Comparison of time and frequency domain analysis of propeller loading noise

Abstract

This study presents a unified analytical framework for investigating the acoustic pressure generated by rotating point forces, with particular emphasis on comparing time and frequency domain formulations. While both approaches yield consistent results, each offers distinct benefits: the time domain provides intuitive insight into the contributions of steady and unsteady loading, whereas the frequency domain facilitates modal decomposition and simplifies the analysis of periodic noise. The model represents uniformly spaced blades as point forces with axial, drag, and radial components. The force magnitude varies in time according to a Gaussian pulse, enabling investigation into the influence of time scale on propeller noise. This formulation is relevant to scenarios such as rotor systems experiencing periodic loading or isolated gust encounters. Both periodic and aperiodic cases are analysed in the time and frequency domains, allowing conditions for constructive and destructive inter-blade interference to be identified. The study also examines the advantages of each formulation. Time domain analysis provides a more intuitive interpretation of aperiodic phenomena but requires careful treatment of retarded time effects. In contrast, the frequency domain is well-suited to periodic problems and naturally aligns with modal acoustic models, though it may obscure certain inter-blade interaction mechanisms. Collectively, these insights advance the understanding of rotor noise generation and its spatial characteristics in installed configurations.

Introduction

This paper provides a comparison of the noise due to rotating point forces formulated in the time and frequency domains within a consistent theoretical framework. We demonstrate that these formulations provide consistent results but offer different insights into the character and mechanisms of sound generation. The noise radiated from a more complex and realistic force distribution may be obtained by the superposition of the sound fields from this idealised problem.

The paper applies these time and frequency solutions to investigate in detail the noise radiation from B evenly spaced rotating point forces, at a particular orientation, radius and rotational speed, whose source strength varies in time as a pulse of varying duration compared to the period of rotation. The solutions are explored and compared for the case where the pulses are emitted periodically with each rotation, and another in which a pulse is emitted only once. The former is representative of installed propeller blades when the blade loading varies periodically, such as in the presence of an upstream obstacle like a boom or another propeller. The latter case is a representation of the blade loading that occurs when a single gust passes over the propeller during a single rotation.

Objectives and scope of the paper

This paper has a number of key objectives that do not appear to have been discussed in previous work.

(1) To review the basic theory of sound radiation due to rotating point forces in both the time and frequency domains, within a consistent theoretical framework aimed at illustrating the differences between them and to highlight their relative advantages and disadvantages.

(2) To extend the early work of Lowson, ¹ and Morfey and Tanna ² to derive time domain expressions for multiple point forces with arbitrary orientation and force time dependence.

(3) To extend the formulations of propeller noise found in the earlier literature to include the effect of a radial component that is characteristic of highly swept blades.

(4) To illustrate the time and frequency domain solutions for a particular example of temporal force distribution that:

(a) Varies periodically on each blade as a signal that is of short duration compared to the period of rotation. Such situations arise in installed propellers in which the blade loading only varies significantly over a duration corresponding to the time over which it passes close to an adjacent propeller or another obstacle.

(b) Only occurs over a single rotation, as might occur when flow distortion passes over the propeller. This solution will be compared to the periodic case of (a).

This is the first paper to compare the frequency and time domain solutions and to demonstrate, by example, the equivalence between them. It is also the first to analyse in detail the role of time scale in propeller noise and to demonstrate the equivalence between periodic and aperiodic noise through a windowing function.

Background

The first analytical study of noise generated by rotating machinery was conducted by Gutin, ³ who correctly identified that, in most cases, the primary source of sound is the force exerted by rotor blades on the surrounding air. His method for predicting the far-field noise at each harmonic frequency, based on the earlier work of Lamb, ⁴ represents the forces acting on the blade surface as rotating point dipole sources. However, Gutin’s analysis was limited to steady forces such as for a propeller operating in a perfectly undisturbed flow. While the force is steady in the rotating reference frame, it appears to accelerate in the stationary frame and was shown by Gutin to be the primary mechanism of sound radiation in the isolated propeller case.

Subsequently, Lowson ¹ discovered that, when applying Lighthill’s ⁵ theory to a moving point source, no sound radiation was predicted for a steady force in motion, suggesting that the role of acceleration in sound generation was not recognised in this early work. To address this deficiency, Lowson extended Lighthill’s model to incorporate convective amplification and the influence of acceleration. A more general formulation of the noise from moving sources was later presented by Ffowcs Williams and Hawkings (FW-H), ⁶ who allowed for a distribution of both steady and unsteady sources. The development of propeller noise theory has proceeded in both time and frequency domains.

Frequency domain

Despite the fundamental governing equations for the radiation from moving sources being in the time domain, ⁶ frequency domain methods are widely used for performing analytical propeller noise calculations.^3,7 This may be due to the periodic nature of the driving forces and propeller rotation. The earliest formulation of propeller noise, presented in the frequency domain, which emphasises the effect of the unsteady blade forces, was also developed by Ffowcs Williams and Hawkings. ⁷ This formulation encapsulates all the principles of propeller loading noise generation, and is discussed in detail in texts by Goldstein, ⁸ Blake ⁹ and Glegg and Devenport. ¹⁰ These works provide an important foundation for subsequent research on frequency domain rotor noise prediction, although the fundamental underlying method remains the same.

More recent work has focused on predicting the force distribution over the rotor blades using classical blade response functions, such as the Sears function, ¹¹ or CFD approaches to infer the source distributions directly. Examples of the former approach, using flat plate theory, are by Xie et al. ¹² and Raposo et al., ¹³ who examine the noise from rotating flat propeller blades. The earlier work of Ffowcs Williams and Hawkings ⁷ restricted the analysis to noise generated by the lift and drag components of the rotating forces; later studies by Xie ¹² and Jiang ¹⁴ extended the formulation to include the radial force component, which may be significant for swept rotor blades.

Time domain

One of the first attempts at a comprehensive time domain representation of the noise due to rotating forces was by Morfey and Tanna, ² who extended the formulation derived by Lowson ¹ to include arbitrary random time force fluctuations. They also derived relationships between total sound power and the mean blade force variations, and scaling laws of the overall sound power with rotational Mach number. However, their analysis was limited to a single rotating point force and excluded the component of the blade forces directed radially from the propeller hub.

An important development in time domain formulations for propeller noise was by Farassat, ¹⁵ who, using the FW-H analogy, ⁶ derived an expression equivalent to Lowson for the far-field acoustic pressure. However, rather than a point force, Farassat allowed for a surface distribution of forces. With appropriate considerations, Farassat’s model is also capable of modelling the noise from a point force. The Farassat 1A formulation ¹⁶ has been implemented in NASA’s ANOPP2, ¹⁷ Penn State’s PSU-WOPWOP, ¹⁸ and numerous other computational aeroacoustic solvers. PSU-WOPWOP, in particular, utilises a time series of aerodynamic induced forces as input, making it especially compatible with the Farassat 1A approach. The time domain model of Farrasat also lends itself to modelling transient phenomena, as is the focus of Brentner’s work. ¹⁸ Other computational models may have adopted Farassat’s formulation for similar practical reasons.

Initial comparison of time and frequency domain approaches

The central difference between the formulations of the far-field acoustic pressure expressed in the frequency domain and that in the time domain is that, at a single frequency, the expression in the frequency domain is in the form of a summation of acoustic modes, each of which radiates with unique directivity patterns and efficiencies. Owing to the cylindrical symmetry of the problem, the modes are in the form of Bessel functions whose order is linked to the observer and source frequency. Thus, the frequency domain solution provides insight into the radiated field at each frequency, linked to the behaviour of its constituent acoustic modes.

By contrast, the time domain solution obscures the frequency dependence of the sound field, which must be revealed by application of the Fourier transform. Time domain approaches capture non-periodic and transient phenomena more naturally than frequency domain methods. ¹⁸ They are therefore better suited for capturing complex blade-vortex interactions, wake effects, and manoeuvring flight conditions. This literature review indicates that frequency domain formulations are generally preferred for analytical propeller noise models, whereas time domain approaches are more commonly adopted in computational methods.

Fundamental equations of sound generation from moving forces

Lighthill ⁵ rearranged the linearised Euler equations to obtain an acoustic wave equation whose right-hand side represents a source of acoustic pressure. In the case of a time-varying point force, the source term takes the form of the divergence of a force f ( x , t) = (f₁( x , t), f₂( x , t), f₃( x , t)) multiplied by a Dirac delta function δ( x − s (t)), where s (t) is the point source position at time t. Expressing this formulation in tensor notation, the acoustic wave equation may be written as,

$\frac{1}{c_0^2}\frac{{\partial }^{2}p(\mathit{x},t)}{\partial t^2}-\frac{{\partial }^{2}p(\mathit{x},t)}{\partial x_i^2}=-\frac{\partial }{\partial x_i}\left\{f_i(t)\delta (\mathit{x}-\mathit{s}(t))\right\},$

(1)

where p is the acoustic pressure at an observer location x = (x₁, x₂, x₃). c₀ is the speed of sound in the acoustic medium, t is the time at the observer location and τ is the time at the source location. A full nomenclature section is provided in Appendix A. The solution to equation (1) can be found through the convolution of the source term with the 3-D free space Green’s function,

$p(\mathit{x},t)=-{\int }_V{\int }_{-\infty }^{\infty }\frac{\partial }{\partial y_i}\left\{f_i(\tau )\delta (\mathit{y}-\mathit{s}(\tau ))\right\}\frac{\delta \left(t-\tau -|\mathit{x}-\mathit{y}|/c_0\right)}{4\pi |\mathit{x}-\mathit{y}|}d\tau d^3\mathit{y}.$

(2)

applying integration by parts, and exploiting that the Green’s function is reciprocal where, ∂G/∂x _i = −∂G/∂y _i ,

$p(\mathit{x},t)=-\frac{\partial }{\partial x_i}{\int }_V{\int }_{-\infty }^{\infty }{f}_i(\tau )\delta (\mathit{y}-\mathit{s}(\tau ))\frac{\delta \left(t-\tau -|\mathit{x}-\mathit{y}|/c_0\right)}{4\pi |\mathit{x}-\mathit{y}|}d\tau d^3\mathit{y}.$

(3)

Since the acoustic medium is at rest, the waves produced at the source time τ propagate spherically outward from the source location at the speed of sound c₀. At a fixed observer location x , the time t at which the sound arrives is found from addition of the source time τ and the propagation time r _s /c₀, where r _s = | x − s (τ)| is the distance between the source and observer positions. The volume integral over y can be determined using the sifting property of the Dirac delta function,

$p(\mathit{x},t)=-\frac{\partial }{\partial x_i}{\int }_{-\infty }^{\infty }{f}_i(\tau )\frac{\delta \left(t-\tau -\frac{r_s}{c_0}\right)}{4\pi r_s}d\tau ,$

(4)

where r _s = x − s . In order to evaluate the integral over t in equation (4) we use the following property of the Dirac delta function ¹⁹

${\int }_{-\infty }^{\infty }\delta (\mathfrak{b}(\tau ))\varphi (\tau )\mathrm{d}\tau =\frac{\varphi ({\tau }_0)}{|\partial \mathfrak{b}({\tau }_0)/\partial \tau |}$

where τ₀ is the only root of $\mathfrak{b}$ , hence $\mathfrak{b}({\tau }_0)=0$ , $\mathfrak{b}=t-\tau -r_s/c_0$ , and τ₀ = t − r _s /c₀. The denominator of equation (5) is given by,

$\frac{\partial \mathfrak{b}}{\partial \tau }=-1+\frac{1}{c_0}\frac{\partial \mathit{s}}{\partial \tau }\cdot \frac{\mathit{x}-\mathit{s}}{r_s}=-1+M_r,$

(5)

where M _r is the component of the source Mach number moving in the direction of the observer. Combining these results into equation (4) yields,

$p(\mathit{x},t)=-\frac{1}{4\pi }\frac{\partial }{\partial x_i}{\left[\frac{1}{r_s(\tau )}\frac{f_i(\tau )}{|1-M_r(\tau )|}\right]}_{\tau =t-r_s(\tau )/c_0}.$

(6)

where the term in square brackets is evaluated at the source time τ = t − r _s /c₀. Equation (6) for the acoustic pressure is equivalent to a particular form of the general expression presented by FW-H, ⁶ where a point force is in arbitrary motion. The divergence may be replaced by differentiation with respect to observer time t using the following approximation, ²⁰

$\frac{\partial }{\partial x_i}=\frac{-1}{c_0}\left(\frac{\partial }{\partial t}\right)\frac{r_{s,i}}{r_s(\tau )},$

(7)

which is derived in Appendix B. r_s,i/r _s is the ith component of the unit vector in the direction of the observer. The application of equation (7) to equation (6) yields the starting point for our frequency domain analysis,

$p(\mathit{x},t)=\frac{1}{4\pi c_0}\frac{\partial }{\partial t}{\left[\frac{1}{r_s}\frac{f_r(\tau )}{|1-M_r(\tau )|}\right]}_{\tau =t-r_s(\tau )/c_0},$

(8)

where f _r is the component of force in the direction of the observer at source time τ,

$f_r(\tau )=f_i(\tau )\frac{r_{s,i}(\tau )}{r_s}.$

(9)

As the variation of r _s with time is neglected, this result is confined to the acoustic far field. The differentiation of equation (8) with respect to observer time t may be performed using the chain and product rules,

$p(\mathit{x},t)=\frac{1}{4\pi c_0}{\left[\frac{1}{r_s}\left(\frac{\partial f_r(\tau )}{\partial t}\frac{1}{|1-M_r(\tau )|}+\frac{\partial M_r(\tau )}{\partial t}\frac{f_r(\tau )}{|1-M_r(\tau ){|}^2}\right)\right]}_{\tau =t-r_s/c_0}.$

(10)

A differentiation with respect to observer time t may be replaced by differentiation with respect to source time τ by noting the relationship between them and following the steps outlined in Appendix B, to give,

$\frac{\partial }{\partial t}=\frac{\partial }{\partial \tau }\frac{1}{1-M_r(\tau )},$

(11)

Employing this result in equation (10), yields equation (12) for the far-field acoustic pressure due to a moving point force, first derived by Lowson, ¹

$p(\mathit{x},t)={\left[\frac{1}{4\pi c_0|1-M_r(\tau ){|}^2}\frac{1}{r_s}\left(\frac{\partial f_r(\tau )}{\partial \tau }+\frac{\partial M_r(\tau )}{\partial \tau }\frac{f_r(\tau )}{|1-M_r(\tau )|}\right)\right]}_{\tau =t-r_s/c_0}.$

(12)

Equation (12) makes explicit the two potential contributions to acoustic radiation. The first arises from the unsteadiness of the force itself and is related to its rate of change with source time, ∂f _r (τ)/∂τ. The second term results directly from the acceleration of the force, ∂M _r (τ)/∂τ. This latter contribution accounts for sound radiation from steady force distributions in motion.

Rotating point force

We now apply the theoretical framework developed in the preceding section to the specific case of a rotating point force. This problem serves as an idealised representation of the sound produced by aerodynamically generated forces acting in the normal direction of the rotor blades’ surface. This simplified problem provides a basis for capturing the fundamental behaviour underlying rotor noise generation. More realistic loading distributions may be constructed through the superposition of multiple point forces.

Coordinate systems

The point force is assumed to rotate in the (y₂, y₃) plane, circling the origin at a constant radius R and angular speed Ω, as sketched in Figure 1. In a cylindrical coordinate system y = (r, θ, z), the source location at any source time τ is given by y (τ) = (R, θ _s (τ), 0). The source azimuthal angle θ _s relative to the y₂ direction can be expressed as θ _s (τ) = θ₀ − Ωτ, where θ₀ denotes the source angle at τ = 0.

Figure 1.

Source coordinates (red) and observer coordinates (Blue).

The force therefore rotates clockwise when viewed from the positive y₁ direction as the source angle θ _s (τ) decreases with increasing source time τ when Ω is positive.

The observer location is expressed in a spherical coordinate system x = (r, θ, β), where r is the distance from the origin, θ is the azimuthal angle measured from the y₂ direction and β is the polar angle from the y₁ direction. These coordinate systems are illustrated in Figure 1. We now confine the analysis to the far-field, r ≫ R, where we may use the Fraunhofer approximation to find the distance between the source and the observer positions at source time τ, r _s as,

$r_s(\tau )\approx r-R\sin \beta \cos (\theta -{\theta }_s(\tau )).$

(13)

where r _s (τ) = | x − y (τ)|.

Force orientation

The point force of magnitude f = | f | acting on the acoustic medium has three perpendicular components acting in the lift f₁, drag f _θ , and radial f _ϕ directions. The lift component f₁ is assumed to act on the fluid in the positive y₁ direction, the drag f _θ in the direction of the source movement, and a third force component f _ϕ acting radially outward from the origin. By geometry, these components are given by,

$\begin{array}{l}{f}_1=f\cos \gamma ,\end{array}$

(14a)

$\begin{array}{l}{f}_{\theta }=f\sin \gamma \cos \varphi ,\end{array}$

(14b)

$\begin{array}{l}{f}_{\varphi }=f\sin \gamma \sin \varphi ,\end{array}$

(14c)

where γ is the stagger angle of the force and ϕ is the sweep angle, which dictates the component of the force acting in the radial direction, as sketched in Figure 2.

Figure 2.

Force component orientations.

The stagger angle γ is the polar angle between the net force magnitude f and the y₁ direction. The angle ϕ is the azimuthal angle between the force direction and the direction of travel. By geometry, the components of r_s,i/| r _s | are given by,

$\begin{array}{l}{r}_1/r_s\left(\tau \right)=\cos \beta ,\end{array}$

(15a)

$\begin{array}{l}{r}_{\theta }\left(\tau \right)/r_s\left(\tau \right)=-\sin (\theta -{\theta }_s\left(\tau \right))\sin \beta ,\end{array}$

(15b)

$\begin{array}{l}{r}_{\varphi }\left(\tau \right)/r_s\left(\tau \right)=\cos (\theta -{\theta }_s\left(\tau \right))\sin \beta ,\end{array}$

(15c)

Frequency domain setup

This section provides an overview of the frequency domain formulation of the sound field due to rotating forces.^9,10 In this derivation, we extend the formulation to include a radial component of the force. Computation of the radiated far-field acoustic pressure using equation (8) requires the evaluation of the force in the direction of the observer f _r , which is obtained by taking the dot product of equations (14a)–(14c) and (15a)–(15c), to give,

$f_r(\tau )=f_1(\tau )\cos \beta -f_{\theta }(\tau )\sin (\theta -{\theta }_s(\tau ))\sin \beta +f_{\varphi }(\tau )\cos (\theta -{\theta }_s(\tau ))\sin \beta .$

(16)

Substituting this result into equation (8) gives a solution for the acoustic pressure of the form,

$p(\mathit{x},t)=\frac{\partial }{\partial t}{\left[\frac{1}{4\pi c_{0}r|1-M_r(\tau )|}(f_1\cos \beta -f_{\theta }\sin (\theta -{\theta }_s(\tau ))\sin \beta +f_{\varphi }\cos (\theta -{\theta }_s(\tau ))\sin \beta )\right]}_{\tau =t-r_s(\tau )/c_0}.$

(17)

The main difficulties in the evaluation of equation (17) for the time domain acoustic pressure are the computation of the retarded time and the differentiation by observer time. These difficulties may be circumvented by conducting the analysis in the frequency domain, which we summarise below.

Periodic noise

In the case where the acoustic pressure varies periodically over the propeller rotation period T = 2π/Ω, such as when the force is steady or varies periodically over T. the acoustic pressure p( x , t) at the observer position x may be expressed as the two sided Fourier series,

$p(\mathit{x},t)=\sum _{n=-\infty }^{\infty }{\widehat{p}}_n(\mathit{x})e^{-in\mathrm{\Omega }t},$

(18)

where ${\widehat{p}}_n(\mathit{x})$ is the complex pressure amplitude associated with the nth harmonic of Ω. Each pressure amplitude may be computed from the transform,

${\widehat{p}}_n(\mathit{x})=\frac{\mathrm{\Omega }}{2\pi }{\int }_0^{T}p(\mathit{x},t)e^{in\mathrm{\Omega }t}\mathrm{d}t.$

(19)

Applying this transform to equation (17), noting that t = τ + r _s /c₀, expanding r _s using the Fraunhofer approximation of equation (13), and applying the Jacobi Anger-expansion to the source time dependence,

$e^{in\mathrm{\Omega }t}=e^{in\mathrm{\Omega }(\tau +r/c_0)}\sum _{m=-\infty }^{\infty }{J}_m\left(\frac{n\mathrm{\Omega }}{c_0}R\sin \beta \right)e^{-im(\theta -{\theta }_0+\mathrm{\Omega }\tau +\pi /2)},$

(20)

where J _m (α) denotes a Bessel function of the first kind of order m and argument α, gives,

$\begin{array}{ll}\hfill {\widehat{p}}_n(\mathit{x})& =\frac{-in{\mathrm{\Omega }}^2}{2\pi }{\int }_0^T\sum _{k=-\infty }^{\infty }\sum _{m=-\infty }^{\infty }\frac{{\widehat{f}}_k{e}^{-ik\mathrm{\Omega }\tau }}{4\pi c_{0}r}\left[\cos \gamma \cos \beta -\sin \gamma \cos \varphi \sin (\theta -{\theta }_s)\sin \beta \right.\hfill \\ \hfill & \left.+\sin \gamma \sin \varphi \cos (\theta -{\theta }_s)\sin \beta \right]e^{in\mathrm{\Omega }(\tau +r/c_0)}{J}_m\left(\frac{n\mathrm{\Omega }}{c_0}R\sin \beta \right)e^{-im(\theta -{\theta }_0+\mathrm{\Omega }\tau +\pi /2)}\mathrm{d}\tau .\hfill \end{array}$

(21)

The term ${\widehat{f}}_k$ is the kth complex force amplitude of the force time series f(τ),

${\widehat{f}}_k=\frac{\mathrm{\Omega }}{2\pi }{\int }_0^{T}f(\tau )e^{ik\mathrm{\Omega }\tau }\mathrm{d}\tau ,f(\tau )=\sum _{k=-\infty }^{\infty }{\widehat{f}}_k{e}^{-ik\mathrm{\Omega }\tau },$

(22)

and we have used the results, ∂/∂t = −inΩ and dt = |1 − M _r (τ)|dτ. This first result is found from the definition of the Fourier harmonics in equation (18) and the second is derived in Appendix B. The integration over τ is readily performed using Kronecker delta function identities to give the final result for the complex amplitude of the sound pressure at frequency n nΩ,

${\widehat{p}}_n(\mathit{x})=-\frac{i{\kappa }_n}{4\pi r}{e}^{i{\kappa }_{n}r}\sum _{m=-\infty }^{\infty }{\widehat{f}}_{n-m}{D}_m(\beta ,{\kappa }_{n}R)e^{-im(\theta -{\theta }_0+\pi /2)},$

(23)

where the acoustic wavenumber κ _n = nΩ/c₀ and m is the acoustic mode order. This result is derived in detail from equation (21) in Appendix C. The acoustic mode order m is the result of the interaction between the nth harmonic frequency and the kth force harmonic, according to the scattering rule,

$m=n-k.$

(24)

D _m is the modal polar directivity function,

$D_m(\beta ,{\kappa }_{n}R)=\underset{D_1}{\underbrace{\cos \beta \cos \gamma J_m\left({\kappa }_{n}R\sin \beta \right)}}+\underset{D_{\theta }}{\underbrace{\frac{m}{{\kappa }_{n}R}\sin \gamma \cos \varphi J_m\left({\kappa }_{n}R\sin \beta \right)}}+\underset{D_{\varphi }}{\underbrace{i\sin \beta \sin \gamma \sin \varphi J_m^{\prime }\left({\kappa }_{n}R\sin \beta \right)}}.$

(25)

This directivity function comprises contributions from the lift D₁, drag D _θ , and radial D _ϕ force components as indicated above. $J_m^{\prime }(\alpha )$ denotes the differential of the Bessel function by its argument ∂J _m (α)/∂α.

Modal interpretation

Equation (23) for the radiated sound field due to a rotating point force with arbitrary orientation takes the form of a summation of acoustic modes of order m. Each of these modes has an amplitude determined by the product of the directivity function D with f _k , where k is due to the interaction between the acoustic mode order m and observer harmonic n, with k = m − n.

A point in the sound field of constant phase is described by,

$\eta (\theta ,r,t)=-m(\theta +{\theta }_0-\pi /2)+{\kappa }_{n}r-n\mathrm{\Omega }t={\eta }_0,$

(26)

which describes a spiral pattern that rotates at an angular velocity −nΩ/m corresponding to an azimuthal velocity of c _θ = −nΩR/m at radius R. A particular mode m radiates most efficiently if its azimuthal phase speed exceeds the speed of sound c _θ > c₀. Rewriting in terms of the source Mach number M _θ leads to the concise expression,

$|M_{\theta }|>\frac{m}{n}.$

(27)

The source Mach number must exceed this value in order to radiate efficiently at a given mode number and observer frequency nΩ.

Another interpretation of the modal efficiency can be drawn from the properties of the Bessel function J _m (κ _n R sin β). When m ≠ 0, this function tends to zero as κ _n R sin β < m. This condition suggests that higher-order modes m radiate less efficiently at the same frequency and observer angle than a mode of lower order. The axi-symmetric mode m = 0 therefore radiates with greatest efficiency, implying that for the axial force, the force harmonics k radiate with the greatest efficiency at the observer harmonic n = k.

The drag component of the acoustic directivity is proportional to the product of the mode order m and the Bessel function J _m (α). As a result, there is no sound radiation in the zeroth mode, and the most efficient radiation occurs in the first-order modes m = ±1. Sound generated by drag forces therefore does not arise from force components at the same frequency as the radiated noise, but rather from force components at multiples of Ω greater or less than the radiated frequency.

We note from equation (25) that the pressure contribution due to the radial component of force radiates 90° out of phase relative to the lift and drag components due to the factor of i. Also of note is that the derivative of the Bessel function satisfies $J_0^{\prime }(0)=0$ , which implies the most efficient modes of the radial force contributions also occur in the modes m = ±1.

Extension to multiple blades

The theory presented above for a single rotating force may be extended to include multiple forces which represent the forces on a propeller consisting of multiple blades. In this case, the force on adjacent blades may be assumed to be identical but shifted in phase such that the phase on the bth blade (b = 0, 1, 2, …, B − 1) is offset by 2πb/B. Equation (18) for acoustic pressure due to a single rotating force, therefore generalises to multiple forces by summation over B forces with the appropriate phase factors,

$p(\mathit{x},t)=\sum _{b=0}^{B-1}\sum _{n=-\infty }^{\infty }{\widehat{p}}_n(\mathit{x})e^{-in(\mathrm{\Omega }t-2\pi b/B)}.$

(28)

This expression is zero unless n = lB, where l is any integer (1, 2, 3 …). Substituting n = lB in the above, the summation over b gives the factor B and hence the expression for the acoustic pressure for B point forces can be expressed as,

$p(\mathit{x},t)=B\sum _{l=-\infty }^{\infty }{\widehat{p}}_{lB}(\mathit{x})e^{-ilB\mathrm{\Omega }t}.$

(29)

and hence, tones are only radiated at multiples of the blade passing frequency BΩ and not the shaft rotation frequency.

Aperiodic noise

We now consider the case when the blade forcing does not repeat on each propeller rotation but instead only occurs on a single rotation. Each blade experiences identical forces but offset in time by bT/B, such that the force on the bth blade is f(τ − bT/B). Each blade encounters the same force pulse as each in turn passes the same angular position. In this paper, we refer to this case as aperiodic. This situation is representative of propeller noise due to a gust passing over each blade. This idealised scenario is chosen to allow direct comparison with the periodic case analysed above. In the aperiodic case, the Fourier series representation of equation (18) for the case of periodic noise must now be replaced by the Fourier transform,

$\widehat{p}(\mathit{x},\omega )=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }p(\mathit{x},t)e^{i\omega t}\mathrm{d}t,$

(30)

where the pressure time series may be recovered from its inverse transform,

$p(\mathit{x},t)={\int }_{-\infty }^{\infty }\widehat{p}(\mathit{x},\omega )e^{-i\omega t}\mathrm{d}\omega ,$

(31)

where ω is the observer angular frequency. Taking the Fourier Transform of equation (17), where the force components are no longer assumed to vary periodically, gives an identical expression to equation (23) for the far-field pressure, but with a modal directivity given by D _m (β, κR). D is defined in equation (25) with the important difference that D is now evaluated over a continuum of frequencies κR, where κ = ω/c₀. In this aperiodic noise case, the frequency of the radiated noise ω is related to the source frequency ζ by the scattering relation,

$\omega =\zeta +m\mathrm{\Omega }$

(32)

where m is the mode order m = (…, −2, −1, 0, 1, 2, …). Following an identical argument to the periodic noise case above, the expression for the far-field pressure due to B blades is therefore of the form,

$\widehat{p}(\mathit{x},\omega )=\frac{-i\omega }{4\pi r{c}_0}{e}^{i\omega r/c0}\sum _{b=0}^{B-1}{e}^{-i\frac{\omega }{\mathrm{\Omega }}(2\pi b/B)}\sum _{m=-\infty }^{\infty }\widehat{f}(\omega -m\mathrm{\Omega })D_m(\beta ,\kappa R)e^{-im(\theta -{\theta }_0+\pi /2)}.$

(33)

where the complex harmonics of the blade force are defined by an analogous Fourier transform,

$\widehat{f}(\zeta )=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }f(\tau )e^{i\zeta \tau }\mathrm{d}\tau .$

(34)

The summation over the blades b, which is in the form of a geometric progression, is therefore performed,

$\sum _{b=0}^{B-1}{e}^{-i\frac{\omega }{\mathrm{\Omega }}(2\pi b/B)}=\frac{1-e^{-i\frac{\omega }{\mathrm{\Omega }}2\pi }}{1-e^{-i\frac{\omega }{\mathrm{\Omega }}\frac{2\pi }{B}}}=\frac{\sin \left(\frac{\omega \pi }{\mathrm{\Omega }}\right)}{\sin \left(\frac{\omega \pi }{\mathrm{\Omega }B}\right)}{e}^{i\frac{\omega \pi }{\mathrm{\Omega }B}\left(1-B\right)}$

(35)

to give the final expression for the pressure frequency spectrum of the form,

$\widehat{p}(\mathit{x},\omega )=\frac{-i\omega }{4\pi r{c}_0}{e}^{i\omega r/c0}\frac{\sin \left(\frac{\omega \pi }{\mathrm{\Omega }}\right)}{\sin \left(\frac{\omega \pi }{\mathrm{\Omega }B}\right)}{e}^{i\frac{\omega \pi }{\mathrm{\Omega }B}\left(1-B\right)}\sum _{m=-\infty }^{\infty }\widehat{f}(\omega -m\mathrm{\Omega })D_m(\beta ,\kappa R)e^{-im(\theta -{\theta }_0+\pi /2)},$

(36)

The pressure frequency spectrum is the result of the product of three factors. One is related to the source spectrum $\widehat{f}(\zeta )$ , which is Doppler shifted according to the mode order m. The second is due to the frequency characteristics of the modal directivity function D _m (β, κR), which is an oscillatory function of frequency as determined by the Bessel functions in equation (25) for D. The third factor arises from the ratio of sine terms, which is identically zero at the harmonics of the shaft rotation frequency ω = nΩ, n ≠ lB, where n and l are any integer, and takes maximum value of B at harmonics of the blade passing frequency ω = lBΩ.

Time domain

The previous section presented the main steps leading to the final expressions in the frequency domain for the far-field acoustic pressure due to a rotating point force with arbitrary orientation. This expression allows for the computation of the acoustic pressure at each frequency individually. In this section, we present the corresponding derivation for the acoustic pressure, but remaining in the time domain to allow the computation of the time variation of the acoustic pressure. The two formulations are related by the Fourier transformation, which we later demonstrate in the results section.

The starting point of this section is the result of Lowson, ¹ as presented in equation (12), where the square brackets indicate quantities evaluated at the source time τ = t − r _s /c₀. This expression is the sum of two terms: the first specifying the contribution due to the rate of change of the force, and the second describing the radiation due to source acceleration. Central to the evaluation of the acoustic pressure in equation (12), is the Mach number component M _r in the direction of the observer. For the rotating force considered in this paper, M is only non-zero in the azimuthal direction θ, and hence from equation (15b), M _r is given by,

$M_r(\tau )=M_{\theta }\frac{r_{s,\theta }(\tau )}{|{\mathit{r}}_s(\tau )|}=-M\sin \beta \sin (\theta -{\theta }_s(\tau )).$

(37)

The acceleration Mach number appearing in equation (12), in the direction of the observer, is therefore,

$\frac{\partial M_r(\tau )}{d\tau }=-\mathrm{\Omega }M\sin \beta \cos (\phi (\tau )).$

(38)

where φ(τ) is the separation angle between the blade and the observer azimuthal angle, defined as φ(τ) = θ − θ _s (τ). Given that θ _s = θ₀ − Ωτ, φ(τ) can also be written as φ(τ) = θ − θ₀ + Ωτ. To maintain clarity, the dependence on τ is omitted. Also appearing in the time domain expression of equation (12) is ∂f _r /dτ, which following equation (16) is given by,

$\frac{\partial f_r}{d\tau }=\frac{\partial f}{\partial \tau }\cos \beta +\sin \beta \left(\frac{-\partial f_{\theta }}{\partial \tau }\sin \phi -\mathrm{\Omega }{f}_{\theta }\cos \phi +\frac{\partial f_{\varphi }}{\partial \tau }\cos \phi -\mathrm{\Omega }{f}_{\varphi }\sin \phi \right)$

(39)

Substituting the individual force and Mach number components specified above gives,

$\begin{array}{ll}\hfill p(\mathit{x},t)& =\frac{1}{4\pi c_{0}r}\left[\frac{{\dot{F}}_1-\dot{F_{\theta }}\sin \phi -\mathrm{\Omega }{F}_{\theta }\cos \phi +{\dot{F}}_{\varphi }\cos \phi -\mathrm{\Omega }{F}_{\varphi }\sin \phi }{{(1+\alpha \sin \phi )}^2}\right.\hfill \\ \hfill & {\left.-\frac{\mathrm{\Omega }\alpha \cos \phi (F_1-F_{\theta }\sin \phi +F_{\varphi }\cos \phi )}{{(1+\alpha \sin \phi )}^3}\right]}_{\tau =t-r_s(\tau )/c_0}\hfill \end{array}$

(40)

where F₁ = f₁ cos β, F _θ = f _θ  sin β, F _ϕ = f _ϕ sin β and α = M sin β. The overdot denotes differentiation with respect to source time τ. Following equation (13), the source time τ and receiver time t are related by,

$t=\tau +\frac{r_s(\tau )}{c_0}\approx \tau +\frac{r-R\sin \beta \cos (\theta -{\theta }_0+\mathrm{\Omega }\tau )}{c_0}.$

(41)

Upon expanding the terms, equation (40) simplifies further to give our final result for the time domain radiated far-field acoustic pressure for a single rotating point force,

$p(\mathit{x},t)=\frac{1}{4\pi c_{0}r}{\left[\frac{\dot{F_1}-\dot{F_{\theta }}\sin \phi +\dot{F_{\varphi }}\cos \phi }{{(1+\alpha \sin \phi )}^2}-\frac{\mathrm{\Omega }(F_1\alpha \cos \phi +F_{\theta }\cos \phi +F_{\varphi }(\alpha +\sin \phi ))}{{(1+\alpha \sin \phi )}^3}\right]}_{\tau =t-r_s(\tau )c_0}$

(42)

The first term in this expression accounts for sound radiation due to the time dependence of the source and is directly related to its rate of change with source time. The second term accounts for sound radiation due to the movement of the steady force and is proportional to its rotational speed Ω. This result represents an extended formulation of the model originally presented by Morfey and Tanna, ² now incorporating the contribution from the radial force component f _ϕ . Equation (42) is completely general and therefore applies equally to both periodic and aperiodic excitations. Equation (42) thus serves as the final expression for the acoustic pressure generated by a single rotating point force of arbitrary orientation.

Since equation (41) is a transcendental equation, there is no closed-form solution to calculate the retarded time τ from a given observer time t. It is therefore more practical to specify the source time τ and compute the corresponding (advanced) observer time t. Equation (42) may be extended to multiple blades by summing the pressure contributions from each blade, but noting that the azimuthal location of the bth blade at τ=0 is given by θ₀ = bπ/ΩB. The corresponding observer time for each blade may be computed using equation (41). We note that care is required in the implementation of this time domain approach since the pressure time series from each blade must usually be interpolated to ensure matching observer times from all blades. Further discussion of this procedure is provided in Section 4.2.

Comparison between time and frequency domain solutions for a Gaussian pulse

In order to demonstrate the equivalence between the time and frequency domain solutions, and to illustrate the behaviour of the radiated noise for transient force variation, it is first necessary to prescribe a force distribution to represent the blade loading.

Force distribution

In this paper, the force applied to the acoustic medium by each blade is assumed to be characterised by a Gaussian pulse with maximum amplitude f₀ and a pulse width of σ, of the form,

$f(\tau )=f_0{e}^{-\frac{1}{2}{\left(\frac{\tau }{\sigma }\right)}^2},$

(43)

where σ is the time at which the pulse is e^−1/2 ≈ 0.6 of its maximum amplitude at τ = 0. This force distribution is chosen to represent a transient force variation that is amenable to both time and frequency analyses. The time domain formulation of equation (42) makes explicit that the radiated acoustic pressure depends on the time rate of change of the force, and hence,

$\frac{\partial f(\tau )}{\partial \tau }=-f_0\frac{\tau }{{\sigma }^2}{e}^{-\frac{1}{2}{\left(\frac{\tau }{\sigma }\right)}^2}.$

(44)

Fourier Transforms

The frequency domain formulation of the radiated pressure requires the frequency spectrum of the force time variation, f(τ). For the aperiodic noise case considered above, in which the force is produced only once on each blade, the Fourier transform equation (34) is applied to the Gaussian distribution equation (43) to find,

$\widehat{f}(\zeta )=f_0\frac{\sigma }{\sqrt{2\pi }}{e}^{-\frac{1}{2}{(\sigma \zeta )}^2},$

(45)

For the case where the transient force is produced by each blade on every rotation, the force f(τ) is periodic and can therefore be represented as a Fourier series with force amplitudes ${\widehat{f}}_k$ and a time period equal to the source rotation time T, from equation (22),

${\widehat{f}}_k=f_0\frac{\sigma \mathrm{\Omega }}{\sqrt{2\pi }}{e}^{-\frac{1}{2}{(\sigma k\mathrm{\Omega })}^2}.$

(46)

In the evaluation of equation (46), the force amplitudes derived from the Fourier Transform of equation (22), we have assumed that the pulse width σ is much shorter than the propeller rotational period T. In practice, some overlapping of the pulses from consecutive rotations may occur, leading to possible error. In contrast, the aperiodic case is valid for any pulse width σ, since the time bounds for the integration extend to τ = ±∞.

Acoustic pressure

Frequency domain (aperiodic)

The final expression for the aperiodic frequency domain formulation of the acoustic pressure to B identical rotating Gaussian point forces is obtained by substituting the force coefficients from equation (45) into equation (36).

$\widehat{p}(\mathit{x},\omega )=-f_0\frac{\sigma }{\sqrt{2\pi }}\frac{i\omega }{4\pi r{c}_0}{e}^{i\omega r/c0}\frac{\sin \left(\frac{\omega \pi }{\mathrm{\Omega }}\right)}{\sin \left(\frac{\omega \pi }{\mathrm{\Omega }B}\right)}{e}^{i\frac{\omega \pi }{\mathrm{\Omega }B}\left(1-B\right)}\sum _{m=-\infty }^{\infty }{e}^{-\frac{1}{2}{(\sigma (\omega -m\mathrm{\Omega }))}^2}{D}_m(\beta ,\kappa R)e^{-im(\theta -{\theta }_0+\pi /2)},$

(47)

To illustrate the general characteristics of the pressure spectrum, Figure 3 shows the mean square pressure spectrum $|\widehat{p}(\mathit{x},\omega ){|}^2$ as a function of frequency ω, evaluated at the location shown in Table 1. The spectral structure is characterised by distinct nulls at integer multiples of the shaft rotational frequency Ω due to perfect cancellation of the noise at these frequencies.

Figure 3.

Pressure amplitude squared $|\widehat{p}(\mathit{x},\omega ){|}^2$ against frequency.

Table 1.

Quantities used in the generation of Figures 3 and 4.

Ω	120π rads−1	σ	0.01 s
θ	π/2 rad	β	π/3 rad
R	0.5 m	c 0	343 ms−1
γ	π/4 rad	ϕ	π/4 rad
r	1 m	B	3

The spectrum is non-zero at ω = 3Ω since B = 3, which reflects the coherent summation of the blade contributions at this frequency. This spectral characterisation provides limited insight into the underlying sound generation mechanisms. Further understanding of the noise characteristics and their sound generation mechanisms may be found by analysis in the time domain.

Time domain

The time domain formulation is easier to perform in the aperiodic case. However, for periodic sources, a comprehensive time domain analysis requires accounting for all source times that could contribute to the pressure at the observer location. While this is straightforward in the present case, it may prove less convenient in more complex or general configurations.

The corresponding time variation of the acoustic pressure in the aperiodic case may be obtained by combining equation (42) with equations (43) and (44) and summing over all blades, to give the final expression of equation (48) for the time variation of acoustic pressure,

$\begin{array}{ll}\hfill p(\mathit{x},t)& =\frac{-f_0}{4\pi c_{0}r}\sum _{b=0}^{B-1}\left[e^{-\frac{1}{2}{\left(\frac{\tau +bT/B}{\sigma }\right)}^2}\frac{(\tau +bT/B)(\cos \beta \cos \gamma +\sin \beta \sin \gamma (-\cos \varphi \sin \phi +\sin \varphi \cos \phi ))}{{\sigma }^2{(1+\alpha \sin \phi )}^2}\right.+\hfill \\ \hfill & {\left.\frac{\mathrm{\Omega }\left(\alpha \cos \gamma \cos \beta \cos \phi +\sin \gamma \sin \beta (\cos \varphi \cos \phi +\sin \varphi (\alpha +\sin \phi ))\right.}{{(1+\alpha \sin \phi )}^3}\right]}_{\tau =t-r_s/c_0}\hfill \end{array}$

(48)

where r _s = R sin β cos(θ − 2πb/Ω + Ωτ). The time and frequency domain formulations for the acoustic pressure due to the rotating forces may now be directly compared.

Figure 4 shows the acoustic pressure at the observer position. In the left-hand plot, the red lines represent the sound experienced at the observer but produced from each point force at source time τ; here, the x-axis corresponds to the source time. The corresponding variation of pressure with observer time, t is shown in black. In this case, the x-axis represents the observer time.

Figure 4.

Acoustic pressure against time normalised by rotation period.

The horizontal displacement between the red and black curves corresponds to the time shift defined by equation (41), which depends on the blade position and hence on t and τ. Although the overall observer time signal would normally be delayed to the right, the two results are aligned here for comparison by shifting the observer time trace to the left by the mean propagation time, r/c₀.

In the right-hand plot, the pressure contributions from each blade, shown previously as black lines, are then summed to yield the total acoustic pressure variation as a function of the observer time. This result is then compared against the inverse Fourier transform of the aperiodic frequency domain prediction plotted in Figure 3, where perfect agreement is observed.

The difference in plotting against source versus observer times becomes more pronounced as the propeller rotational speed and radius increase. Additionally, at high blade counts or for extended pulse durations, significant cancellation effects occur between the blades. Since the pulse duration is greater than the rotational time T, the periodic result is not applicable in this case, and it is not shown in this subsection.

Comparison of the periodic and aperiodic solutions

In this section, we compare the time variation of the acoustic pressure for two cases: when a pulse is generated on each blade only on one rotation (aperiodic) and when it is repeated on every rotation (periodic) to make clear the significant differences between them.

Frequency domain (Periodic)

To determine the discrete pressure spectrum in equation (29), for the periodic case, we substitute n = lB and incorporate the expression for the force amplitudes f _k from equation (46) into equation (23) yielding the final expression,

${\widehat{p}}_{lB}(\mathit{x})=-f_0\frac{\sigma \mathrm{\Omega }}{\sqrt{2\pi }}\frac{i{\kappa }_{lB}}{4\pi r}{e}^{i{\kappa }_{lB}r}\sum _{m=-\infty }^{\infty }{e}^{-\frac{1}{2}{(\sigma (lB-m)\mathrm{\Omega })}^2}{D}_m(\beta ,{\kappa }_{lB}R)e^{-im(\theta -{\theta }_0+\pi /2)},$

(49)

By applying the Fourier series given in equation (29) to the complex pressure amplitudes obtained in equation (49), the time history of the acoustic pressure can be reconstructed. Figure 5(a) illustrates the influence of inter-blade interaction on the radiated acoustic pressure. The parameters used in the generation of Figure 5 are again taken from Table 1, with the exception that the pulse width has been reduced to σ = 0.002 to maintain the accuracy of equation (46). In the periodic case, a substantial degree of cancellation occurs due to overlapping contributions from successive blades. In the aperiodic case, similar cancellation is observed near t/T = −0.25, leading to close agreement between the two cases in this time frame. However, at the beginning and end of the pulse, when only the first and last blades are active, there are no concurrent contributions from other blades to cancel the radiated sound. As a result, these regions exhibit higher acoustic pressures.

Figure 5.

Acoustic Pressures at σ = 0.002 and σ = 0.0005.

In Figure 5(b), a smaller value of σ = 0.0005 is now chosen to isolate the acoustic contributions of each blade. This narrow pulse width reduces inter-blade interactions, resulting in perfect agreement between the periodic and aperiodic cases at the times when the pulses occur.

These results clearly demonstrate that, when observing the time interval between τ = 0 and −T, the aperiodic pulse consistently produces acoustic pressure that is greater than or equal to, the periodic case.

Sound fields

The observations made in Section 4.3.1 are further supported by contours of acoustic pressure. In this section, we compare the acoustic pressure fields generated by 8 rotating point dipoles in the plane of rotation (corresponding to β = π/2), where any observed acoustic pressure must result solely from the drag or radial components of the force. The snapshot is taken at an observer time such that all the blades have passed the pulse location and a small amount of propagation time has elapsed.

The left-hand plot of Figure 6 illustrates the acoustic pressure field for the aperiodic case, while the right-hand panel displays the corresponding periodic case. A consistent colour scale is used across both plots to facilitate direct comparison of pressure magnitudes. Due to the high blade count, the periodic case exhibits significant inter-blade cancellation, leading to lower overall pressure levels. In contrast, the aperiodic case shows noticeably higher pressure amplitudes at the beginning and end of the pulse series, where such cancellation is absent. θ = 0° is located on the left-hand side where y₂ = 0. All quantities are consistent with Table 1.

Figure 6.

Acoustic pressure fields at B = 8 σ = 0.002.

Window interpretation

It is worthwhile to highlight some additional relationships between the periodic and aperiodic results. Notably, the time domain signal of the aperiodic acoustic pressure can be interpreted as the convolution of the periodic pulse with a rectangular window function of length T. If we define the window function as a rectangle function of length T,

$\text{rect}(\alpha )=\left\{\begin{array}{c}\hfill 1\text{if}|\alpha |<T/2\hfill \\ \hfill 0\text{if}|\alpha |>T/2\hfill \end{array}\right..$

(50)

Then the product of the windowing function with the periodic acoustic pressure signal results in an aperiodic pressure signal,

$p_{(a)}(t)=p_{(p)}(t)\cdot \text{rect}(t).$

(51)

Here we introduce the subscripts (a) and (p) to denote whether the quantities are aperiodic or periodic, respectively. The convolution theorem states that the Fourier transform of the product of two time domain functions is equal to the convolution of their individual Fourier transforms. To express the periodic sound in the frequency domain, we define it using a Fourier series,

$p_{(p)}(\mathit{x},t)=\sum _{n=-\infty }^{\infty }{\widehat{p}}_n(\mathit{x})e^{-in\mathrm{\Omega }t}.$

(52)

Then, by applying the definition of the Fourier transform, we obtain,

$\widehat{p}(\mathit{x},\omega )=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }\sum _{n=-\infty }^{\infty }{\widehat{p}}_n(\mathit{x})e^{it(\omega -n\mathrm{\Omega })}\mathrm{d}t.$

(53)

In the second part of the equation, performing the integral over t leads to a Dirac delta function, 2πδ(ω − nΩ). We then obtain,

$\widehat{p}(\mathit{x},\omega )=\sum _{n=-\infty }^{\infty }{\widehat{p}}_n(\mathit{x})\delta (\omega -n\mathrm{\Omega }),$

(54)

as the Fourier transform of a periodic signal. The Fourier transform of the function rect(α) is well established,

$\frac{\sin (\omega T/2)}{\pi \omega }=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }\text{rect}(t)e^{i\omega t}\mathrm{d}t,$

(55)

We then find the Fourier transform of the aperiodic signal as the convolution of these two results,

${\widehat{p}}_{(a)}(\omega ,\mathit{x})={\int }_{-\infty }^{\infty }\sum _{n=-\infty }^{\infty }\frac{\sin ((\omega -\alpha )T/2)}{(\omega -\alpha )\pi }{\widehat{p}}_n(\mathit{x})\delta (\omega -n\mathrm{\Omega })\mathrm{d}\alpha ,$

(56)

${\widehat{p}}_{(a)}(\omega ,\mathit{x})=\sum _{n=-\infty }^{\infty }{\widehat{p}}_n(\mathit{x})\frac{\sin ((\omega -n\mathrm{\Omega })T/2)}{(\omega -n\mathrm{\Omega })\pi }$

(57)

This method is valid only when the acoustic pressures generated by each blade are temporally separable. If the force pulses are sufficiently long, they may extend beyond the boundaries of the window defined by the rect function, resulting in clipping effects.

All the other parameters used in the generation of Figure 7 are consistent with Table 1. Even with this short pulse time, there is an amount of clipping at the ends of the force pulse. This artificial truncation leads to the exaggerated high-frequency response in the calculated spectrum of the windowed case. An advantage of the windowed approach is that it can be much quicker to calculate than the standard aperiodic approach, as fewer Bessel functions need to be calculated. A small disagreement is also observed at low frequencies, possibly due to the fact that the lowest harmonic captured by the Fourier series occurs at a frequency of Ω. To improve the method, a windowing function could be applied directly to the force on each blade rather than to the radiated sound. This approach would more accurately represent the aperiodic case being modelled, though its implementation would introduce additional complexity.

Figure 7.

Windowed versus Aperiodic case, B = 4, σ = 0.0005.

Conclusions

This study provides a comprehensive comparison of time domain and frequency domain approaches for calculating the sound pressure generated by rotating point forces, with applications to installed subsonic rotors. The analytical models derived in this work account for axial, drag, and radial force components. We demonstrate by example the equivalence of the time and frequency domain solutions, which has not been explicitly demonstrated in previous work. However, each approach offers different insights into the structure and nature of the radiated sound.

The radial force component, often neglected in previous studies, is shown to be a potential sound source, particularly in propellers with high rake or skew. Time domain methods proved computationally advantageous, especially for the aperiodic case, while frequency domain methods are more approachable for the periodic case and for modal analysis.

Advantages of each method

The time domain formulation naturally separates the contributions of unsteady and quasi-steady source terms, allowing more straightforward derivation of scaling laws with respect to the pulse duration σ and rotor speed. This could also be applied to calculate sound power scaling laws, individual to each component of the force. These scaling laws are less apparent in the frequency domain, where such scaling dependencies are obscured by requiring infinite summations to be performed over modes. When operating in the time domain, it is clear that for long pulse durations, the contributions from each of the blades are offset in time, potentially leading to significant inter-blade cancellation.

A possible disadvantage of the time domain approach is the requirement to determine the source–observer time relationship. Due to the nature of this relationship, a computational approach is required. In this work, we have used an interpolation-based method.

In the frequency domain, the use of a Fourier series inherently accounts for force contributions over an infinite sequence of periodic intervals, thereby simplifying the modelling of steady forces and periodic forces. By contrast, time domain analysis requires a sufficiently long time window to fully capture these periodic effects. This limitation can, however, be partially mitigated by exploiting the periodic nature of the source–observer time relationship, provided that a sufficiently long time window is kept.

Additionally, the analysis of directivity is less straightforward compared to the frequency domain. While the solution for the pressure at a single observer time may be trivial, practical applications would consider a range of observer times to obtain meaningful information. This limitation could potentially be addressed through appropriate temporal averaging or through a Fourier transform to compare the directivity of certain frequencies.

The frequency domain offers its own advantages in other areas, such as the decomposition of acoustic radiation into frequency-dependent modes, which can yield valuable insights into the spatial structure of the radiated sound. This modal representation also facilitates integration with duct acoustics models, where mode-based analysis is common.

Computational considerations

Once the retarded times have been determined, the time domain formulation is generally more computationally efficient, especially in the aperiodic case, as it avoids the need to evaluate infinite summations involving Bessel functions. In the frequency domain, careful attention is required to ensure that all significant modes are included without overextending the summation range.

The frequency domain is more efficient at analysing periodic noise than the aperiodic case. The periodic noise is limited to harmonics of the blade passing frequency. Although there is still an infinite summation of Bessel functions, the sum is limited to discrete frequencies. In contrast, the continuous spectrum associated with the aperiodic case is significantly more computationally demanding to evaluate.