This work identifies and explores several aeroacoustic metrics that allow for urban air mobility (UAM) vehicle noise prediction. An increase in production and use of UAM and distributed electric propulsion vehicles within populated civilian areas stands behind the need to minimize the noise produced by these vehicles. The Federal Aviation Administration's (FAA's) strict noise regulations on UAM aircraft compels designers to place a significant emphasis, early in the design phase, on the characterization and analysis of the external noise generated by these vehicles, namely, to ensure their design viability. To accomplish this, the present study focuses on the analysis and interpretation of predicted noise signals using a set of characteristic metrics that can be instrumental at guiding the design process. Following a thorough review of metrics standardized by the International Civil Aviation Organization as well as the FAA, seven general metrics are identified, evaluated, and discussed in the context of UAM noise prediction. When used in conjunction with a modern surface-vorticity panel code, these metrics are shown to provide an effective assortment of tools to concisely describe UAM-based acoustic signal properties.

## I. INTRODUCTION

A substantial challenge posed to UAM vehicle design teams stands in collecting measured or predicted aeroacoustic signals and then reliably determining their favorable and adverse characteristics. From a historical perspective, research related to acoustics and sound generation affecting an urban environment may be traced as far back as 500 B.C., particularly, to Pythagoras and his intuitively conceived correlation between the length of a string and the pitch or frequency that it produces. He finds, for example, that by doubling the length of a string, the pitch is reduced by one octave relative to its original value.^{1} One hundred and fifty years later, the Greek architect Polykleitos participates in the design of the Theater of Epidaurus, one of the most well-preserved urban assembly sites of ancient Greece. Roman architect and engineer Vitruvius later notes that this esthetic structure is constructed in such a way to ensure that the acoustics are evenly distributed and distinct up to the uppermost seats, and that the voice meets “no obstruction.” During that era, it is also noted how bronze sounding walls and vessels can be used to reinforce the tones generated by humans.^{2} In the 6th century A.D., Roman philosopher Boethius publishes his work on the relation linking music, science, and human perception of sound. Through it, he particularly calls attention to the frequency's bearing on the sound perceived by humans.^{3}

Skipping forward to the early part of the 16th century, French mathematician Mersenne, who is often referred to as the “father of acoustics,” manages to develop preliminary expressions describing the frequency of an audible tone and the frequency ratio between a note and its octave.^{4} Further in the 16th and 17th centuries, the work of Hooke and Savart is known for ushering the development of what is now known as the Savart wheel, specifically, the first device capable of producing sound at a prescribed frequency.^{5} In roughly the same period, attempts are made to experimentally measure *c*, the speed of sound in air. However, disparate results are reported and this essential calculation is refined incrementally.

For example, in the mid-1600s, Gassendi attempts to calculate sound propagation using firearms; he emerges with a value of approximately *c* = 478 m/s. Although he arrives at an imprecise value, he notes that the speed of sound does not depend on the pitch itself, contrary to a statement made by Aristotle a millennium earlier. Following Gassendi's experiment, Mersenne reports a slightly rectified value of 450 m/s.^{6} This is followed by Borelli and Viviani, who jointly improve this measurement to 350 m/s. Around the same period, Bianconi independently deduces that the speed of sound must increase with temperature.^{7} Meanwhile, a similar attempt at predicting the speed of sound through air is made by Newton. By assuming that air particles undergo a simple harmonic motion, Newton predicts a speed of sound equaling the square root of pressure over density, $(p/\rho )1/2$, thus leading to slightly lower values than expected.^{1} His expression is soon adjusted by Laplace who, assuming isentropic gas behavior, determines the correct form of $c=(\gamma p/\rho )1/2$, where *γ* is the ratio of specific heats.^{8}

In hindsight, the development of calculus by Newton and Leibniz during this epoch may be viewed as being invaluable at providing the necessary tools to formulate, derive, and solve the first wave equation. The required effort is actually carried out by D'Alembert in the mid-1740s, the dawn of theoretical acoustics.^{9} Both Lagrange and Euler then independently extend the newly conceived calculus relations to pave the way for modern acoustics. As such, in the wake of Lagrange, Euler, and Newton, research in acoustics continues to flourish into the 19th century, particularly, through the efforts of von Helmholtz, Rayleigh, Stokes, and Kirchhoff, to name only a few.

For instance, around the mid-1800s, German physiologist and physicist von Helmholtz manages to achieve a deeper understanding of the physiology of the human ear.^{10} By observing the manner by which resonators in the cochlea are affected by alternating frequencies and tones, von Helmholtz points out how the quality of a tone results in unique resonator responses.^{1} This is followed by the construction of the Fast Fourier Transform (FFT), an efficient formulation of the Discrete Fourier Transform (DFT) by Gauss,^{11} which is popularized by Cooley and Tukey in 1965.^{12} Then in 1877, Lord Rayleigh publishes his *Theory of Sound* in two impressive volumes, with the second part being devoted to the theory of sound propagation in fluids.^{13} While this work is being brought to fruition, Stokes continues to advance his theory on hydrodynamics, including his seminal contributions to the development of the Navier–Stokes equations in 1880.^{14} Along similar lines, Kirchhoff, a German physicist with groundbreaking formulations in circuitry, spectroscopy, and blackbody radiation, makes substantial contributions in linear acoustics and mathematical physics. Among his many notable achievements, one may point out the Kirchhoff–Helmholtz integral and the Kirchhoff–Fresnel approximations. These would later prove to be instrumental in the analysis of single-edge diffraction and directional sound propagation.

Moving on to the 1950s, and prompted in part by the need to mitigate jet engine noise, Lighthill pioneers several predictive formulations in aeroacoustics.^{15,16} His concepts quickly gain popularity and are later viewed as being rather foundational to the advancement of theoretical and computational acoustic analysis. Shortly thereafter, Lighthill's work is extended by Curle,^{17} Ffowcs Williams,^{18} and Lowson,^{19} eventually leading to the groundbreaking article by Ffowcs Williams and Hawkings^{20} in 1969. The latter describes the acoustic contributions of sound emanating from a solid surface of arbitrary shape and propagating in a general direction through a fluid medium;^{20} today, the resulting formulation is universally referred to as the Ffowcs Williams–Hawkings (FWH) equation. Four decades later, Farassat^{21} obtains a closed-form solution to the FWH equation, which is quickly superseded by several reformulations,^{22–24} namely, the so-called Formulation 1, Formulation 1A, and Formulation 3. Due to their effectiveness and versatility, the last two formulations stand at the basis of several present-day subsonic and supersonic acoustics simulation packages, being embedded into ASSPIN/ASSPIN2, ANOPP/ANOPP2, PSU-WOPWOP, and other predictive codes in the literature.^{25–27}

With the growing number of airports and aircraft operating amid civilian populations, the need to predict, control, and suppress aircraft noise continues to receive attention in the acoustics research community. Examples include recent studies by McKay and Kingan,^{28} Jia and Lee,^{29} Lee and Lee,^{30} Gwak *et al.*,^{31} Smith *et al.*,^{32} Krishnamurthy *et al.*,^{33} Jeong *et al.*,^{34} Dbouk and Drikakis,^{35} Fenyvesi *et al.*,^{36} Ko *et al.*,^{37} and Yucel *et al.*^{38} With similar objectives in mind, the present work seeks to address the two-pronged nature of this problem. Not only must an aircraft designer identify the physical characteristics of a given signal, but also account for its human-perceived impact. In fact, the former issue is covered extensively in the fields of linear acoustics and signal analysis, particularly, through signal decomposition tools such as the Fast Fourier Transform (FFT) method. Such a technique adequately resolves the frequency spectrum of a signal, while its pressure magnitude is further simplified using suitably scaled measurements that are expressed in terms of the decibel “dB” unit. The latter enables us to precisely assign values to physical noise intensities on a logarithmic scale that is congruent with the human auditory range. First envisioned by Bell, this effective unit has been widely adopted since the 1931 edition of the *NBS Standards Yearbook*.^{39}

In complementing advances in acoustic measurement technology, the need to properly model the physiologically perceived noise is addressed through the works of several scientists; these include von Bekesy, Rhode, Robles, Fletcher, and Munson. On the one hand, von Bekesy's nonlinear model of the inner ear earns him a 1962 Nobel prize in physiology.^{40} Moreover, his framework proves instrumental to the experimental investigations of Rhode and Robles,^{41} whose measurements help to elucidate the nonlinearities affecting the growth response of particular signal frequencies and intensities. On the other hand, von Bekesy's results serve a vital role in corroborating the pioneering work of Fletcher and Munson,^{42} whose identification of equal-loudness contours for the human ear in 1933 remains one of the most cited works on the subject. Replicated presently in Fig. 1, the loudness curves are prescribed not necessarily by the signal frequency but, rather, by the signal intensity at a given frequency, as perceived by listeners. Consequently, each curve in Fig. 1 constitutes a line of equal-loudness, in units of “phons,” as a function of both frequency and intensity. To some degree, the ensuing graphs provide a visual depiction of the formidable nonlinearities inherent to the auditory system. In actuality, the availability of Fletcher and Munson's curves would later incentivize the development of several useful metrics by which objective qualities of human-perceived signals produced by aircraft and industrial machinery could be quantified.

In this vein, and following the experimental work conducted by Fletcher and Munson,^{42} other studies that are inspired by technological advancements have provided the basis for establishing additional aeroacoustic metrics. These have gradually gained popularity and have been incorporated into aircraft regulatory policies, mainly, those that are prescribed and implemented by both the Federal Aviation Administration (FAA) and the International Civil Aviation Organization (ICAO). These metrics include, for example, the Power Spectral Density (PSD) spectrum (including the A-weighting correction for a given signal), the Proportional Band Spectrum (PBS), the Overall Sound Pressure Level (OASPL), the Sound Exposure Level (SEL), the Perceived Noise Level (PNL), the Tone-corrected Perceived Noise Level (PNL-T), and the Effective Perceived Noise Level (EPNL). Such metrics serve to quantify the physical properties of a signal while accounting for the shift in intensity or perceived annoyance according to the physiological sensitivity of an individual's auditory system.

Additionally, the coupling of these two approaches, i.e., the physical description of a particular signal as well as its human-perceived characterization, will be jointly pursued to establish a systematic procedure for evaluating several complementary metrics that are capable of providing an effective description of the external noise generated by a specific vehicle design.

## II. METRICS: SPECIFIC FORMULATIONS AND SIGNIFICANCE

### A. Identifying metrics of broad interest

The multi-faceted field of aeroacoustics extends into almost every type of fluid flow application, with an equivalent number of metrics describing each problem. Because of the subject's broadness, it is helpful to identify several metrics that are capable of providing a complete but concise characterization of an externally observed signal. To be specific, the metrics must describe both physical and perceived properties at an observer's location. Recognizing that the latter is a function of the former, we turn our attention first to metrics that describe a signal's physical constitution.

With the end goal of designing an aircraft that can successfully meet acoustic regulations, we begin by performing a review of metrics standardized by the ICAO and FAA. Through this effort, a total of seven metrics are identified as measurements of interest; four of those describe a signal's physical makeup, and three metrics attempt to evaluate the human-perceived signal. These metrics, when viewed collectively, can help to convey a clear and concise description of the effect of an aeroacoustic signal.

In the interest of clarity, the computations used to illustrate the variety of metric characteristics in Secs. III and IV are based on a representative aircraft shown in Fig. 2. The vehicle in question is an open-source model that is publicly supplied as an Open Vehicle Sketch Pad (OpenVSP) geometric file of a UAM concept from NASA's library. The corresponding Revolutionary Vertical Lift Technology (RVLT) vehicle consists of an electric Vertical Take-Off and Landing (eVTOL) aircraft carrying eight 5-bladed propellers that operate at 1420 RPM, with a blade-tip Mach number of 0.5. This UAM vehicle is simulated in forward-flight mode at a zero degree incidence angle and a cruise speed of 20 m/s. Acoustic pressure-time values are collected at 36 observation points, which are distributed within a plane passing through the vehicle's centroid and parallel to the ground. Each observer is equidistant from the aircraft's geometric center with a $10\xb0$ separation about the aircraft assuming that the $0\xb0$ observer is positioned at the rear of the aircraft. An analysis of the vehicle's acoustic signature requires the extraction of the acoustic pressure from the recorded pressure-time data. The acoustic pressure may be defined as $ptot\u2212pstat=p\u2032$, where $ptot,\u2009pstat$, and $p\u2032$ represent the total pressure, static pressure, and acoustic pressure, respectively. It should be noted that at substantial observer distances, $pstat$ will consist of the atmospheric pressure only. However, if the observer falls in close proximity to the source, the static pressure may deviate from the atmospheric pressure value due to steady, normal forces acting on the vehicle's body. This additional shift is included within $pstat$.

In analyzing the makeup of the acoustic signature produced by such a vehicle, both the frequency-domain and time-domain components will be quantified using the PSD spectrum as well as the PBS, OASPL, and SEL distributions. In an effort to depict the human perceived characteristics, three additional metrics are considered, namely, the PNL, PNL-T, and EPNL. Additionally, these seven metrics are considered one-by-one and described in more detail.

### B. Power Spectrum Density (PSD) spectrum

The PSD quantifies the physical variance of acoustic pressure as a function of frequency in units of Pa^{2}/Hz due to the presence of sound waves. The PSD decomposition of the signal permits the extraction of *L*(*f*), the frequency band-dependent Sound Pressure Level (SPL). The frequency dependent SPLs are calculated directly from the PSD via

where $\Delta fref$ denotes the frequency bin width, and $pref$ refers to the standard acoustic reference pressure ($2\xd710\u22125$ Pa). This parameter allocates the distribution of pressure magnitudes into the corresponding frequency bins or bands that form the signal. For propeller-driven vehicles, such as the vehicle shown in Fig. 2, peaks arise along frequencies pertaining to primary propeller angular rates, and these are typically accompanied by diminishing peaks at blade passage harmonic frequencies.

To further illustrate this point, the PSD SPLs from a single observer taken at 50 m from the RVLT vehicle is visualized in Fig. 3(b) side-by-side with the acoustic pressure time-history, which is provided in Fig. 3(a). Therein, a principal peak can be seen at roughly 250 Hz, with several diminishing peaks both above and below this frequency. For vehicles carrying multiple propeller blade sizes or operating speeds, the PSD is capable of identifying the frequency at which the largest amount of acoustic power is being expended. The PSD calculation often serves as the first step in signal analysis due to its ability to accurately account for the physical frequencies that constitute a signal.

The PSD is calculated by first Fourier transforming an acoustic pressure-time signal into its complex Fourier coefficients; the latter enable us to extract the sound wave amplitude and corresponding phase angle at a given frequency. Due to the discrete nature of acoustic pressure data, it is necessary to employ the Discrete Fourier Transform (DFT) procedure. The DFT converts a sequence of *N* acoustic pressure amplitudes that are equally spaced in time, $p\u2032n={p\u20320,p\u20321,p\u20322,\u2026,p\u2032N\u22121}$, into another sequence of *N* frequency-dependent, complex valued acoustic pressure data points, $Pk={P0,P1,P2,\u2026,PN\u22121}$. The Discrete Fourier Transform is subsequently processed by evaluating

where *k*, *n*, and *i* stand for the Fourier coefficient index, the discrete acoustic pressure-time point index, and the defining root of imaginary numbers ($i2\u2261\u22121$), respectively.

#### 1. DFT and FFT

In practice, the DFT can be characterized by a unique matrix that encapsulates the overall transformation. However, because of the computational expense of this technique, the Fast Fourier Transform (FFT) algorithm is often substituted to reduce the ensuing effort, namely, by taking advantage of the symmetry underlying the DFT matrix. Credit for the FFT's modern formulation and popularization is often given to Cooley and Tukey,^{12} whose work in the mid-sixties has become recognized among the most significant and widest-reaching in computational science. For the reader's convenience, more detail on the DFT matrix manipulation, which is followed by the FFT algorithm and its implementation in the context of this study, are provided in Appendices A and B, respectively. As shown therein, the PSD can be resolved by scaling the resultant sequence $P\u2032k$ using

where *f _{k}* represents the

*k*th frequency band and

*T*refers to the total signal time period. In Eq. (3), $|Pk|2=ak2+bk2$, whereas

*a*and

_{k}*b*stand for the real and imaginary parts of the Fourier coefficients, respectively. As for $\varphi k$, which represents the phase at a given frequency

_{k}*k*, it may be readily determined from

Note that the bin-width in the PSD scheme is simply $\Delta f=1/T$. It may be added that only coefficients up to $N/2$ are used in the PSD calculation. Since the Discrete Fourier Transform for real signals, such as acoustic pressure-time signals, is mirrored on the left and right-hand sides of the ($N/2$)th point, only $N/2$ data points are required in the PSD computation.

### C. *A*-weighting (*W*_{A})

_{A}

While the PSD results lead to deeper insight into the physical makeup of the signal, it does not account for the human perception of the sound. Since the human-perceived intensity of a signal remains dependent on both the signal magnitude and frequency, this effect is accounted for using a weighting function. At the time of this writing, weighting functions are standardized by the International Electrotechnical Commission (IEC), and the most suitable function to adequately adjust for human sound perception is referred to as *A-*weighting. Its incorporation into PSD calculations can be undertaken rather straightforwardly by calculating

and

where $K1=2.242\u2009881\xd71016,\u2009K2=1.562\u2009339,\u2009f1=20.598\u2009997,\u2009f2=107.652\u200965,\u2009f3=737.862\u200933$, and $f4=12\u2009194.217.$ These values denote experimental fits to average measurements of the ear's sensitivity to specific frequencies. Thus, in the process of accounting for physiological perceptions, the A-weighting serves to gradually attenuate acoustic pressure amplitudes at frequencies that fall above and below 1000 Hz, i.e., the frequency at which the human ear tends to be most sensitive.

By way of illustration, Fig. 4(a) is used to depict the unweighted PSD SPL for the representative RVLT vehicle, whose A-weighted counterpart is provided in Fig. 4(b). Clearly, the A-weighting rectifies the PSD SPL distribution quite appreciably, namely, in a manner that is commensurate with its perceived impact on human hearing.

### D. Proportional Band Spectrum (PBS)

When analyzing broadband noise, such as jet engine mixing and turbulence, as opposed to signals comprised of several discrete signals (e.g., those generated by propeller-driven aircraft), the PSD does not produce well-defined peaks such as those observed in Fig. 4(a). For such cases, it is useful to evaluate the integrated acoustic pressure over larger bands of frequency. This metric, which is referred to as the PBS, is illustrated in Fig. 5 following its conversion to PBS SPLs. To account for the human perceived characteristics of the signal, the PBS is often A-weighted analogously to the A-weighted PSD. The PSD SPLs are calculated in the same fashion as the PSD levels using Eq. (1). In the interest of clarity, the A-weighted PBS SPLs are shown in Fig. 5(b) next to their unweighted counterparts in Fig. 5(a).

In practice, the PBS is commonly used to analyze noise distributed over a wide frequency range. It is comprised of 40 frequency bins, each spanning 1/3-octave. The bins exhibit a frequency range of 1–10 000 Hz, where the 30th band extends around a central frequency of 1000 Hz. Using the unweighted PSD level spectrum in Fig. 4(a) as an example, its corresponding PBS level spectrum in Fig. 5 displays the individual contributions of each 1/3-octave frequency band to the total signal SPL.

Interestingly, an octave band is defined as one for which the upper edge of the band is twice the frequency of its lower edge, specifically, $f(u)=2f(l)$, where the superscripts *u* and *l* refer to “upper” and “lower,” respectively. The relation between upper and lower edges of $1/N0$-octave frequency bands can be written as $f(u)=21/N0f(l)$. In this context, the edge ratio for 1/3-octave bands may be determined from $f(u)=21/3f(l)$. As for the band central frequency, it is given by $f(c)=21/6f(l)$ or $f(c)=2\u22121/6f(u)$, where the superscript *c* stands for “center.”

In Fig. 5, vertical lines are drawn to depict the centers of the proportional frequency bands. Note that the American National Standards Institute (ANSI) defines the PBS bands using a base-10 rather than a base-2 formulation, as shown above.^{39} The relation between upper and lower bands on a base-10 scale is then given by

As a result, the band central frequencies may be deduced from

Moreover, the value assigned to a given frequency bin, *m*, can be obtained by integrating the unweighted PSD over the frequency range of each bin. One gets

where *m* and *f* specify the band index and frequency, respectively. Similarly, for a discrete PSD, this integral becomes

where $\u27e8p2\u27e9m$, *N _{m}*, and $\Delta f$ denote the squared mean pressure, number of PSD points, and PSD frequency resolution within each bin

*m*, respectively. For convenience, it is customary to present the PBS on a logarithmic SPL scale using units of dB rather than $Pa2$, as shown presently. The conversion from $Pa2$ to $dB$ follows from

where *L _{m}* refers to the

*m*th band SPL and $pref=2\xd710\u22125\u2009Pa$.

### E. Overall Sound Pressure Level (OASPL)

The OASPL is commonly used to quantify the overall loudness of a signal at some observer location. On the one hand, the unweighted OASPL reflects the equivalent SPL that can be obtained by incorporating the total energy carried by all of the discrete spectral SPLs at every frequency in the signal into a single, well-behaved, cumulative wave. The resulting integration across all frequencies returns a single OASPL value at each observer location for a given event, such as an aircraft flyover. On the other hand, the A-weighted OASPL ($OASPLA$) represents the perceived SPL as physiologically interpreted by human listeners; we subsequently use metrics such as the PNL, PNLT, and EPNL to provide further detail on the character of the signal. Forthwith, the OASPL and $OASPLA$ may be extracted from the weighted and unweighted PSD spectra, respectively. Their calculation follows the same procedure arising in the proportional band analysis, except for using an infinite bandwidth. This can be accomplished by taking

where *M* denotes the total number of bins contributing to the PSD. In decibel units, the OASPL can be calculated from

The OASPL commonly serves two principal objectives, with the first being the visualization of a power distribution in a field of observers, and the second being an OASPL-time history for SEL computation, which is an essential metric for small aircraft certification. The generation of such power distribution plots requires the use of multiple observers that are distributed in an effective manner within the field of interest. In this fashion, a visual representation of the bulk direction of the signal power can be realized around the source.

Using the metrics described above, both OASPL and $OASPLA$ maps are readily generated and showcased in Fig. 6. The corresponding graphs depict three particular OASPL/$OASPLA$ lines that are taken at observer distances of 50, 100, and 150 from the RVLT aircraft's centroid. Based on these distributions, one may directly infer that, at the 50 m distance from the vehicle, the peak OASPL and $OASPLA$ values of 38 and 35 dB occur at $\theta =0\xb0$ and $(90\xb0,270\xb0)$, respectively. It can also be inferred that the minimum OASPL and $OASPLA$ of 28 and 19 dB appear at $\theta =(55\xb0,305\xb0)$ and $180\xb0$ symmetrically about the vehicle's axis, respectively.

The drops in decibel values at larger observer distances across all observers stand in agreement with the inverse-square law of outwardly propagating spherical waves. These OASPL maps prove very useful at characterizing the sound field around a vehicle, as it will be further demonstrated using three detailed UAM configurations in Sec. III below.

### F. Sound Exposure Level (SEL)

The SEL denotes the first of the two metrics used for ICAO and FAA certifications, with the other being the EPNL. The SEL is typically applied to small aircraft and follows directly from an $OASPLA$ time-history. For clarity, a representative SEL field is computed for the representative RVLT aircraft shown in Fig. 7. Since the SEL characterizes the perceived intensity or loudness from a single noise event, it returns a constant “lumped” SPL that carries the same amount of energy in one second as the original noise event. It is therefore essential to the certification of lightweight, propeller-driven airplanes and lightweight helicopters. To compute the SEL, one evaluates the following expression:

where *T* = 1 s represents a normalizing time constant and $OASPLA$ alludes to the A-weighted OASPL. As for *t _{i}* and

*t*, they denote the initial and final time steps of the original signal, respectively. The time-dependent $OASPLA$ metric can be derived from an acoustic pressure-time signal discretized into $0.5\u2009s$ intervals for which a single $OASPLA$ value is calculated at each interval. For the case of an observer that is moving with the vehicle, the SEL expression can be simplified into

_{f}### G. Perceived Noise Level (PNL) and Tone-Corrected Perceived Noise Level (PNL-T)

To measure the objective annoyance of an event, the PNL is commonly employed. The PNL and PNLT differentiate themselves from previous metrics by taking into account the additional annoyance created by the presence of pure tones within a signal. The PNL is measured in perceived noise level decibels (PNLdB). Numerically, the PNL equates to a random noise-band of width 1/3-octave, whose center frequency is 1000 Hz, and which is considered by listeners to be equally noisy. Typically, the original signal is discretized into multiple time segments, where a PNL is calculated for each segment. This, in turn, provides a PNL-time history of the signal. This particular procedure is then repeated for the evaluation of the EPNL, whose calculation requires a tone-corrected PNL time-history, commonly referred to as the PNL-T or PNLT. The PNLT increases the noise level to account for the impact of discrete tones within the spectra, particularly, those that are deemed to be more annoying, on average, than broadband noise of the same magnitude. The representative PNL and PNLT fields for the RVLT aircraft are shown in Fig. 8, with black and blue lines denoting PNL and PNLT values, respectively. The determination of PNL and PNLT values are standardized by the ICAO and follow the strategy described below (Fig. 9).

#### 1. PNL/PNLT calculation

The PNL evaluation can be initiated by first determining the perceived noisiness, N, in units of “noys,” within PBS bands between 50 and 10 kHz:

where both *M*(*a*) and SPL(*a*) represent empirically derived frequency-dependent coefficients, and with *a* serving as the particular coefficient index. These are based on the findings of Kryter and Pearsons,^{43} whose work in the 1960s have led to the development of the standard method by which the perceived noisiness is evaluated today. In the above, the corresponding coefficients are illustrated in Fig. 10. Once the values of N are found for each 1/3-octave band, the total perceived noisiness can be determined from

where *i* is the band index. The PNL is then calculated using

The PNLT follows from ten standardized steps that are outlined algorithmically below.

### H. Effective Perceived Noise Level (EPNL)

The EPNL, which serves as the second ICAO certification metric, is primarily employed in the certification of large airplanes and heavy helicopters. Although the relatively small-size RVLT vehicle does not belong to this category, its EPNL is computed for the chosen stationary observer field and presented in Fig. 11 for the sake of completeness. The EPNL consists of a single value quantifying the relative noisiness of an individual aircraft fly-by event. According to the ICAO Volume 1, Annex 16,^{44}

“… EPNL shall consist of instantaneous perceived noise level, PNLT, corrected for spectral irregularities (the correction, called ‘tone correction factor,’ is made for the maximum tone only at each increment of time) and for duration.”

and

“Three basic physical properties of sound pressure shall be measured: level, frequency distribution, and time variation. More specifically, the instantaneous SPL in each of 24 one-third octave bands of the noise shall be required for each one-half second increment of time during the airplane flyover.”

It can therefore be seen that the EPNL considers a specific PNLT-time history, in one-half second increments, and integrates the portion of the signal that falls within 10 dB of the peak. Mathematically, this operation corresponds to

where *T* consists of a normalizing constant of 1 s. The integration limits *t _{i}* and

*t*represent the times that bracket the portion of the signal that falls within 10 dB of the peak PNLT measurement. Note that all portions between

_{f}*t*and

_{i}*t*are included, regardless of whether the signal drops below 10 dB of the peak measured value at any point between

_{f}*t*and

_{i}*t*. Similarly to the SEL, for a fixed observer in the vehicle's frame of reference, the PNLT term inside the integral remains constant, and thus simplifies the resulting equation to the form

_{f}## III. DEMONSTRATIVE CASES

To illustrate how these metrics can be employed in a real-world application, several UAM type aircraft are analyzed using an acoustic module developed in-house for a surface-vorticity flow solver by Ahuja *et al.*^{45,46} The advantages of utilizing surface-vorticity and vortex methods have been demonstrated in several related applications by Wu *et al.*,^{47} Ahuja and Hartfield,^{48} Faure and Leogrande,^{49} Faure *et al.,*^{50} and Fenyvesi *et al.*^{36} The solver itself has also undergone stringent validation studies by NASA and UAM researchers.^{51–53} In what follows, numerical measurements are extracted from the solver and further analyzed using several of the metrics previously defined. When taken collectively, these metrics serve to fully characterize the impact of noise by providing a coherent landscape of the acoustic signatures produced around each representative vehicle.

### A. Case 1: Full UAM vehicle with isolated propeller

The first example that we consider focuses on a single five-bladed propeller on a contemporary UAM aircraft that is shown in Fig. 12(a). The prototype chosen for this simulation corresponds to the publicly available NASA OpenVSP model of the Joby S4 UAM eVTOL vehicle. The model is transferred from OpenVSP into the solver as a Plot3D geometric file with both solid and thin-surface representations of the propeller blades. More specifically, the fuselage, wings, and nacelles are Boolean-united in OpenVSP and then imported into the flow solver. The propeller is operated at a speed of 955 RPM with a vertical pitch of $\alpha =0\xb0$ and a tip Mach number of $Ma=0.4$. As usual, the propeller's pitch represents the angle subtended between the rotor disc's horizontal plane of rotation and the blade's chord line.

The acoustic signal histories are generated at 36 microphones that are distributed in the form of a circular array with a radial range of 50 m within the plane of the aircraft and centered about the vehicle's centroid. Each microphone is separated from its neighbor by an azimuthal angle of $10\xb0$, where acoustic pressure-time signals are recorded for each observer and then post-processed to retrieve the seven characteristic metrics at each observer location. By way of illustration, a sample acoustic pressure-time signal recorded at the angle of maximum in-plane OASPL (290°) is provided in Fig. 12(b). The cumulative weighted and unweighted OASPL values of each observer are subsequently displayed on a polar chart in Fig. 13(a). Therein, the locus of lowest and largest OASPL values are marked.

Along with the in-plane data, additional measurements are obtained in the out-of-plane field, as shown in Fig. 13(b). This is accomplished by lowering the first observer circle with a 50 m fixed radius to vertical distances of 50, 100, and 150 m below the aircraft. Although the general trends remain similar, moving below the aircraft reduces the sharp deviations in the curves that are caused by direct proximity to the propeller location. This may be attributed to the geometric smoothing of the distances from the source as the measurement plane is progressively shifted below the aircraft. Nonetheless, the locations of minimum and maximum OASPL remain virtually unchanged, although the angular distributions of the OASPL and $OASPLA$ become gradually more uniform with successive increases in the vertical distance from the plane; such a feature could not be captured strictly through in-plane measurements.

While OASPL measurements convey information about the physical distribution of a rotor's radiated acoustic power, an observer's experience of the signal intensity can be quite different when taking into account the increased sensitivity of human hearing to certain frequency ranges. This sensitivity is captured most aptly by the $OASPLA$, whose values are calculated as per Sec. II C and overlaid in both Figs. 13(a) and 13(b) (using blue-colored lines). Interestingly, the $OASPLA$ curves lead to generally comparable patterns to those of their unweighted counterparts, albeit at substantially lower magnitudes, especially around the $180\xb0$–$0\xb0$ line. Moreover, the location of maximum $OASPLA$ shifts slightly from 60° to 20°.

Based on the resulting visual representations, one may readily identify the points around the aircraft that receive the most acoustic power, and those at which observers may experience the most intense sound projected by the propeller. Note that air flow is convected from the $180\xb0$–$0\xb0$ line into the propeller facing the $180\xb0$ station. As expected, the loudest signals of approximately 65 dB are detected at essentially right angles to the propeller blades, looking at the tips of the blades. This is expected because the normal velocity of the blade (toward the observer) is the highest and thus produces the largest signal intensity at this particular location.

It should be noted that the total acoustic signal is comprised of both large and small sinusoidal waves of different frequencies. These correspond to the primary propeller frequency of 955 RPM, the five-bladed passage frequency, and their respective harmonics. The corresponding characteristics of the acoustic pressure-time signal [shown in Fig. 12(b)] are reflected in the frequency domain analysis of this signal, which is provided using both unweighted and A-weighted PSD computations in Fig. 14(a).

Based on this graph, it is apparent through the distinct local maxima of the PSD spectrum that the largest physical contribution stems from the propeller revolution rate (15.9 Hz). Nonetheless, the most significant perceived contribution to the overall noise appears at 429.4 Hz. A closer inspection of the PSD spectrum shows secondary contributors at frequency multiples of 5 times the propeller revolution rate (79.5, 159, 239 Hz,…). These are possibly caused by the aircraft's five-bladed propeller, which in turn translates into a blade passage frequency that is five times higher than the overall propeller revolution rate.

In complementing this analysis, Fig. 14(b) is used to display the 1/3-octave frequency bands, whose center frequency is denoted by vertical dashed lines. These bands range from approximately 1 to 400 Hz and capture the frequency band's proportionate decibel contribution to the OASPL. Graphically, one may infer that the largest physical (unweighted) contribution originates from the band around 15.9 Hz, and that the largest perceived (A-weighted) frequency band contributions stem from the 1/3-octave band centered near 429.4 Hz, thus lending additional support to the findings based on the PSD spectrum.

In the interest of completeness, the remaining four characteristic metrics for vehicle 1, namely, the PNL, PNLT, SEL, and EPNL, are evaluated and shown in Fig. 15. Therein, the PNL and PNLT are paired side-by-side with the SEL and EPNL measurements at three different distances from the vehicle's rotor. Graphically, one can infer from Fig. 15(a) that the inclusion of a tone-correction in the PNLT leads to a higher perceived loudness level at all observer locations. This behavior could have been, perhaps, anticipated because the presence of a pure tone in a signal creates the sensation of a louder noise. In fact, tone-correction effects are particularly prominent in Fig. 15(b) where contours of SEL and EPNL measurements are furnished. Since only the EPNL incorporates a tone-correction, it is markedly larger at nearly all detection points. Moreover, it may be seen that the overall shapes of the contours in Fig. 15 tend to resemble those of the $OASPLA$ in Fig. 13. The increased sound directionality and resulting differences also lead to a distinct shift from 60° in the locus of peak values relative to the OASPL contours. More specifically, the maximum values and locations of the PNL, PNLT, SEL, and EPNL are found to be 54.6 PNLdB (20°), 57.8 PNLdB (30°), 57.2 dB (20°), and 67.8 dB (30°), respectively.

### B. Case 2: Full UAM vehicle with multiple in-phase propellers

In the second case study, the remaining five propellers on the S4 UAM vehicle are enabled. Moreover, all propellers are operated synchronously at 955 RPM (15.9 Hz). In this case, the tip Mach number is kept at $Ma=0.4$ and the vehicle is operated in forward-flight mode. Hover modes for this vehicle are also directly possible and additional case studies can be undertaken using the present framework. This UAM vehicle configuration is shown in Fig. 16(a).

Once again, 36 stationary microphones are seeded in the form of a circular array at different radii with ranges of 50, 100, and 150 m, respectively; these microphones are placed in the plane defined by the forward velocity vector and an adjacent line running through the vehicle's centroid in level flight. As before, each microphone is separated from its neighbor by an azimuthal angle of 10°. For each observer, an acoustic pressure-time signal is recorded [see Fig. 16(b)] and then transformed into the frequency domain using both unweighted and A-weighted PSD spectra, as shown in Fig. 17(a). Consistently with the single propeller case, we find in Fig. 17(a) that most of the signal power stems from the propeller's RPM (15.9 Hz), with secondary contributions caused by harmonics associated with the blade passage frequency. In this configuration, as in the first case, spikes along multiples of the blade passage frequency, specifically at 15.9, 79.6, 159, 239, and 318 Hz, are detected. The corresponding PBS plots in Fig. 17(b) confirm that, despite the simultaneous operation of six identical propellers, the largest perceived (A-weighted) contributions to the PBS signal strength correspond to the 1/3-octave band centered at approximately 251 Hz, with a close second at around 80 Hz. Comparing Cases 1 and 2, where the maximum OASPL measurements return 65.6 and 77.3 dB, respectively, one may infer that the simultaneous operation of several propellers and methodical superposition of their collective SPL values clearly lead to a stronger signal strength across all frequencies than using a single propeller.

Moreover, and similarly to Case 1, we find that the largest physical contributor to the acoustic signal differs from the largest perceived contributor to the noise signal. The aircraft propeller noise, when heard by humans, is clearly dominated by its higher frequency tones that are more easily sensed by humans. These higher tones are intimately related to the number and size of propellers on the aircraft. For a given design team, the analysis may lead to a decision to reduce or increase the number of propellers in order to mitigate the perceived intensity of higher frequency tones.

In addition to acoustic pressure-time, PSD, and PBS computations, OASPL/$OASPLA$ maps are generated and displayed in Fig. 18(a) for the multi-propeller driven case; these are produced at fixed radii of 50, 100, and 150 m, respectively. For this case study, the use of multiple radii for generating OASPL maps helps to ascertain the effect of distance on signal strength and the expected attenuation as the distance from the source is increased. As before, the air flow is directed axially from the 180°–0° station into the propeller that faces it. Unsurprisingly, the loudest signal (83 dB) is observed once again at nearly right angles to the propeller blades while facing the tips of the blades, specifically at 70° and 290°. However, in contrast to Case 1, the presence of multiple propellers leads to a small OASPL spike in Fig. 18(a) that corresponds to an observer location of $0\xb0$, immediately behind the aircraft. The source of this peculiarity may be attributed to both constructive and destructive interferences, which are brought to bear through the use of multiple propellers. Moreover, and in accordance with the inverse-square-distance law for spherical pressure wave strength propagation, we see in Fig. 18(a) a distinct 6 dB loss in signal strength each time that the radial distance to the observer is doubled. This occurs when the radius is increased from 50 to 100 m; we also incur an additional 3.5 dB loss in signal strength as the distance from the source is increased from 100 to 150 m. These values help to confirm that the mathematical framework that we have implemented is capable of accurately predicting the power-distance relations for acoustic pressure signals. It should also be noted that not only does the use of multiple propellers affect the OASPL magnitude, it radically changes the shape of the $OASPLA$ distribution around the vehicle (c.f. blue-colored contours). Unlike Case 1, which yields $OASPLA$ contours that resemble those of their unweighted counterparts, a visible difference in the form of distinctive ripples in the $OASPLA$ lines can be detected in front of the aircraft. The constructive and destructive interference of multiple propellers, as well as observer proximity to certain sources, may be responsible for this $OASPLA$ behavior. Clearly, the sound coupling from multiple propellers has a nonlinear effect on the overall perceived signal distribution.

Furthermore, and in conformity with Case 1, out-of-plane OASPL data are obtained and shown in Fig. 18(b) for measurements taken in three descending horizontal planes of 50, 100, and 150 m below the aircraft. We note in this regard that as the observer field shifts vertically downwardly away from the aircraft, the angular locations of the minimum and maximum OASPL values appear to be different from those of Case 1 as well as the in-plane computations of Fig. 18(a). Caused mainly by the combined effects of multiple propellers, the maximum OASPL is seen to shift to the $0\xb0$ location, immediately behind the aircraft. Conversely, the minimum OASPL may be seen to move to around 100° and 260° symmetrically about the 180°–0° line. This is not surprising because increasing the distance below the aircraft makes the distance between the propellers and the vehicle's centroid negligible relative to the overall distance to the 50 m circle of detection. Moreover, consistently with Case 1, as we shift to lower planes of detection, the tendency to approach a rather uniform intensity at a given radius around the vehicle's centroid can be seen. As a result, the shifting of the maximum OASPL to the rear of the aircraft, where sound generated from the various propellers seems to converge, can be observed first at 50 m, and then quite evenly all around the aircraft as the distance reaches 150 m. Similar arguments can be used to explain the redistribution of OASPL intensities along the 90° and 270° stations below the aircraft. Another interesting feature appears in the $OASPLA$ lines of Figs. 18(a) and 18(b). Particularly, a similar shift toward a uniform distribution of signal intensity may be observed in the $OASPLA$ values, although the perceived intensity spikes present in the in-plane data are still clearly visible in the out-of-plane data as well.

This frequency dependent sensitivity is further reflected through computations of the corresponding PNL, PNLT, SEL, and EPNL, which are provided in Fig. 19. Here too, when compared to Case 1, it appears that the presence of multiple propellers leads to a markedly different sound structure including signal perception levels all around the vehicle. Similarly to the results obtained for a single propeller, and despite the virtual invariance of the locations producing the highest and lowest values, the inclusion of tone-corrections leads to an unmistakable increase in perceived loudness for all observers around the aircraft, including those stationed at the locations of peak intensity. In this case, the maximum values of the calculated PNL, PNLT, SEL, and EPNL are found to be 61.7 PNLdB, 63.4 PNLdB, 62.8 dB, and 73.4 dB, respectively. Due to the inherent symmetry of sound sources with respect to the fuselage, which is aligned with the $180\xb0$–$0\xb0$ direction, all peak values are detected evenly at both 70° and 290°.

### C. Case 3: DEP vehicle with multiple pusher propellers

The third case study is performed on a pusher-propeller UAM vehicle using NASA's OpenVSP file of the Kittyhawk KH-H1 DEP concept. The purpose here is to demonstrate the difference in acoustic signatures from two dissimilar conceptual UAM/DEP designs underlying Cases 2 and 3.

It should be noted that the KH-H1 vehicle employs eight pusher-type propellers in the forward-flight mode shown in Fig. 20(a). In this configuration, the two pairs of inboard wing propellers are operated at 1430 RPM (23.4 Hz) with a slightly reduced tip Mach number of $Ma=0.2$. As for the wingtip propellers, they are operated at a higher revolution rate of 1670 RPM (27.8 Hz) with a slightly increased tip Mach number of $Ma=0.25$. Finally, the canard propellers are operated at the highest revolution rate of 1910 RPM (31.8 Hz) with the highest tip Mach number of $Ma=0.28$. All propellers are assumed to remain in phase during their forward-flight mode operation.

As usual, a sample acoustic pressure-time signal that is representative of the highest acoustic pressure level at a fixed radius of 50 m is provided in Fig. 20(b). Furthermore, both weighted and unweighted PSD and PBS distributions are displayed side-by-side in Figs. 21(a) and 21(b). Despite the emergence of several contributing harmonic tones, one can possibly anticipate seeing three distinct spikes of equal magnitude, i.e., for each of the three different propeller RPMs being used. These are realized in the three spikes detected at the left of the PSD plot. In fact, by turning our attention to the PBS distribution in Fig. 21(b), it is encouraging to see that these three particular tones correspond to the largest physical contribution to the signal intensity, which is reflected in the frequency band that is centered at 25.1 Hz. However, when human perception in the form of an A-weighted PBS distribution is accounted for, it can be seen that the largest human-perceived detection band shifts toward the band centered at 251 Hz. Overall, we observe a change in the power distribution shape, which is based on the change in aircraft geometry and propeller location. Given the forward swept wings on the aircraft, it becomes rather apparent that propeller location and aircraft shape have noticeable effects on the overall distribution of signal power in the proximity of the aircraft. It is also interesting to note that the distribution of PSD peaks corresponds to the propeller's revolution rate, number of blades, and associated harmonic frequencies. For example, in contrast to Case 2, where a handful of distinct spikes in the PSD plot can be attributed to one primary propeller revolution rate, the presence of multiple revolution rates in Case 3 leads to a significantly more crowded PSD distribution; this is evidenced by the substantially larger number of spikes observed in Fig. 21(a). Without further investigation, it is actually difficult to attribute any particular spike to a single factor. Rather, it is likely that the emergence of each spike is due to a cumulative effect entailing both constructive and destructive interference of blade passage frequencies, vehicle geometry, and distance from the various propellers.

Along with frequency domain metrics, the computed in-plane and out-of-plane OASPL, both weighted and unweighted, are further reported on polar charts in Figs. 22(a) and 22(b). Here too, the flow is directed axially toward the propeller face along the $180\xb0$–$0\xb0$ horizontal line. On the one hand, for the in-plane measurements in Fig. 22(a), the signal is physically strongest at right angles to the aircraft and propeller blades, specifically at 90° and 270°, where the OASPL reaches a value of 53.0 dB at a radius of 50 m. The corresponding $OASPLA$ peaks at a much lower value of 29.1 dB, which consists of a whopping 23.9 dB drop. On the other hand, for the out-of-plane measurements in Fig. 22(b), which are collected at 50 m below the aircraft, the maximum OASPL and $OASPLA$ decrease to 50.9 dB and 44.5 dB at 0°, respectively. Overall, the reshaping of the OASPL distribution in the out-of-plane measurements seems consistent with the trends characterizing Case 2 in Fig. 18(b). In both instances, both unweighted and A-weighted OASPL contours approach a progressively more uniform distribution around the vehicle with successive increases in the vertical distance from the source.

Finally, as for the metrics that account for human perception and tonality effects, the relevant characteristics are provided in Fig. 23. In comparison to the OASPL contours, one can observe noticeable deviations in the overall distribution of the perceived intensity of the vehicle's acoustic signal, specifically, in relation to the distribution of raw physical sound intensity in Fig. 22. Similarly to Case 2, the visual representations in Fig. 23 confirm that the perceived loudness of a signal becomes highly angle dependent, judging by the sharp distortions of the contour plots at distinct observer locations. In fact, this heightened spatial sensitivity of the metrics may be attributed to the presence of several additional pure tones stemming from different propeller revolution rates and blade speeds at different observer locations. Although the angles at which the maximum and minimum values of these metrics remain largely unchanged, the deviations in perceived signal intensities can differ by 5 to 10 dB depending on the observer's angular orientation, notwithstanding the distance from the vehicle. Here too, because of the strict symmetry of the propellers about the fuselage's *X*-axis, the peak values of all four properties are detected equally at both 70° and 290°. As for their peak values, they are found to be 31.7 PNLdB, 34.4 PNLdB, 39.1 dB, and 44.4 dB for the PNL, PNLT, SEL, and EPNL signals, respectively.

### D. Comparative summary

For the reader's convenience, some of the characteristic metrics associated with the three illustrative cases, which include spatial locations and values of corresponding extrema, are provided in Table I. These refer to the $LPSD,\u2009LPBS$, OASPL, SEL, PNL, PNLT, and EPNL properties, as defined in Sec. II. The inclusion of A-weighting accounts for three additional variations, namely, the $LPSDA,\u2009LPBSA$, and the $OASPLA$. The resulting side-by-side comparison enables us to identify the common features as well as the dissimilarities among the three cases in question. For example, a cursory examination of the maximum PSD values confirms that the most physical power stems from the propeller's primary rotation rates. When the frequency-dependent perception of the human ear is accounted for using the A-weighting function, we find that the largest perceived peak values reside in the higher frequency ranges that humans are more sensitive to. Specifically, we find $LPSDAmax$ at values of 254, 255, and 168 Hz for Cases 1, 2, and 3, respectively. Each of these maxima correspond to a multiple of the blade passage frequency for the case in question.

Properties: . | Case 1 [Fig. 12(a)] . | Case 2 [Fig. 16(a)] . | Case 3 [Fig. 20(a)] . |
---|---|---|---|

Number of propellers | 1 | 6 | 8 |

Blades per propeller | 5 | 5 | 3 |

Propeller revolution rate | 15.9 Hz | 15.9 Hz | 23, 28, 32 Hz |

Blade tip Mach number, $Ma$ | 0.4 | 0.4 | 0.20, 0.25, 0.28 |

$LPSDmax$ (dB) (frequency) | 67.3 (16 Hz) | 75.4 (16.6 Hz) | 49.9 (23.4 Hz) |

$LPSDAmax$ (dB) (frequency) | 33.0 (429 Hz) | 50.9 (255 Hz) | 21.1 (168 Hz) |

$LPBSmax$ (dB) (band center frequency) | 67.5 (15.8 Hz) | 76.4 (15.8 Hz) | 50.5 (25.1 Hz) |

$LPBSAmax$ (dB) (band center frequency) | 37.5 (398 Hz) | 51.3 (251 Hz) | 23.4 (251 Hz) |

$OASPLmax$ (dB) (observer angle) | 68.0 (60°) | 77.3 (70°, 290°) | 53.0 (90°, 270°) |

$OASPLmin$ (dB) (observer angle) | 51.5 (170°) | 51.1 (170°, 190°) | 35.8 (170°, 190°) |

$OASPLAmax$ (dB) (observer angle) | 47.2 (20°) | 52.8 (70°, 290°) | 29.1 (70°, 290°) |

$OASPLAmin$ (dB) (observer angle) | 36.5 (140°) | 29.9 (180°) | 16.3 (170°, 190°) |

10 s $SELmax$ (dB) (observer angle) | 57.2 (20°) | 62.8 (70°, 290°) | 39.1 (70°, 290°) |

10 s $SELmin$ (dB) (observer angle) | 46.5 (140°) | 39.9 (180°) | 26.3 (170°, 190°) |

$PNLmax$ [PNLdB] (observer angle) | 54.6 (20°) | 61.7 (70°, 290°) | 31.7 (70°, 290°) |

$PNLmin$ [PNLdB] (observer angle) | 42.3 (140°) | 32.1 (180°) | 0 (160°, 200°) |

$PNLTmax$ [PNLdB] (observer angle) | 57.8 (30°) | 63.4 (70°, 290°) | 34.4 (70°, 290°) |

$PNLTmin$ [PNLdB] (observer angle) | 43.3 (30°) | 34.0 (180°) | 1.3 (160°, 200°) |

10 s $EPNLmax$ [EPNdB] (observer angle) | 67.8 (30°) | 73.4 (70°, 290°) | 44.4 (70°, 290°) |

10 s $EPNLmin$ [EPNdB] (observer angle) | 53.3 (140°) | 44.0 (180°) | 11.3 (160°, 200°) |

Properties: . | Case 1 [Fig. 12(a)] . | Case 2 [Fig. 16(a)] . | Case 3 [Fig. 20(a)] . |
---|---|---|---|

Number of propellers | 1 | 6 | 8 |

Blades per propeller | 5 | 5 | 3 |

Propeller revolution rate | 15.9 Hz | 15.9 Hz | 23, 28, 32 Hz |

Blade tip Mach number, $Ma$ | 0.4 | 0.4 | 0.20, 0.25, 0.28 |

$LPSDmax$ (dB) (frequency) | 67.3 (16 Hz) | 75.4 (16.6 Hz) | 49.9 (23.4 Hz) |

$LPSDAmax$ (dB) (frequency) | 33.0 (429 Hz) | 50.9 (255 Hz) | 21.1 (168 Hz) |

$LPBSmax$ (dB) (band center frequency) | 67.5 (15.8 Hz) | 76.4 (15.8 Hz) | 50.5 (25.1 Hz) |

$LPBSAmax$ (dB) (band center frequency) | 37.5 (398 Hz) | 51.3 (251 Hz) | 23.4 (251 Hz) |

$OASPLmax$ (dB) (observer angle) | 68.0 (60°) | 77.3 (70°, 290°) | 53.0 (90°, 270°) |

$OASPLmin$ (dB) (observer angle) | 51.5 (170°) | 51.1 (170°, 190°) | 35.8 (170°, 190°) |

$OASPLAmax$ (dB) (observer angle) | 47.2 (20°) | 52.8 (70°, 290°) | 29.1 (70°, 290°) |

$OASPLAmin$ (dB) (observer angle) | 36.5 (140°) | 29.9 (180°) | 16.3 (170°, 190°) |

10 s $SELmax$ (dB) (observer angle) | 57.2 (20°) | 62.8 (70°, 290°) | 39.1 (70°, 290°) |

10 s $SELmin$ (dB) (observer angle) | 46.5 (140°) | 39.9 (180°) | 26.3 (170°, 190°) |

$PNLmax$ [PNLdB] (observer angle) | 54.6 (20°) | 61.7 (70°, 290°) | 31.7 (70°, 290°) |

$PNLmin$ [PNLdB] (observer angle) | 42.3 (140°) | 32.1 (180°) | 0 (160°, 200°) |

$PNLTmax$ [PNLdB] (observer angle) | 57.8 (30°) | 63.4 (70°, 290°) | 34.4 (70°, 290°) |

$PNLTmin$ [PNLdB] (observer angle) | 43.3 (30°) | 34.0 (180°) | 1.3 (160°, 200°) |

10 s $EPNLmax$ [EPNdB] (observer angle) | 67.8 (30°) | 73.4 (70°, 290°) | 44.4 (70°, 290°) |

10 s $EPNLmin$ [EPNdB] (observer angle) | 53.3 (140°) | 44.0 (180°) | 11.3 (160°, 200°) |

Moreover, a direct correlation may be seen to exist between a vehicle achieving higher peak OASPL values and possessing more propellers or higher tip Mach numbers. This anticipated dependence helps to explain the reason for the 11 dB $OASPLmax$ increase between Cases 1 and 2, where the latter employs six propellers instead of one; it also helps to justify the largest $OASPLmax$ of 77.3 dB being associated with Case 2, and for the lowest $OASPLmax$ of 53.0 dB stemming from Case 3, whose respective blade tip Mach numbers average at 0.4 and 0.24, respectively. In fact, the same trend may be seen to affect the remaining metrics that include $SELmax$ and $EPNLmax$, with their highest values of 62.8 dB and 73.4 EPNdB corresponding to Case 2. The latter, unlike Case 3, exhibits the lowest $SELmax$ and $EPNLmax$ values of 39.1 dB and 44.4 EPNdB, respectively. The finding that Case 3 produces the lowest metrics despite possessing the most propellers suggests that the overall perceived intensity is more dependent on the blade tip Mach number than the number of propellers used. In this vein, UAM designers seeking to minimize the overall acoustic output (and reliability) may be better served using a large number of propellers at relatively low blade-tip speeds than a small number of propellers running at larger speeds. One also notes that an aircraft with a symmetric distribution of propellers, such as Cases 2 and 3, gives rise to symmetric distributions of $OASPLmax,\u2009OASPLAmax$, and $SELmax$.

Further, by examining the minimum values of the various metrics in Table I, one may identify, for all cases considered, a consistency in locating the lowest metrics directly in front of the aircraft, around the 180° observer station. The reason for the quietest signal to appear along the axis of the vehicle, directly in front of it, and not behind it, may be partly attributed to the forward-flight motion, the direction of fluid convection, and the normal orientation of the 180°–0° line with respect to the propeller blades.

## IV. CONCLUSION

In this work, a set of seven particularly useful acoustic metrics that are relevant to the analysis of SPLs, frequency dependent spectrum levels, power distributions, and perceived noisiness produced by an aircraft in flight are identified and defined. It is also shown how the corresponding calculations can be practically implemented in conjunction with a robust surface-vorticity solver. Presently, these metrics are evaluated and discussed in the context of several UAM vehicles with highly dissimilar and complementary acoustic signal characteristics. The analysis provided through our designated metrics leads to useful insights into the topological characteristics of a given signal, such as both physical and perceived signal power levels being most sensitive to the blade tip Mach number. We also identify a highly directional dependence of perceived signal strengths on the spatial angle from the vehicle, with the latter being attributable to vehicle geometry and proximity to different propellers. This behavior is adequately captured by A-weighting and tone-corrections, including PNLT computations, which account for an increase in perceived intensity by 2–3 dB.

Although the evolution of a symmetrical signal distribution around the aircraft is unsurprising, one may not have anticipated the directionality of the signal and the nonlinear manner by which perceived signals depend on propeller locations, revolution rates, and the resulting constructive and destructive interference of associated tones.

Based on the fundamental metrics that are evaluated and discussed here, the present analysis leads to numerical predictions of both physical (unweighted) and perceived (weighted) qualities of aircraft generated noise. Also provided are visual representations of the spatial distributions of the acoustic power within the observer field that are computed over particularly meaningful bandwidths. In this manner, the effective combination of several figures of merit and graphical aids stands to supply the aircraft design teams with an assortment of measures that can facilitate the identification of both actual and perceived noise levels and locations early in the design phase. Through the analysis of the presented metrics, it may be further concluded that both physical and perceived intensities of a given signal depend more appreciably on the blade tip Mach number than on the number of propellers; this feature seems to be well aligned with the reliance of modern UAM vehicles on multi-propeller rotor systems as opposed to the traditional single-propeller helicopter design. Moreover, as the distance from the aircraft is increased, the location of the propellers becomes immaterial, especially that the signal tends to become progressively more uniform at remote locations. Given such predictive capability at their disposal, design teams can proceed to assemble the information necessary to make incremental modifications that enable the aircraft in question to meet FAA and ICAO regulations. Ultimately, with an efficient implementation of these metrics into a fast-performing and reliable flow solver, aeroacoustic analysis can be readily integrated into the early design phases of a developmental program, thus ensuring a seamless transition from a conceptual vehicle design to a commercial platform. Efforts in this direction are presently under way, as illustrated in a companion paper by Ahuja *et al.*^{54}

## ACKNOWLEDGMENTS

The authors wish to acknowledge the cooperation and support of the NASA researchers and the NASA SBIR Office on this effort. The primary work for this study was performed within the context of the NASA STTR 80NSSC20C0586 Phase I contract activity.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Daniel S. Little:** Conceptualization (lead); Data curation (lead); Formal analysis (lead); Funding acquisition (supporting); Investigation (lead); Methodology (lead); Resources (equal); Software (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). **Joseph Majdalani:** Conceptualization (equal); Data curation (supporting); Formal analysis (equal); Funding acquisition (supporting); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (supporting); Supervision (lead); Validation (supporting); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **Roy Hartfield:** Conceptualization (equal); Funding acquisition (lead); Project administration (equal); Resources (equal); Supervision (equal); Visualization (equal). **Vivek Ahuja:** Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (supporting); Visualization (supporting); Writing – original draft (supporting); Writing – review & editing (supporting).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

### APPENDIX A: DFT MATRIX

As previously defined, the DFT algorithm takes a collection of *N* points and transforms its components into their Fourier coefficient counterparts, ${f0,\u2009f1,f2,\u2026,fN\u22121}\u2192{f\u03020,\u2009f\u03021,f\u03022,\u2026,f\u0302N\u22121}$, where *f _{n}* and $f\u0302n$ refer to the

*n*th sample data and Fourier coefficients, respectively. This transformation may be accomplished by taking,

and, inversely,

The exponential coefficient contained in this expression may be replaced by a complex frequency coefficient, $\omega n$, where

Based on this definition, one can compute a DFT matrix encapsulating the Fourier transformation, which we will call **F**. The DFT can then be written as $f\u0302=F\xb7f$ or, explicitly,

In practice, $F\xb7f$ can be quite expensive to compute, having a size that is proportional to $O(N2)$.

### APPENDIX B: FFT ALGORITHM

This section illustrates the advantage of using the Fast Fourier Transform. Based on its fundamental definition, the DFT calculation requires $O(N2)$ calculations, whereas the FFT calculation entails an almost linear relation between data points and computational cost, specifically, $O(N\u2009\u2009log\u2009N)$. For this reason, the difference between the standard DFT and FFT algorithm remains negligible at small values of *N*; however, as *N* increases beyond approximately 100, the advantage of using the FFT method can translate into several orders of magnitude in run-time savings.

For example, given a typical 10 s audio signal, which is sampled at 440 kHz, one recovers $N=4.4\xd7105$ data points to manipulate. Comparing the FFT and traditional DFT formulations, the increase in computational efficiency consists of a factor of roughly 100 000 times. The FFT algorithm takes advantage of the extensive symmetry throughout the DFT matrix.

When it was first being studied, the FFT was only applied to specific values of *N*, specifically, for even values of *N*. When extending these datasets where $N\u22602k$, a transform known as the chirp-*z* transformation, or zero-padding may be applied to the given datasets. For convenience, we find it useful to define a special FFT algorithm that is ideally suited for cases with $N=2k$.

To illustrate this operation, consider a dataset of size *N*, where $N=210=1024$. The DFT matrix can be factorized and the DFT rewritten using

In the above, $I$ represents an identity matrix, $DN$ refers to a diagonal matrix of size *N*, and $FN/2$ denotes a DFT matrix of size *N*/2; finally, $feven$ and $fodd$ stand for the even and odd indexed data points of the original data vector $f$. It may be instructive to note that the process of converting $FN\u2192FN/2$ may be carried out until $N/k=2$, where *k* is the number of times the method is employed. As such, two iterations of this process yield

For datasets of size $N=210$, one gets: $F1024\u2192F512\u2192F256\u2192\cdots \u2192F4\u2192F2$. When carried out to completion, this action specifically reduces the computational cost by almost a whole factor of *N*. In standard notation, this transformation can be expressed as

Several variations of this method can be found in the literature, including those using factors 4 and 8 instead of 2.^{55}