1 Introduction
There has been an increasing body of work on the use of high-blade-number asymptotic approaches to study propeller noise and, perhaps more importantly, understand the physics underlying the source generation mechanisms. The initial work focused on single-rotating propellers. The far-field radiation can be calculated, numerically, from the analytic ‘frequency-domain’ solutions of Hanson (Reference Hanson1983) and Parry (Reference Parry1988). However, these methods involve, for each individual tone, a double integration over the blade planform area (chord and span) in which the integrand includes Bessel functions and complex phase terms. Using a frequency-domain approach, Parry & Crighton (Reference Parry and Crighton1986, Reference Parry and Crighton1989a) showed that subsonic propeller noise could be predicted using a simplified formula that removed the need for an integration over the blade span and showed how the noise is dominated by radiation from the blade tips, with the result linked precisely to the tip geometry. Crighton & Parry (Reference Crighton and Parry1991, Reference Crighton and Parry1992) applied the approach to supersonic propellers where the noise is dominated to leading order by radiation from a sonic radius with a lower-order term arising from the blade tips. For both the subsonic and supersonic cases, the asymptotic approach produced closed-form algebraic results in the form of simple algebraic expressions, which were, nonetheless, still accurate and showed how the noise was related to the geometry and aerodynamics of a particular propeller radius.
In a novel time-domain analysis of the thickness noise of supersonic single-rotating propellers, Amiet (Reference Amiet1988) showed that the sound radiated to the far field was dominated by regions of the blade that satisfied two criteria. First, that the Mach number of the blade at that location was precisely unity when resolved in the direction of the observer and, second, that the leading edge of the propeller blade was normal to a line drawn between it and the observer. We will refer to these criteria as the ‘sonic’ and ‘normal-edge’ conditions, respectively. Amiet argued that a time-domain analysis, rather than a frequency-domain approach, was essential to properly understand the underlying physics and to obtain the dominant source locations. Parry (Reference Parry1995) showed that Amiet's criteria could be obtained in the frequency domain. For other work on asymptotic analyses of single-rotating propeller noise, we refer the reader to the discussions in Kingan & Parry (Reference Kingan and Parry2019) and Parry & Kingan (Reference Parry and Kingan2019).
There has been little work on the application of asymptotic analyses to contra-rotating propellers. From the numerical point of view, Envia (Reference Envia2015) extended previous work on single-rotating propellers to the contra-rotating case, with the radiation expressions, as before, requiring a high-fidelity calculation of the local source region. For analytical and/or semi-numerical approaches, Kingan & Parry (Reference Kingan and Parry2019) used the framework provided by the frequency-domain description of Hanson (Reference Hanson1985), Parry (Reference Parry1988, Reference Parry1997) and Parry & Crighton (Reference Parry and Crighton1989b), regarding convected wake interactions, and extended the two-dimensional surface asymptotic approach of Parry (Reference Parry1995) to contra-rotating propellers, including the effects of blade sweep, quasi-three-dimensional blade unsteady response and sub- or supercritical wake–blade interactions. The analyses once again produced simplified, largely algebraic, results and showed that the noise is dominated in most cases by localised critical points on the surface of revolution described by the rear blade leading edges. In certain cases – dependent on blade sweep and the order of the circumferential spinning mode – radial or azimuthal one-dimensional continuums of critical points can be present or, exceptionally, a two-dimensional continuum of critical points.
Kingan & Parry (Reference Kingan and Parry2019) validated their asymptotic approach extensively by comparing their results with full numerical calculations – of a real engineering test case – over a range of combination frequencies and azimuthal modes, including the zero-mode case, and for both straight and swept propeller blades. For all cases, there was good agreement between the asymptotic results and the numerical calculations (that had, themselves, been validated via many previous comparisons with a wide range of experimental data), showing that the asymptotics could be applied to prediction and design (see Kingan & Parry, Reference Kingan and Parry2020a).
Parry & Kingan (Reference Parry and Kingan2019) explored the underlying physics and showed that, in the frequency domain, the locations of the interaction sources for each combination tone can be described by an ‘event line’ – where the phrase ‘event’ was taken from the paper of Tyler & Sofrin (Reference Tyler and Sofrin1962) on rotor–stator interaction – which represents the source locations at a fixed reception time and spins around the annulus at the speed of the acoustic mode (associated with the combination tone). Parry & Kingan (Reference Parry and Kingan2019) explained that, at the critical source locations, this event line has a resolved Mach number of unity in the direction of the far-field observer and is normal to a line drawn between it and the observer, i.e. the event line (for each individual combination tone) satisfies the two criteria of Amiet (Reference Amiet1988). Indeed, perhaps surprisingly, Parry & Kingan (Reference Parry and Kingan2019) showed that the two criteria are still satisfied even when the relevant acoustic mode number is zero and the event line spins around the annulus at infinite speed.
Regarding analytical work on contra-rotating propeller noise in the time domain, the authors are aware only of the work of Kingan & Parry (Reference Kingan and Parry2020b), who considered the interactions between convected wakes from the front propeller and a blade of the rear contra-rotating row. Kingan & Parry (Reference Kingan and Parry2020b) hypothesised that the sharp peaks in the radiated acoustic waveform would be significantly enhanced when the reception time, for a particular wake–blade interaction, became stationary in terms of radius. They showed that, at such a radius, the local trace velocity was precisely sonic when resolved in the direction of the observer.
Here, our aim is to understand the physics of convected wake interactions on contra-rotating propellers in the time domain, rather than just for individual frequencies. In particular, we focus on the location and motion of the interaction ‘events’ as a function of time and space, both radially and circumferentially. We emphasise that the analytical framework for contra-rotating propeller noise that underpins the analysis – namely that of Hanson (Reference Hanson1985), Parry (Reference Parry1988, Reference Parry1997) and Parry & Crighton (Reference Parry and Crighton1989b) – has been well validated against a range of experimental data. Nonetheless, from the point of view of steady and unsteady aerodynamics, it is important to point out that the general approach neglects a number of effects such as the presence of swirling or radial flow between the two blade rows, the mutual interference that occurs between the vortical and potential fields generated by the front and rear blade rows, the end effects (at the hub and tip) that contribute to the unsteady response, and the thickness and camber of the blades. Most of these effects could be included through extensions to the approach. Despite these limitations, there are detailed discussions of the experimental validation cases in Kingan & Parry (Reference Kingan and Parry2019) as well as of validation of the asymptotic approach via comparisons with full numerical calculations. To demonstrate the accuracy of the asymptotic (or many-bladed) approach, one of the comparisons from that paper is shown here, in figure 1, for the first interaction tone produced by a contra-rotating propeller with front and rear blade numbers of 12 and 9, respectively.
Figure 1. Plot of sound pressure level (SPL) versus 
$\theta$ for the first interaction tone. The solid black line denotes numerical solutions whilst interior stationary point solutions are denoted by the blue dash-dotted lines and boundary critical point solutions are denoted by the red dotted line. The shaded regions correspond to locations where no critical points exist, and the grey dashed lines denote interpolated levels within the exclusion zones. (From Kingan & Parry, Reference Kingan and Parry2019.).
The paper is laid out as follows. In § 2 we present the three-dimensional coordinate system and the aerodynamic and acoustic parameters. The locations of the convected wake interactions are discussed in § 3 in terms of the ‘firing order’ as a function of time and circumferential location; examples are discussed for various blade number combinations and architectures; then, we analyse the variation in interaction location with radius. In § 4 the interactions are discussed relative to a far-field observer.
2 Coordinate system
In order to allow comparison with the results of Kingan & Parry (Reference Kingan and Parry2019, Reference Kingan and Parry2020a,Reference Kingan and Parryb) and Parry & Kingan (Reference Parry and Kingan2019) we will use their notation where possible, though we will define the parameters here explicitly because of some differences between the frequency-domain and time-domain publications. The architecture involves two rotating frames, and one absolute frame, as well as important reference points such as the front rotor wake, the rear blade leading edge and the two propeller pitch change axes, and it will be necessary to use all of these frames and locations. The analysis uses a cylindrical coordinate system, 
$(x,r,\phi )$, where 
$x$ is the axial coordinate, which is parallel to the propeller axis, 
$r$ is the radial coordinate and 
$\phi$ is the azimuthal angle. The coordinate system is shown in figure 2. Subscripts 1 and 2 denote parameters associated with the upstream and downstream rotors, respectively, and subscripts 
$x$, 
$t$ and 
$r$ are used for the Mach numbers in the axial, rotational (at the tip radius) and helical (at radius 
$r$) directions, respectively. In particular, the rotors move with Mach number 
$M_x$ in the positive 
$x$ direction; the upstream and downstream rotors rotate in the negative and positive 
$\phi$ directions with tip rotational Mach numbers 
$M_{t_1}$ and 
$M_{t_2}$, respectively.
Figure 2. Propeller coordinate system.
The effect of the flow induced by the rotors is neglected (which is entirely in keeping with the helicoidal surface theory of Hanson (Reference Hanson1983)) so that the stagger angle, 
$\alpha _i$, 
$i=1,2$, of each blade is defined by 
$\tan \alpha _i = r M_{t_i}/M_x$ and, for each row, 
$M_{r_i} = \sqrt {M_x^2+r^2 M_{t_i}^2}$ is the section relative, or helicoidal, Mach number. The propeller blades have leading-edge sweep 
$s(r)$ measured from the pitch change axis and each propeller has 
$B_i$ blades with front and rear pitch change axes separated by a distance 
$g$ in the axial direction. The observer lies in the 
$\phi = 0$ plane at polar angle 
$\theta$ from the 
$x$ axis. For convenience, distances have been normalised on the tip radius 
$R$, velocities on the speed of sound 
$c_0$ and time on 
$R/c_0$. The hub radius is 
$r_h$. For a contra-rotating propeller, the radiated tones can be described as a double Fourier series of combination tones 
$(n_1,n_2)$ at frequencies 
$\eta = n_1 B_1 M_{t_1} + n_2 B_2 M_{t_2}$ and with azimuthal order 
$\nu = n_2 B_2 - n_1 B_1$. Source time is denoted by 
$\tau$ and the specific times at which the wake centreline of the front reference blade interacts with the rear reference blade are given by 
$\tau _0$ where, in the general case of swept blades, 
$\tau _0=\tau _0 (r)$. We suppose that the pitch change axis of the rear row reference blade lies at 
$\phi =0$ when 
$\tau =0$. It is also convenient to introduce the term 
$S(r)=s(r)/M_{r_2}(r)$.
3 Physics of the interaction in the time domain
3.1 Precession of the interaction
In this section we consider the interactions between specific blades on the rear rows and the wakes from specific blades of the front row. Here, an interaction occurs when the centreline of the wake from an upstream blade impinges on the leading edge of a downstream blade. In reality, of course, since the wake and blade leading edge are of finite thickness, this interaction occurs over a finite time. Tyler & Sofrin (Reference Tyler and Sofrin1962) refer to these interactions as blade–vane events or, more simply, just as ‘events’. In the general case, the wakes from the front propeller do not all interact simultaneously with the rear propeller blades for the simple reason that the number of wakes, 
$B_1$, is not the same as the number of blades, 
$B_2$, on the rear row. In the time domain, the interactions can be viewed as occurring at locations that ‘jump’ around the circumference, sometimes by considerable distances. For rotor–stator interactions, Tyler & Sofrin (Reference Tyler and Sofrin1962) referred to this effect as the ‘firing order’. Here we will suppose that this sequence of events represents the precession of the interaction around the circumference and consider it in detail.
In order to determine the speed at which the individual interactions precess around the circumference, we start by assuming that, in figure 3, the blades are numbered in the positive 
$\phi$ direction on each row and that the interaction between blade 
$0$ on the front row and blade 
$0$ on the rear row occurs at source time 
$\tau = 0$. For the purposes of the analysis that follows, we assume that the blade numbers 
$B_1$ and 
$B_2$ are relatively prime. If 
$B_1$ and 
$B_2$ have a common factor then there will be more than one interaction occurring simultaneously and the analysis applies over an appropriate subsection of the annulus, with the results then repeating over each subsection. Of course, the blades chosen as the zeroth blades on each row are arbitrary. The pitch is given by 
$P_1 = 2{\rm \pi} r/B_1$ and 
$P_2=2{\rm \pi} r/B_2$ on the front and rear row, respectively, so that the azimuthal gap between blade 
$j_1$ on the front row and blade 
$j_2$ on the rear row is
\begin{equation} \Delta \phi_{j_1,j_2} = j_1 \frac{P_1}{r} - j_2 \frac{P_2}{r} = 2 {\rm \pi}\frac{\left(\,j_1 B_2 - j_2 B_1\right)}{B_1 B_2}. \end{equation}
Figure 3. Two-dimensional cascade definition.
As, with our normalisation, the relative angular speed is 
$(M_{t_1}+M_{t_2})$, the time it takes from the interaction between the datum blades to that of the interaction between blade 
$j_1$ on the front row and blade 
$j_2$ on the rear row is
\begin{equation} \Delta \tau_{j_1,j_2} = \frac{\Delta \phi_{j_1,j_2}}{(M_{t_1}+M_{t_2})} = \frac{2 {\rm \pi}(\,j_1 B_2 - j_2 B_1)}{B_1 B_2 (M_{t_1}+M_{t_2})}. \end{equation} In that time, the interaction has moved from 
$\phi _2=0$ to 
$\phi _2 = 2 {\rm \pi}j_2/B_2$, where 
$\phi _2=\phi -M_{t_2}\tau$ is the azimuthal angle in the frame of reference of the rear row, so we can deduce that the effective speed with which the interaction has moved in the absolute frame is 
$M_{j_1,j_2} = M_{t_2} + (2 {\rm \pi}j_2 / B_2 \Delta \tau _{j_1,j_2} )$. Then, after using (3.2), we obtain the effective interaction speed and interaction location as
\begin{equation} M_{j_1,j_2} = \frac{j_2 B_1 M_{t_1} + j_1 B_2 M_{t_2}}{j_1 B_2 - j_2 B_1}, \quad \phi_{j_1,j_2} =\frac{2 {\rm \pi}(\,j_2 B_1 M_{t_1} + j_1 B_2 M_{t_2})}{B_1 B_2 (M_{t_1}+M_{t_2})}. \end{equation} The result for interaction speed in (3.3a,b) is very familiar and, suitably normalised, looks like the standard relationship for circumferential mode-phase speed 
$\eta /\nu$ in turbomachinery (see e.g. Tyler & Sofrin, Reference Tyler and Sofrin1962) in terms of an angular frequency 
$\eta =n_1 B_1 M_{t_1}+n_2 B_2 M_{t_2}$ and mode number 
$\nu =n_2 B_2 - n_1 B_1$. However, the standard relationship applies to the frequency domain and to a range of different combination tones 
$(n_1,n_2 )$ each of which has its own frequency and azimuthal order so that every one rotates at a different speed. Here it is important to realise that (3.3a,b) is a time-domain result and that 
$(\,j_1,j_2)$ is used to refer to interactions between specific blades with the fixed values 
$j_1$ and 
$j_2$ being determined by the blade numbers. In order to determine which interactions occur in which order, it follows from (3.1) and (3.2) that, after the 
$0$–
$0$ front and rear blades interact, the next interaction occurs a short time 
$\Delta \tau$ later where
\begin{equation} \Delta \tau = \frac{2 {\rm \pi}}{B_1 B_2 (M_{t_1} + M_{t_2})} \times \min_{\begin{array}{c} {\scriptstyle 0 \leq\, j_1 \leq B_1-1} \\ {\scriptstyle 0 \leq\, j_2 \leq B_2-1} \\ {\scriptstyle j_1 B_2 - j_2 B_1 > 0} \end{array}} \{\,j_1 B_2 - j_2 B_1\} , \end{equation}
so that 
$\Delta \tau$ is the time taken between successive wake interactions (and it is important to note that those interactions do not necessarily occur between adjacent blades). The values 
$(\,j_1,j_2)$ that are obtained as the solution to (3.4) are related to the blade numbers 
$B_1$ and 
$B_2$ and can be treated as fixed for that blade number combination.
By considering the blade row speeds and blade spacings, and using a little simple arithmetic, it is easy to show that the next interaction will occur between blades 
$(2j_1,2j_2)$ at azimuthal location 
$\phi _{2j_1,2j_2} =4 {\rm \pi}(\,j_2 B_1 M_{t_1} + j_1 B_2 M_{t_2})/[B_1 B_2 (M_{t_1}+M_{t_2})]$, and so on, so that the effective speed at which the interaction location moves – or precesses – is continuous around the annulus. If we suppose that 
$l$ and 
$m$ are the values of 
$j_1$ and 
$j_2$ that are required in (3.4), then the speed (or tip Mach number) at which the interaction events precess around the annulus is
\begin{equation} M_p = \frac{m B_1 M_{t_1} + l B_2 M_{t_2}}{l B_2 - m B_1}. \end{equation} In practice, the interactions are discrete but the idea that the interactions are moving continuously becomes more acceptable for higher blade numbers. Indeed, the assumption of a continuum of interactions is completely consistent with the many-bladed asymptotic approach, used in the frequency-domain analyses of Kingan & Parry (Reference Kingan and Parry2019, Reference Kingan and Parry2020a) and Parry & Kingan (Reference Parry and Kingan2019), in which the blade numbers 
$B_1 \rightarrow \infty$, 
$B_2 \rightarrow \infty$ because then the time interval between interactions 
$\Delta \tau \rightarrow 0$.
3.2 Examples
We will consider a few simple examples to illustrate the values of 
$l$ and 
$m$. Note first, though, that the integers 
$l$ and 
$m$ can be negative, where negative values are obtained if the blades are counted in the opposite (or negative 
$\phi$) direction, provided that we still maintain 
$lB_2-mB_1>0$ to ensure that we are moving forwards in time. Of course, the interactions occur in the same ‘firing order’ in both forward and reverse cases; the point is that the interaction disturbances can merely be viewed as moving in either the positive or negative direction.
First, we examine the contra-rotating propeller studied by Kingan & Parry (Reference Kingan and Parry2020b) in their time-domain analysis. The propeller had 10 blades on the front row and a single blade on the rear row, and tip rotational Mach numbers 
$M_{t_1}=M_{t_2}=0.7$. That case is straightforward, as all the interactions (events) must occur at the leading edge of the same rear blade so that 
$m=0$; the interactions must also be caused by the wakes from successive blades on the front row so that 
$l=1$ and 
$M_p=M_{t_2}$. We emphasise that the interactions are not continuous; from (3.4) they occur discretely at source time intervals of 
$\Delta \tau = 2{\rm \pi} /B_1(M_{t_1} +M_{t_2})={\rm \pi} /7$, given as the period 
$T$ by Kingan & Parry (Reference Kingan and Parry2020b). In order to demonstrate the effect of the source we show, in figure 4, a full numerical calculation for this contra-rotating propeller, in the time domain, with the pressure per unit span, 
$P$, radiated from the mid-span (
$r=0.7$) of the blade. Figure 4 has been recalculated and is plotted here in terms of (leading-edge) source time 
$\tau$ rather than reception time 
$t$ (as it was in figure 5(a) of Kingan & Parry (Reference Kingan and Parry2020b)). The source period is normalised on 
$T$ so that pulses occur at multiples of 
$\tau /T=1$, precisely as expected. The calculation also serves to demonstrate the sharpness of the interaction impulses in a practical wake–blade interaction case. For this configuration, the interactions can be considered to be rotating with the rear blade leading edge and at the speed of the rear blade row.
Figure 4. The pressure field radiated from 
$r=0.7$ for a straight propeller, plotted in terms of leading-edge source time 
$\tau\!$.
Second, we consider the examples used by Kingan & Parry (Reference Kingan and Parry2019) in demonstrating the accuracy of their asymptotic, frequency-domain analysis of contra-rotating propellers. Whilst their example architectures had different degrees of sweep, they all had 12 blades on the front row and 9 blades on the rear row. In a fixed reference frame, the front blades are thus 
$30^{\circ }$ apart and the rear blades are 
$40^{\circ }$ apart. Assuming that the zeroth front and rear blades are aligned at 
$\tau =0$ then, taking into account the direction of motion of the two blade rows, a little arithmetic can be used to show that the next interaction will occur between the third blade of the front row and the second blade of the rear row. (Of course, since for this case there is a common factor of 3 in the blade numbers, the interactions repeat spatially every 
$120^{\circ }$.) The relevant values are thus 
$l=3$ and 
$m=2$, and we obtain the precession speed of 
$M_p = 8 M_{t_1}+9M_{t_2}$ (which is much faster than that of the case with a single-bladed rear row) and the time interval of 
$\Delta \tau = 2 {\rm \pi}/36(M_{t_1}+M_{t_2})$ between interaction events. By counting blades in the opposite direction, an alternative solution is also obtained with 
$l=-1$ and 
$m=-1$ for which the interactions precess in the reverse direction at 
$M_p = -(4 M_{t_1}+3M_{t_2})$, which is less than half that of the precession speed in the forward (or positive 
$\phi$) direction. The interval between interaction events is the same as that of the forward propagating sequence of interactions.
Purely for completeness, we will also consider a third case relating to a rotor–stator interaction. For such architectures, the stator numbers are usually selected to be more than twice the number of rotors in order to ensure the first interaction tone cannot propagate. We thus pick the numbers 
$B_1=18$ and 
$B_2=40$. In a fixed reference frame, the rotors are thus 
$20^{\circ }$ apart and the stators are 
$9^{\circ }$ apart. Once again, we assume that the zeroth front and rear blades are aligned at 
$\tau =0$. The next interaction is then easily shown to occur between the fifth blade of the front row and the eleventh blade of the rear row. (Of course, since for this case there is a common factor of 2 in the blade numbers, the interactions repeat spatially every 
$180^{\circ }$.) The relevant values are thus 
$l=5$ and 
$m=11$, and we obtain the precession speed of 
$M_p = 220M_{t_1}$ and the time interval of 
$\Delta \tau = 2 {\rm \pi}/360 M_{t_1}$ between interaction events. Alternatively, we can use 
$l=-4$ and 
$m=-9$ and obtain 
$M_p = -81M_{t_1}$ and the same value of 
$\Delta \tau\!$.
3.3 Interaction location as a function of radius
The analysis of Kingan & Parry (Reference Kingan and Parry2020b) showed that the front and rear reference blades, on a contra-rotating propeller, interact at source times that vary across the blade span due to the effect of blade sweep. Their derivation (given as (2.11) in Kingan & Parry (Reference Kingan and Parry2020b)) is straightforward, as it is based solely on blade geometry and speed. With our normalised parameters and definition of 
$\tau =0$, the expression simplifies and the reference blade interaction source times are 
$\tau _0(r)=S(r)$, where both 
$\tau _0$ and 
$S$ are zero at the hub: 
$r=r_h$.
The precession speed is given by (3.5), so that the azimuthal location at which the interactions take place can be expressed as 
$\phi _{ev}=M_{p}(\tau - \tau _0 )$, where we have used the subscript ‘
$ev$’ to reflect the fact that the wake–blade interactions are referred to by Tyler & Sofrin (Reference Tyler and Sofrin1962) as ‘events’. We will require the radial derivative of 
$\phi _{ev}$ in the following section and we get
\begin{equation} \frac{\textrm{d}\phi_{ev}}{\textrm{d}r} = -M_p S'(r). \end{equation}4 Interactions as viewed by the observer
For the observer-based analysis, we follow much of the approach given in § 4 of Parry & Kingan (Reference Parry and Kingan2019) except, crucially, for the fact that the Mach number of the interaction disturbance, or event, is dependent not on the front and rear propeller blade Fourier harmonics 
$(n_1,n_2)$ but on a precession speed that is fixed for a given propeller architecture and is given here by (3.5). As the propeller is advancing with Mach number 
$M_x$, and the disturbance rotates (or precesses) around the annulus with Mach number 
$M_p$, the combined Mach number 
$M_{obs}$ of an ‘event’ at radius 
$r$ towards an observer in the far field at polar angle 
$\theta$ can be obtained, using simple geometry, as 
$M_{obs}=M_x \cos \theta - \mathbb {M}$, where 
$\mathbb {M} = r M_p \sin \theta \sin \phi _{ev}$ is the resolved component of the rotating disturbance's precession speed towards the observer. The combined speed of the disturbance, in the direction of the observer, is thus sonic when 
$\mathbb {M} = -(1-M_x \cos \theta )$ or
\begin{equation} \sin \phi_{ev} = -\frac{(1-M_x \cos \theta )}{r M_p \sin \theta}. \end{equation}The location of an interaction event, at the leading edge of a propeller blade, is given by
\begin{equation} x_{ev} = -M_x S + M_x\tau, \quad y_{ev} = r \cos \phi_{ev}, \end{equation}
where 
$x$ and 
$y$ are Cartesian coordinates in the horizontal plane in figure 2. For an observer in the far field, the dot product of the tangent vector to the event line with the position vector of the event location, relative to the observer, is given by
\begin{equation} \ell = \frac{\textrm{d}x_{ev}}{\textrm{d}r} \cos \theta + \frac{\textrm{d}y_{ev}}{\textrm{d}r} \sin \theta , \end{equation}
and we find, after differentiating (4.2a,b) with respect to the radius 
$r$, and also using (3.6) and (4.1), that 
$\ell =0$, and the event line is normal to the line between it and the far-field observer, when
\begin{equation} \frac{\sin \theta \cos \phi_{ev}}{\textrm{d}S/\textrm{d}r} = 1. \end{equation} In their time-domain analysis, Kingan & Parry (Reference Kingan and Parry2020b) (see (4.6) and (4.7) and the accompanying discussion therein) showed that the term on the left-hand side of (4.4) represented the Mach number of the wake centreline/leading-edge impingement point (or trace Mach number along the reference blade's leading edge) resolved in the direction of the observer. (They used a fixed coordinate system so their Mach number was defined relative to the fluid. That is not the case here.) For a swept blade, the Kingan & Parry (Reference Kingan and Parry2020b) analysis showed, further, that the condition (4.4) – representing a trace Mach number of unity towards the observer – occurred when the reception times were stationary with respect to the radius 
$r$ and, as a consequence, it coincided with an enhanced accumulation of noise from across the blade span and to strong peaks in the resultant noise waveform received by the observer. In the time domain, therefore, (4.4) can be viewed as a condition, or requirement, for enhanced peaks in the radiated sound field.
We have thus shown that this enhanced noise radiation in the time domain corresponds, not just to a wake/blade leading-edge trace Mach number of unity, resolved in the direction of the observer, but also to the criteria of Amiet (Reference Amiet1988), which relate to the shape and motion of the disturbance event line that precesses around the circumference. The first of these criteria is the sonic condition, in which the disturbance moves at a Mach number of unity in the direction of the observer. Note that this Mach number relates to that of the event line and not of the impingement point along the reference blade's leading edge. The second criterion is the normal-edge condition, in which the tangent to the event line is normal to a line drawn between it and the observer.
For the case in which the two blade rows have equal numbers (
$B_1=B_2$), the interactions occur simultaneously at all the rear blade leading-edge locations. To understand this case, we must proceed with care and consider the limit as the two blade numbers approach each other. Formally, at each radial station, we consider the interactions between two cascades translating in opposite directions with speeds 
$M_{t_1}$ and 
$M_{t_2}$. This approach is completely in keeping with that taken in the strip-theory model used for aerodynamic interactions by Parry & Crighton (Reference Parry and Crighton1989b), Parry (Reference Parry1997) and Kingan & Parry (Reference Kingan and Parry2019, Reference Kingan and Parry2020a,Reference Kingan and Parryb). Specifically, we suppose that the cascade pitches are 
$P_1 = P + \delta P$, 
$P_2 = P$ and 
$P=O(B^{-1})$, with 
$\delta P = O(B^{-2})$, 
$B \rightarrow \infty$. (Here we have specified the magnitude of 
$\delta P$ for definiteness but, in fact, we merely require 
$\delta P = o (B^{-1} )$.) Then, we find that 
$l=m=1$ and, as might be expected, the precession speed 
$M_p=(M_{t_1}+M_{t_2})P/\delta P \rightarrow \infty$ as 
$\delta P \rightarrow 0$ (or 
$B \rightarrow \infty$). Despite the fact that the precession speed is infinite, we also find from (4.1) that, provided the observer is not positioned on the propeller axis (i.e. 
$\theta \neq 0$), the sonic condition is still satisfied when 
$\phi _{ev} =0, {\rm \pi}$ (and the event line passes through the 
$x$–
$y$ plane) and, from (4.4), that the normal-edge condition is then satisfied at observer locations 
$\theta = \pm \sin ^{-1}\textrm {d}S/\textrm {d}r$. When the observer is positioned on the propeller axis (
$\theta = 0$), 
$\phi _{ev}$ is indeterminate and we thus find that the two criteria are satisfied at all azimuthal locations provided that 
$\textrm {d}S/\textrm {d}r=0$. For swept propellers, the enhanced noise thus comes from locations distributed around a ring (or rings), at a radius (or radii) satisfying 
$\textrm {d}S/\textrm {d}r=0$ and, for straight propellers (for which 
$\textrm {d}S/\textrm {d}r=0$ everywhere), it comes from the entire nominal propeller disk. These time-domain results echo completely those found in the frequency domain by Parry & Kingan (Reference Parry and Kingan2019) for the zero mode case.
5 Conclusions
This paper has discussed the location of convected-wake interactions on a contra-rotating propeller in real time. In the general case, these interactions occur at discrete, and equal, time intervals at different locations around the circumference. However, as the blade numbers tend to infinity, the process tends to a continuum of interactions. In this limit, we showed how the interaction location precesses around the circumference at a speed different from that of the front or rear blades and given by a combination of the two blade numbers and the two tip speeds. The relevance of the calculation has been demonstrated by application to recent frequency- and time-domain work on contra-rotating propellers and also to a rotor–stator type configuration. We have exploited previous time-domain work on the radial variation in the location of the wake interactions on contra-rotating propellers to produce a complete (radial and azimuthal) space–time expression for the interaction location. The radial variation in this location can be considered as a time-domain version of an event line. An analysis from the viewpoint of a far-field observer showed that the sonic and normal-edge conditions are satisfied by the time-domain event line at locations in the nominal propeller disk corresponding precisely to those found in the far field, by Kingan & Parry (Reference Kingan and Parry2020b), to have stationary reception times relative to the radius 
$r$ and, thereby, to produce enhanced peaks in the radiated waveform.
The paper thus shows that, for a contra-rotating propeller in the time domain, as well as the frequency domain, the dominant noise generation regions are given, once again, by the sonic and normal-edge criteria. Those criteria were first discovered by Amiet (Reference Amiet1988) in a time-domain analysis of the thickness noise from a single-rotating propeller. There the criteria applied along the line defined by the propeller's leading edge. We have shown the criteria apply equally well to wake interactions on a contra-rotating propeller – provided that the source line can be suitably defined. That line is the time-domain event line which spins around the annulus at a speed given by the precession speed of the interactions.
Throughout, our aim has been to ensure that the time-domain and frequency-domain analyses are completely consistent and that the physics underlying the noise generation are fully understood, particularly in relation to the dominant source locations.
Declaration of interests
The authors report no conflict of interest.
