depth. A more complex treatment of the ground was developed by Donato (1977). It uses two parameters: an effective flow resistivity and the ground’s rate of change of porosity with depth. These equations for the impedance of ground for these two models are stated in the standard for its measure (ANSI/ASA S1.18, 2010). The real and imaginary parts of the ground impedance for the one-parameter model, R2 and X2, can be found in equations (15) and (16) where f is frequency (Hz),  is the flow resistivity of the ground (kPa s/m2), and the specific acoustic impedance of the air is used to normalize the ground impedance by dividing by the product of air density, 1 (kg/m3) and sound speed in air, c1 (m/s).   ܴଶ ߩଵܿଵ ൌ 1 ൅ 9.08 ൬ 1000݂ ߪ ൰ ି଴.଻ହ   (15)   ܺଶ ߩଵܿଵ ൌ 11.9 ൬ 1000݂ ߪ ൰ ି଴.଻ଷ   (16) The two parameter model of the ground uses an additional parameter that characterizes the change in pore density as a function of depth, e (m-1), along with the ratio of specific heat, , to compute the ground impedance using equations (17) and (18):   ܴଶ ߩଵܿଵ ൌ 1 ඥߨߛߩଵ ඨ ߪ௘ ݂   (17)   ܺଶ ߩଵܿଵ ൌ 1 ඥߨߛߩଵ ඨ ߪ௘ ݂ ൅ ܿଵߙ௘ 4ߨߛ݂  (18) It should be noted that the effective flow resistivity e has the same units as the one-parameter model, but it will have a different value for the same ground type due to the additional parameter for

3-5    porosity. Typical surfaces for which the parameters for both the one and two parameter model of the ground can be found in Table 1 and Table 2, respectively (ANSI/ASA S1.18, 2010). TABLE 1 Parameter Values Obtained Using One-Parameter Model Description of surface Flow resistivity, σ (kPa s/m2) Dry snow, newly fallen 0.1 m over about 0.4 m older snow 10 - 30 Sugar snow 25 – 50 In forest, pine, or hemlock 20 - 80 Grass: rough pasture, airport, public buildings, etc. 150 - 300 Roadside dirt, ill-defined small rocks up to 0.1 m mesh 300 - 800 Sandy silt, hard packed by vehicles 800 - 2500 Asphalt, sealed by dust and light use 30000 Upper limit set by thermal-conduction and viscous boundary layer 2 x 105 to 1 x 106   TABLE 2 Parameter Values Obtained Using Two Parameter Model Description of Surface Effective Flow Resistivity, e (kPa s/m2) Change in Pore Density, e (/m) Thick newly fallen snow 5 0 Floor of pine forest 7.5 16 Cricket field 70 25 Grass-covered field 100 250 Institutional grass 100 3 – 50 Lawn 182 40 Hay-covered field 188 50 Meadow with grass 8-10 cm high 227 121 For grass-covered surfaces, the one and two parameter impedance models behave similarly and have been shown experimentally to predict the measured spectra (Attenborough, 2011); however, these models did not perform as well as more complex models of ground impedance for different ground types. One modification to Delany and Bazley’s one-parameter model that improved its prediction when compared to measurements of sound propagation over forest floors was to assume the top layer of the ground had an effective depth down to a hard-backed layer. Given that the impedance of the ground is Zc = (R2 + iX2)/1c1, a modification to the ground impedance using the hard-backed layer model can be determined using:   ܼଶ ൌ ܼ௖ cothሺെi݇ଶ݀ሻ (19) where k2 is the wave number in the ground and d (m) is an estimate of the depth to a hard-backed layer below the surface with impedance Zc. The real and imaginary parts of the wavenumber, k2, in the ground, 2 and 2 are found using equations (20) and (21) where the components are normalized by the wave number in the air, k1.   ߙଶ ݇ଵ ൌ 1 ൅ 10.8 ൬ 1000݂ ߪ ൰ ି଴.଻଴   (20)

3-6      ߚଶ ݇ଵ ൌ 10.3 ൬ 1000݂ ߪ ൰ ି଴.ହଽ   (21) Table 1 from the ANSI standard is based on work by Embleton et al. (1983) which in turn is based on fitting data using various source-receiver geometries and the one parameter impedance model. Using the hard-backed layer model of the ground impedance for fitting measurements over different impedance ground results in the values in Table 3. These values come from use of the study by Attenborough et al. (2011) using data from Taraldsen and Jonasson (2011). TABLE 3 Parameter Values for Hard-Backed Layer Version of One-Parameter Model Description of Surface Effective Flow Resistivity,  (kPa s/m2) Effective Depth (m) Snow 10 – 50 0.1 – 0.5 Forest floors (pine and beech) 40 – 140 0.01 – 0.1 Grass (lawn, pasture, institutional) 50 – 400 0.01 – 0.04 Compacted grassland, bare ground 140 - 2000 0.01 – 0.04 Attenborough’s study (2011) compared various models to characterize ground impedance. The complexity of the models varied from needing only one input parameter, like Delany and Bazley’s flow resistivity above, to needing seven input parameters. Types of modeling varied from only characterizing the flow resistivity to including the depth to another layer below the surface as is shown in Equation (19). The one and two parameter models above showed good agreement with measurements over grassland, with the two parameter model performing best. One conclusion in the paper states that models requiring more than two parameters should be discounted for routine prediction of ground impedance. While it was shown that higher-fidelity models fit measured data with less error, the information needed to characterize the ground (pore shape, depth to hard-backing layers, etc.) will not be available in general for modeling terrain in a typical study. Hence, modeling the ground using the single parameter of flow resistivity is the most common method used in noise models. Sensitivity studies will be applied in this project to determine whether modeling the different surfaces with more than one parameter will be beneficial to the improvements sought for AEDT’s modeling of ground impedance. 3.2. Diffractions In addition to the reflected ray caused by the source-ground-receiver geometry, additional rays due to diffraction are generated when ground impedance discontinuities exist (Boulanger et al., 1997), (Daigle et al., 1985), (DeJong, 1983), (Plovsing, pg. 27-28, 2001), (Menge et al., pg. 96-98, 1998). Consider Figure 2, where an additional path is generated by a refraction point between the boundary with impedance Z1 and impedance Z2.

3-7      FIGURE 2. Basic ray geometry with reflection and single impedance discontinuity. In such a case, the acoustic field, p, as a function of angular frequency, , at the receiver location can be described by (Boulanger et al., 1997):   ݌ ݌ଵ ൌ 1 ൅ ݎଵ ݎଶ ܳଵ݁ ௜௞ሺ௥మି௥భሻ ൅ ሺܳଶ െ ܳଵሻ ݁ ି௜గସ √ߨ ݎଵ ݎଷ ൈ ሾܨଷଵ േ ܨଷଶ݁ ௜௞ሺ௥మି௥భሻሿ  (22) where, k is the wave number in air, r3 is the length of the diffracted path (where r3 = rc + rd), Q1 is the spherical wave reflection coefficient for the boundary with impedance Z1, Q2 is the spherical wave reflection coefficient for the boundary with impedance Z2, and p1 is the acoustic pressure at the receiver location due to the direct ray given by   ݌ଵ ൌ ݌଴ ݁ ௜௞௥భ ݎଵ   (23) and   ܨ௜௝ ≝ ܨ ቈට݇൫ݎ௜ െ ݎ௝൯቉  (24) for i,j = 1,2 or 3 with   ܨሺݔሻ ൌ න ݁௜௪మ݀ݓ ௜௡௙ ௫   (25) Reflected Ray Receiver Source Diffracted  Ray Direct Ray Z1 Z2 θ rb r1 rd Zair ra rc Image Source Air Ground

3-8    being the Fresnel integral. The plus sign between the Fresnel integrals in Equation (22) is used when the specular reflection occurs on the boundary with impedance Z1 and the minus sign is used when the specular reflection occurs on the boundary with impedance Z2. The Geometrical Theory of Diffraction (GTD) was formulated by Keller (1962) as it applied to optics. Rasmussen (1984) developed the methodology for acoustics while working for the Danish Acoustical Institute. The premise of the theory is to treat the space above an impedance discontinuity or barrier as a screen full of apertures. The solution is then carried out using ray theory to solve for the pressure from the source and its image at each aperture and then treat each aperture as a source whose pressure needs to be solved for at the receiver. The solution for the problem is then a sum of the pressures from all the apertures in the screen at the receiver. Figure 3 demonstrates the geometry described for an impedance discontinuity with the source above one surface type and the receiver above another. The GTD method can be applied to more complex terrain. In particular, it is used to correctly predict sound fields at receivers behind barriers (DELTA, 1993).   FIGURE 3. Geometry for the geometric theory of diffraction. When more than one impedance discontinuity exists between the source and receiver, as in Figure 4. Equation (22) produces poor results (Boulanger et al., 1997), therefore, in such cases, a ground impedance averaging approach is used. In this approach, an area-average impedance is computed by weighting the impedance of each ground type by the proportion of a Fresnel ellipse that the particular ground type occupies. The ground impedance averaging approach can be used for cases with a single impedance discontinuity. Both methods are presented here for completeness. The Fresnel ellipse is defined by the intersection of the boundary with a Fresnel ellipsoid defined by:   |ܵܲ| ൅ |ܴܲ| െ |ܴܵ| ൌ ܨఒߣ (26) where |SP| is the distance from source to point P, |RP| is the distance from receiver to point P, |SR| is the distance from source to receiver, P are points with a path length of /3 greater than the secularly reflected Air Ground Source Receiver Z 1 Z 2

3-9    path, and ܨఒ is an empirical constant, here set to 1/3 after Boulanger (1997). Assuming that the z-axis corresponds to height, the y-axis corresponds to the horizontal distance shown in Figure 4, the x-axis is perpendicular to the figure and a flat boundary, then the Fresnel ellipse can be defined by:   ݕଵ,ଶ ൌ ܤܣ േඨ 1 ܣ െ ൬ ܿ ⋅ ݏ݅݊ߠ ܣܾܽ ൰ ଶ   (27) and   ݔଵ,ଶ ൌ േܾඨ1 െ ሺݕ௠ܿ݋ݏߠ െ ܿሻ ଶ ܽଶ െ ݕ௠ଶ sinଶ ߠ ܾଶ   (28) where   ܽ ൌ ݎଶ ൅ ܨߣ2   (29) and   ܾ ൌ ඨݎଶܨߣ2 ൅ ൬ ܨߣ 2 ൰ ଶ   (30) and   ܣ ൌ ൬ܿ݋ݏߠܽ ൰ ଶ ൅ ൬ݏ݅݊ߠܾ ൰ ଶ   (31) and   ܤ ൌ ܿ ⋅ ܿ݋ݏߠ/ܽଶ (32) and   ݕ௠ ൌ ܤܣ.  (33) In this case, the ground effect can be described by the excess attenuation, ܧܣ ൌ 20 ⋅ ݈݋݃|݌ଵ ݌⁄ |, as given by (Boulanger et al., 1997):   ܧܣ ൌ 20 ⋅ log ൜ߤ ฬ1 ൅ ݎଵݎଶ ܳଵ݁ ௜௞ሺ௥మି௥భሻ ฬ ൅ ሺ1 െ ߤሻ ൈ ฬ1 ൅ ݎଵݎଶ ܳଶ݁ ௜௞ሺ௥మି௥భሻ ฬൠ  (34) where  is calculated from the proportion of the Fresnel ellipse covered by the ground type with impedance Z1.

3-10      FIGURE 4. Example source-receiver geometry over strip of different impedance.   3.3. Boundary Element Method The boundary integral equation method (BEM) is a general numeric solution of the Helmholtz harmonic wave equation. This is the Fourier transform of the wave Equation (1). An implementation of this method was used by Boulanger to model propagation above a plane with a strip of different impedance material normal to the direction of propagation. The solution to the problem presented in Figure 4 is given by   Pሺr,r0ሻൌ P1ሺr,r0ሻ ‐ ikሺ 1ܼ2 ‐ 1 Z1ሻනGZ2ሺrs,rሻPሺrs,r0ሻdsሺrsሻ௦   (35) where P1(r,r0) is the pressure at the receiver if the surface had the homogeneous impedance Z1 and r0, r, and rs are the position vectors of the source, the receiver, and a point in the boundary, respectively. GZ2(rs,r) is the Green’s function associated with propagation over a boundary of impedance Z2, and s is the surface of the strip. Boulanger noted that the BEM calculations were within 2 dB of the measured data for all frequencies in the range from 300 to 20000 Hz with measurements over a continuous sand surface. When compared with the de Jong method, the BEM gave characteristically similar results over an impedance discontinuity. This model was not applied to more complex geometries in Boulanger’s paper due to limitation of computer processing and BEM techniques available at the time. More work has been done in the time since to allow for more complex usage of the technique. This coupled with the increase in computer speed will allow for more complex usage of BEM. The details which are employed by the computational solvers are beyond the scope of this review; however, there are many tutorials that explore the uses of the methods of these codes. It was the intent of the authors to utilize a BEM code named Helm3D in order to compare the ground attenuation over mixed impedance surfaces to the Fresnel zone approach detailed above, but further investigation revealed that the BEM was not practical for long range Reflected  Ray Receiver Source Diffracted  Ray A Direct Ray Z1 Z2 θ Zair Image Source Diffracted  Ray B Z1 Air Ground

3-11    modeling due to the computation resources involved. However, it is noted that the method does agree well with other models, but is not a practical solution to the problem at hand. 3.4. Parabolic Equation The Parabolic Equation refers to a method of solving the Helmholtz equation, similar to the Boundary Element Method. There are two formulations for solving the problem: a Crank-Nicolson approach and a Green’s function approach. It is left to the reader to study the theory as it is beyond the scope of this report. Two good references for studying the method are Rosenbaum (2011) and Arranz (1996). Gilbert and White (1989) showed the degree to which the Parabolic Equation can match measurements if a refracting atmosphere is taken into account. In their paper, the estimate of the atmospheric profile from measurements between source and receiver was compared to the differences between predictions and measurements at discrete frequencies. The spread of as much as +/- 10 dB in the comparisons highlights the difficulty of including refraction in long range propagation models. The good agreement with measurements made over finite impedance surfaces (Arranz, 1996) and comparison with Rasmussen’s method make it an important theory to consider in this project (Arranz p. 91, 1996). One important note on the usage of the Parabolic Equation method: it is limited in its applicability for when the angle of elevation from the receiver to the source is less than 30 degrees; thus, it is not valid for areas below the source. As with BEM, computation times for running the Parabolic Equation method can be excessive. 3.5. Review In summary, all of the models described above are formulated for a constant atmosphere. No attempt was made to expand this review to account for aspects of propagation beyond how it is affected by the impedance of the ground over which the sound travels. The algorithms in AEDT are set up to account for individual aspects of propagation; therefore, as an example, finding a relationship that represented downwind propagation over hard ground would not be consistent with AEDT methodology. Furthermore, convolving different aspects of propagation would complicate implementation because it would also be necessary to include a relationship for upwind propagation over hard ground. The methods described above have been shown to agree reasonably well with measurements and each other for a range of conditions. The ray theory presented at the beginning of this chapter has been used extensively since its formulation. Combined with the one-parameter model of the ground it agrees well with measurements in controlled experiments at distances between source and receiver that allow for discounting of any refraction or turbulence from the atmosphere. Attenborough (2011) showed that for short-range distances (15 m) over grassland, the theory agrees well with measurements. Disagreement with measurements over different surfaces was shown to be a result of the ground model used and not the application of the theory itself. For example, measurements of sound propagation over forest floors agreed with theory best when using a hard-backed layer version of the one-parameter ground model. The detail needed to properly characterize the ground in order to invoke a complicated model of ground impedance beyond the one-parameter model is most likely not going to be available for most noise studies at US airports; however, the effects and sensitivity to the use of more complex ground models will be evaluated as part of this project. Sensitivity studies outlined below should indicate the effects of using such models. At greater distances, using commercial jets as the noise source, Plotkin (2000) showed that using ray theory together with the one-parameter model of the ground for propagation over soft terrain, whose flow resistivity had been measured, correctly accounted for excess ground attenuation out to 2000 ft for

3-12    one-third octave band noise measurements from 50 to 10000 Hz. This confirms the applicability of both ray theory and the one-parameter ground model for propagation over mowed grassland. 3.5.1. Single Impedance Discontinuity Sound propagating over large distances from an elevated source and receiver often encounter variations in the acoustic impedance of the ground boundary. For example, sound waves may encounter water, sand, pavement, lawn and then pavement again as the sound propagates from an aircraft approaching a coastal airport to the houses of local residents. Many approaches have been developed to account for single impedance discontinuities in outdoor sound propagation (Hothersall and Harriott, 1995) and (Boulanger et al, 1997). These include exact solutions to the Helmholtz equation, numerical solutions of the Parabolic Equation (PE), and semi- empirical, heuristic solutions. To obtain exact solutions, limiting the geometric assumptions, such as requiring both source and receiver to be near the ground, make the exact solutions inappropriate to use for most aircraft sound propagation problems. Even though computing power continues to increase, the number of computations required for numerical solutions such as Finite Difference or Boundary Element Models are still impractical for modeling large areas, as is required for modeling most transportation noise sources. The most accurate solutions that are still practical for engineering (but not research) sound propagation models are those that fall into the semi-empirical, heuristic solutions. Most of these semi-empirical solutions begin with wedge diffraction theory (e.g., Koers, 1983; and de Jong, 1983) or ground impedance averaging (Slutsky and Bertoni, 1987; Boulanger et al., 1997). Solutions using wedge diffraction theory model the impedance discontinuity by assigning one side of the wedge with one impedance value and the other side of the wedge with a second impedance value. The wedge angle is then adjusted to get the limiting case of a flat surface (Naghieh and Hayek, 1981). Solutions using impedance averaging ignore the oscillations due to diffraction and assume a monotonic transition between one impedance and the other dominating the sound propagation. The transition is determined by the ground in a region around the specular reflection point is generally modeled as a Fresnel zone with path differences ranging from /2 to /4 (Hothersall and Harriott, 1995). It is important to note that Naghieh and Hayek’s solution is actually an exact solution not relying on heuristics (Naghieh and Hayek, 1981). However, the solution is not well suited for engineering models and is referenced here only for the theoretical value. 3.5.2. Multiple Impedance Discontinuities When more than one impedance discontinuity is present, it is useful to consider how the discontinuities are distributed. If the discontinuities are periodic, with uniform spacing and varying between two impedance values, then a solution to the Helmholtz equation proposed by Nyberg (1995) can be used provided that the sum of the lengths of the two impedance regions is much less than a wavelength. The solution is valid for high frequencies only when the ground impedance changes rapidly as a function of distance. For long distances, low frequency sound tends to dominate, so this limitation may not pose a significant issue in many cases; however, in most real world applications, ground impedance does not vary periodically nor does it vary only between two impedance values. (Consider the example at the beginning of Section 3.5.1) Therefore, although Nyberg’s solution provides very good results for the specific cases that it can model, it is not considered a useful approach where ground

3-13    impedances are likely to change more or less randomly both in their geometry as well as their impedance values. It should be noted that for the periodic case, Boulanger et al. (1997) found that their impedance averaging approach did not perform as well as the Nyberg approach in predicting measurements and that the de Jong approach performed even less well. (It is not clear if the poor performance of the ground impedance averaging was due to sign errors in the printed equations 13 and 15, but is assumed that this is not the case based on the good performance elsewhere.) When more than one impedance discontinuity is present in a non-periodic form, de Jong’s semi- empirical model and ground impedance averaging should be considered. Hothersall and Harriott (1995) found good agreement between de Jong’s approach and a more accurate Boundary Element Model for both a single discontinuity and for two discontinuities defined by a strip in between the source and receiver. The impedance averaging approach had good agreement on average, but inherently did not predict the fine oscillations. In their analysis, Hothersall and Harriott found that de Jong’s approach performed better when at least the source or receiver was significantly above the ground but poorer when both source and receiver were close to the ground. Boulanger et al. (1997) also found that better agreement was obtained when the impedance weighted averaging was applied on the pressures (Equation (34)) rather than performing a weighted average of the excess attenuation. It is important to note that accounting for mixed impedances that run parallel to the sound propagation path requires that ground characteristics in a region surrounding the ray are examined. This level of detail is beyond the scope of an engineering sound propagation model and is therefore not included in this discussion. 3.5.3. Practical Considerations Sound propagation models developed for engineering use must balance model fidelity with computational speed. The de Jong model has been shown to produce good agreement when a single impedance discontinuity is present between source and receiver. However, with each additional diffraction modeled, the number of computations increases exponentially because each of these new paths can reflect and diffract again. For AEDT purposes, it is necessary to use a method that can maintain a reasonable level of accuracy while at the same time prevent excessive computations. In reality, every estimate of ground impedance is in effect a form of ground impedance averaging, since the practitioner is not able to account for every difference in rock and binder distribution in a section of pavement or the differences in clay-to-soil ratios or root distribution in a field. Ground impedances are averaged over an area before the data is even brought into the model. While these averages can be empirically based, they do reinforce the idea that averaging ground impedances is not only reasonable but is necessary to a large extent. The main drawback to ground impedance averaging is the loss of the fine oscillations due to the diffractions. However, when considering that transportation noise models use one-third octave band data and not, for example, power spectra, it cannot be assumed that a model implementation using de Jong’s approach for multiple discontinuities would show a significantly better match to the measured data. 3.5.4. Recommendations It is recommended that de Jong’s approach should be used when a single discontinuity is present, as this represents a reasonable compromise between accuracy and computational speed. When more than one impedance discontinuity is present it is recommended to use a ground impedance averaging approach whereby the effects on the pressures have a weighted averaged applied rather than applying a weighted average to the excess attenuation. This approach is also consistent with the practice of only modeling the most significant geometric diffractions as well.

3-14    Isolating the efficacy of a theory in how it compares to measurements is difficult at large distances because propagation over the intervening distance between source and receiver does not account for effects not addressed by the theory under study. As noted above, the objectives of this project will be met if the effects of ground impedance can be isolated and quantified.