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\DOI{10.5802/crmeca.312}
\datereceived{2025-05-25}
\daterevised{2025-06-30}
\dateaccepted{2025-06-30}
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\dateposted{2025-07-18}
\begin{document}

\begin{noXML}

\CDRsetmeta{articletype}{research-article}

\title{Doppler effect considered as a distortion of time, application
to an engineering case}

\alttitle{L'effet Doppler consid\'{e}r\'{e} comme une distorsion du
temps, application \`{a} un cas d'ing\'{e}nierie}

\author{\firstname{Jean} \lastname{Varnier}}
\address{French Aeronautical and Astronautical Association, 3AF, 75016 Paris, France}
\address{National Office for Aerospace Studies and Research, ONERA, 92320 Ch\^{a}tillon, France} 
\email{varnierjean948@gmail.com}

\keywords{\kwd{Aeroacoustics}
\kwd{Doppler effect}
\kwd{Time approach}
\kwd{Sound power spectrum}
\kwd{Sonic boom} 
\kwd{Atmospheric propagation}}

\altkeywords{\kwd{A\'{e}roacoustique}
\kwd{Effet Doppler}
\kwd{Approche temporelle}
\kwd{Spectre de puissance acoustique}
\kwd{Bang sonique}
\kwd{Propagation atmosph\'{e}rique}}

\begin{abstract} 
The ground-based listening of aircraft, rockets, or space launchers,
and the simulation of recorded acoustic spectra with a view to civilian
or military applications, raise the problem of the distortion by the
Doppler effect of the spectrum emitted by the sound sources, in
addition to other problems related to the sound propagation in
atmosphere. We propose a solution to the direct  problem---the
characteristics of the sound source and its trajectory are 
known---based on an original time approach to the Doppler effect. This
study shows in fact that it is not necessary to distinguish between
frequency and sound level variations, both being directly related to
the ratio of the emission to the reception durations of a chosen signal
sequence. The developed method has a character of generality and can be
applied without restriction in most cases (e.g. variable speed of the
mobile, curved trajectory, atmosphere with wind and temperature
gradients) using an acoustic propagation code, but remains obviously
applicable in an isotropic environment to simple or complex cases, for
example when the observer is moving.
\end{abstract}

\begin{altabstract} 
L'\'{e}coute au sol d'avions, de fus\'{e}es ou de lanceurs spatiaux,
ainsi que la simulation de spectres acoustiques enregistr\'{e}s en vue
d'applications civiles ou militaires, soul\`{e}vent le probl\`{e}me de
la distorsion par effet Doppler du spectre \'{e}mis par les sources
sonores, en plus d'autres probl\`{e}mes li\'{e}s \`{a} la propagation
du son dans l'atmosph\`{e}re. Nous proposons une solution au
probl\`{e}me direct -- les caract\'{e}ristiques de la source sonore et
sa trajectoire \'{e}tant connues -- bas\'{e}e sur une approche
temporelle originale de l'effet Doppler. Cette \'{e}tude montre en
effet qu'il n'est pas n\'{e}cessaire de distinguer les variations de
fr\'{e}quence et de niveau sonore, toutes deux \'{e}tant directement
li\'{e}es au rapport des dur\'{e}es d'\'{e}mission et de r\'{e}ception
d'une s\'{e}quence de signaux choisie. La m\'{e}thode
d\'{e}velopp\'{e}e pr\'{e}sente un caract\`{e}re g\'{e}n\'{e}ral et
peut \^{e}tre appliqu\'{e}e sans restriction dans la plupart des cas
(par exemple, vitesse variable du mobile, trajectoire courbe,
atmosph\`{e}re avec gradients de vent et de temp\'{e}rature) \`{a}
l'aide d'un code de propagation acoustique, mais est \'{e}videmment
applicable en environnement isotrope \`{a} des cas simples ou
complexes, par exemple lorsque l'observateur est en mouvement.
\end{altabstract}

\maketitle

\end{noXML}

\section{Introduction}\label{sec1}
In the field of aeroacoustics, and more particularly in ground
listening of aircraft or launch vehicles in flight, we are generally
confronted with the following phenomena:
\begin{itemize}
\item
the non-linearity of the sound source's trajectory;
\item
the variation of its speed over time;
\item
the anisotropic nature of the atmospheric propagation.
\end{itemize}

It is known, in particular, that the frequency spectrum emitted by the
source is modified by the Doppler effect, and that the same is true for
the sound level received at the listening point. The effects of
atmospheric absorption also affect frequency and amplitude, but remain
negligible at the frequencies of interest and at the distances and
altitudes involved in the example discussed here.

In the 2000s, ONERA was led to address this issue as part of the Ariane
5 program of CNES and during studies on the noise environment of
Military Aviation. It turned out that the classical geometric acoustics
approach was not suited to the problem conditions, and that NASA's
engineering approaches used complex and more or less empirical
formalisms that were difficult to transpose to the experimental cases
studied.

For these reasons, we were finally led to develop an original method,
which was the subject of a first communication at the European Congress
of Acoustics in  2008~\cite{1}, and which is developed here.

The purpose of this account is to describe this theoretical approach
and the experimental case on which it was tested with a view to its
validation.

\section{State of art and aim of the study}\label{sec2} 
The Doppler effect of moving acoustic sources, first discovered in 1842
by Christian Doppler, and more precisely developed in 1847 in the
classical form of a frequency shift of a signal emitted by a moving
harmonic sound source and perceived by a motionless  observer~\cite{2},
appears in various forms in many fields. From a theoretical point of
view, the geometric approach to the Doppler effect is described for
example in  Ref.~\cite{3}, the relativistic approach, valid in
astronomical optics, in  Ref.~\cite{4}. A time approach based on
successive pulses emitted by the sound source is sometimes used, for
educational purposes, to introduce the Doppler effect in the case where
the source and the observer are located on the same  axis~\cite{5}. In
the continuation, our discussion concerns an application developed in
an operational context in the domain of aeronautics.

In the field of aeroacoustics, the study of the Doppler effect is most
often applied to the case of subsonic motion, the supersonic case
generating certain particularities---duplication of the Doppler
frequency already pointed out  in~\cite{6}, requiring some theoretical
complements. In particular, it should be noted that despite a classical
approach found in the literature for a long  time~\cite{7,8}, the
harmonic sound sources and the Doppler effect are not responsible of
the sonic boom, except in the purely theoretical case of a mobile
reduced to a sound source of volume zero. In reality, it is the nose of
the supersonic body which creates the shock front, so called Mach cone,
responsible of the sonic boom, a phenomenon which is clearly not
concerned by the Doppler  effect~\cite{9}. 

In the 1960's, the literature mainly provided semi-empirical approaches
applied to listening aircraft or rockets in flight, with a view to
predict the spectrum and sound levels from jet noise received off the 
ground~\cite{10,11}. These models are generally accompanied by
simplifying hypotheses (rectilinear trajectory, constant speed of the
sound source, isotropic atmosphere) and applied to subsonic cases.

Otherwise, it is remarkable to note that while all publications agree
on the calculation of frequencies shifted by the Doppler factor (see 
Section~\ref{sec3}), there is a great disparity concerning the
influence of the Doppler effect on sound levels. For example, in 
Ref.~\cite{12} we find a ``convective amplification factor'' where the
Doppler factor is raised to the power 4, this exponent being reduced to
1.5 in another  study~\cite{13}. Only~\cite{11} recommends, without
further explanation, using the inverse of the Doppler factor to correct
the sound level calculated at the reception point. As for the recent 
study~\cite{14}, it takes into account spatial decrease of acoustic
intensity, atmospheric attenuation and Doppler-shifted frequencies, but
does not consider or address a Doppler correction of the calculated
sound levels, which suggests either that it is forgotten or that it is
assumed to be negligible compared to other parameters.

In fact, even in a fundamental work such as~\cite{3}, the formula
relating to the variation in sound level due to the Doppler effect is
particularly complex and given without any explanation. It is also
remarkable that no educational work on the Doppler effect, to our
knowledge, mentions this question other than by attributing it to the
variation in distance. Note that it is not easy to find studies in
recent publications that include theoretical considerations on the
Doppler effect.

In a first step, the objective of this study was to attempt to lighten
the apparent disparities and uncertainties mentioned above and to free
ourselves from the constraints that impose simplifying assumptions
concerning the medium of sound propagation. In this perspective, we
were led to imagine a time-domain approach that seems, at least from a
theoretical point of view, simpler and more general than classical
geometric approaches. 

In particular, the speed of a sound source is likely to vary rapidly,
its trajectory may be curved and not inscribed in a plane. The model
must be able to calculate in frequency and amplitude the spectrum of
the noise received at the listening point, apply indifferently to the
subsonic case and the supersonic case, and be coupled with an acoustic
propagation code in order to adapt to any atmospheric environment,
standard, statistical or taken from a weather report.

\section{Geometric approach}\label{sec3} 
Considering the review above and the disparities we have encountered,
we do not think it is useless to recall the physical and mathematical
bases of the Doppler effect in aeroacoustics.

Figure~\ref{fig1} is a classic illustration of the Doppler effect in
the subsonic case, considering a succession of spherical waves emitted
at equal time intervals, i.e.\ the period of the harmonic or impulse
source S moving at speed $V$, and opposite listening directions, where
the perceived frequencies are continuously variable when the observer
is not on the axis of the sound source's movement. The geometric
representation of  Figure~\ref{fig1} is convenient, because it allows
to visualize the apparent modification of the wavelength of the signal
according to a listening direction in relation to the sound source:
either  $\lambda_1$ in upstream direction (direct frequency, the source
is getting closer), $\lambda_2$ in downstream direction (reverse
frequency, the source is moving\break away).

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig1}Subsonic case: apparent wavelengths in opposite
listening directions.}
\vspace*{9pt}
\end{figure}

Figure~\ref{fig2} pulled from~\cite{9} illustrates the supersonic case,
where the Doppler effect only affects the sound sources within the
acoustic zone, included in a ``sound cone''. The two propagation
directions shown represent the doubling of Doppler frequencies: the
one, emitted in forward direction, is called direct, the other, emitted
in backward direction, is called  reverse~\cite{6}. The two frequencies
are perceived simultaneously by the same observer $\Delta$ when he is
inside the sound cone. Both frequencies are confused and in theory
become infinite on the sound cone (see  Figure~\ref{fig3}), also called
Mach cone in the  literature~\cite{2,7,8}. Knowing that the apparent
wavelengths $\lambda_1$ and $\lambda_2$ become zero on this cone 
(Figure~\ref{fig4}), a focus of the sound waves occurs, but in fact the
sonic boom really heard corresponds to the shock fronts, so called
N-wave, constituting the supersonic wake of  Figure~\ref{fig2}. 

\begin{figure}
\includegraphics{fig02}
\caption{\label{fig2}Supersonic case: shock wake and acoustic zone.
Direct wave (blue arrow) and reverse wave (red arrow).}
\vspace*{9pt}
\end{figure}

\begin{figure}
\includegraphics{fig03} 
\caption{\label{fig3}Doppler frequencies generated by a harmonic sound 
source of frequency 100~Hz moving at supersonic speed $(M = 2)$.}
\vspace*{9pt}
\end{figure}

\begin{figure}
\includegraphics{fig04}
\caption{\label{fig4}Supersonic case: sound cone, direct$^{1}$ and
reverse$^{2}$ apparent wavelengths.}
\end{figure}

Recall that the aperture $\alpha$ of the sound cone shown in 
Figure~\ref{fig4} is such that: 
{\begin{equation}\label{eq1}
\sin \alpha=c / V= 1 /M
\end{equation}}\unskip
where $c$ is the speed of sound in the ambient medium and $M$ is the
Mach number. 

The Doppler factor $\mathrm{DF}$, which we define as the ratio of the frequency
$f$ perceived at the listening point linked to the fixed  reference
point and the frequency $f_0$ emitted by the moving sound source, is
given by the classic formula:
{\begin{equation}\label{eq2}
\mathrm{DF}= f /f_0 =
1 / (1 + M\cos\theta)
\end{equation}}\unskip
with the orientation conventions of  Figure~\ref{fig5}, the speed  $V$
of the source being subsonic or not. This formula is not much explicit
but becomes evident, considering  Figure~\ref{fig1}, in the form:
{\begin{equation}\label{eq3}
\lambda_0/\lambda=c/(c + V\cos\theta)
\end{equation}}\unskip
where angle $\theta$ characterizes the listening direction with respect
to the velocity vector $V$, see  Figure~\ref{fig5} where the sound
source S is represented at the moment when the wave considered reaches
the listening point E.

\begin{figure}
\includegraphics{fig05} 
\caption{\label{fig5}Direction of propagation $\theta$ for a listening 
point E linked to the fixed reference point.}
\end{figure}

What happens when the velocity of the source tends towards the speed of
sound? As shown in  Figure~\ref{fig6}, all the sound waves emitted
forward in the preceding moments focus on a plane passing through S and
perpendicular to the trajectory. This plane is equivalent to a sound
cone whose half-angle at the apex $\alpha$ is $\rmpi/2$. The signal is
perceived instantly at point E, in theory at any distance from the
source: thus, everything happens as if the emission point was located
upstream at infinity. We verify that formulas~(\ref{eq2})
and~(\ref{eq3})  give an infinite Doppler  frequency  $f$ and a zero
wavelength $\lambda$, the acoustic intensity also being infinite in
theory. 

\begin{figure}
\includegraphics{fig06}                                                                           
\caption{\label{fig6}Sonic case: the sound waves emitted previously
focus on a plane perpendicular to $V$ passing through S.} 
\vspace*{8pt}
\end{figure}

\section{Time approach}\label{sec4}
Figure~\ref{fig7} shows two acoustic beams emitted by source S at A at
time $t_{\mathrm{A}}$ and at B at time $t_{\mathrm{B}}$. The first beam
reaches listening point E at time $t_{\mathrm{E}1}$, the second at time
$t_{\mathrm{E}2}$. 

\begin{figure}
\includegraphics{fig07} 
\caption{\label{fig7}Case of a listening point E at infinity.}
\end{figure}

Now suppose that E is far enough away so that the two beams are nearly
parallel and that triangle AHB, where HB represents the path difference
between the two beams, can be compared to a right triangle (note that
the time interval $t_{\mathrm{B}}$--$t_{\mathrm{A}}$ can be chosen
arbitrarily small).

We can pose the following identities:
{\begin{equation}\label{eq4}
\begin{array}{c}
\mathrm{HB} = - \mathrm{AB}\cos \theta\vspace*{2pt}\\
\mathrm{AB} = M c (t_{\mathrm{B}} - t_{\mathrm{A}})\vspace*{2pt}\\
\mathrm{HB} + \mathrm{BE} = \mathrm{AE}\vspace*{2pt}\\
\mathrm{AE} = c (t_{\mathrm{E}1}- t_{\mathrm{A}})\vspace*{2pt}\\
\mathrm{BE} = c (t_{\mathrm{E}2}- t_{\mathrm{B}}).
\end{array}
\end{equation}}\unskip

Substituting the values  of the segments in the first equality, we
obtain after simplification by $c$:
{\begin{equation}\label{eq5}
(t_{\mathrm{B}}-t_{\mathrm{A}})/(t_{\mathrm{E}2}-t_{\mathrm{E}1}) =
1/(1+M\cos\theta)=\mathrm{DF}.	
\end{equation}}\unskip

Thus the ratio of the time intervals at emission and reception, inverse
of the frequency ratio, is equal to the Doppler factor $\mathrm{DF}$ and is
identified with Equation~(\ref{eq2}). This calculation can therefore be
considered as a demonstration of the general formula from a virtual
displacement of the emission point.

We can notice that the first two equalities of~(\ref{eq4}) allow us to
write the geometric relation:
{\begin{equation}\label{eq6}
\cos \theta = (\mathrm{BE} - \mathrm{AE}) /\mathrm{AB}
\end{equation}}\unskip

We now consider the case where the observation point is at a finite
distance from the trajectory of the source  (Figure~\ref{fig8}). The
atmosphere is assumed to be homogeneous ($c$ is constant), the
trajectory rectilinear and the speed $V$ of the source S constant. The
harmonic source emits spherical waves of which two rays passing through
the listening point E are indicated.

\begin{figure}
\includegraphics{fig08}
\caption{\label{fig8}Case of a listening point $\mathrm{E}$ at a finite
distance.}
\end{figure}

Let $\Delta t_0$ be the time taken by S to travel to AB, $\Delta t_1$
the propagation time between A and E, $\Delta t_{2}$ the propagation
time between B and E. The rays therefore arrive at E at  times $\Delta
t_1$ on the one hand, and $\Delta t_0+ \Delta t_2$ on the other hand.
The ratio between the reception time $(\Delta t_0+ \Delta t_2) - \Delta
t_1$ and the emission time $\Delta t_0$ is therefore equal to:
{\begin{equation}\label{eq7}
1 + (\Delta t_2-\Delta t_1)/
\Delta t_0= 1/\langle \mathrm{DF}\rangle
\end{equation}}\unskip
In light of Equation~(\ref{eq5}), we postulate that this ratio is equal
to the inverse of the average value, noted  $\langle
\mathrm{DF}\rangle$, of the Doppler factor at the listening point when
the source travels the distance AB.

The trigonometric relationships in triangle ABE allow us to write:
{\begin{equation}\label{eq8}
\mathrm{AB}/\sin
(\theta_1-\theta_2) =
\mathrm{AE} /\sin
\theta_2 =
\mathrm{BE}/\sin
\theta_1    
\end{equation}}\unskip
knowing that:
{\begin{equation}\label{eq9}
\begin{array}{c}
\mathrm{AB} = 
V \Delta t_0 = M c \Delta t_0\vspace*{2pt}\\
\mathrm{AE} = 
c \Delta t_1\vspace*{2pt}\\
\mathrm{BE} = 
c \Delta t_2
\end{array}
\end{equation}}\unskip

Equation~(\ref{eq8}) can be written:
{\begin{equation}\label{eq10}
M \Delta t_0/\sin (\theta_1 - \theta_2) = 
\Delta t_1/\sin \theta_2 = 
\Delta t_2/\sin\theta_1
\end{equation}}\unskip
We can deduce from  Equation~(\ref{eq8}) the geometric identity:
{\begin{equation}\label{eq11}
(\mathrm{BE} - \mathrm{AE})/\mathrm{AB} =
(\sin \theta_1 - \sin \theta_2)/\sin
(\theta_1 - \theta_2) 
\end{equation}}\unskip
and from  Equations~(\ref{eq7}) and~(\ref{eq10}) the identity:
{\begin{equation}\label{eq12}
1 / \langle \mathrm{DF} \rangle = 
1 + [(\sin\theta_1 - \sin \theta_2)/\sin
(\theta_1 - \theta_2)] M  
\end{equation}}\unskip
By analogy with Equations~(\ref{eq2}) and~(\ref{eq6}), we can assume
that the expression in brackets is equal to the average value of $\cos
\theta$ between points A and B, which according to~(\ref{eq11}) can be
written in two different ways:
{\begin{equation}\label{eq13}
\langle\cos \theta\rangle =
(\mathrm{BE} - \mathrm{AE}) /
\mathrm{AB} = (\sin
\theta_1 - \sin \theta_2)/\sin
(\theta_1 - \theta_2)
\end{equation}}\unskip
The sine in the denominator, unusual in the expression of the average
value of a cosine, corresponds to the fact that the variation of the
angle $\theta$ is not linear along the segment AB.

Since Equation~(\ref{eq8}) is true for all values of the angles
$\theta_1$ and $\theta_2$, we can verify Equation~(\ref{eq11}) by
arbitrarily fixing one of the two angles. It is convenient to choose
$\theta_1= \rmpi/2$ which corresponds to the origin O of the coordinate
system in  Figure~\ref{fig6}. For simplicity, we also set $\theta_2 =
\theta$. If  $x$ is the abscissa of the current point on the trajectory
and  $h$ the ordinate of the listening point E, we immediately 
have:
{\begin{equation}\label{eq14}
\cos \theta = x/\sqrt{(x^2 + h^2)} 
\end{equation}}\unskip
The average value of the cosine over the interval Ox is then obtained
by integration as a function of the variable  $x$ and division by $x$,
the value $-h$ of the integration constant being given by the initial
conditions  $\theta= \rmpi/2$ and  $x = 0$. We arrive at:
{\begin{equation}\label{eq15}
\langle\cos \theta\rangle =
\sqrt{(1 + h^2/x^2)} - h / x
\end{equation}}\unskip
expression which tends towards 0 when  $x$ tends towards 0. Noting that
$h / x = \tan \theta$, we can still write:
{\begin{equation}\label{eq16}
\langle \cos \theta\rangle = 1 /\cos
\theta - \tan \theta
\end{equation}}\unskip
We verify that this is also the form that Equation~(\ref{eq9}) takes
when we set $\theta_1= \rmpi/2$ and $\theta_2=\theta$, which
demonstrates the proposition. As expected, we can write 
Equation~(\ref{eq12}) in the form:
{\begin{equation}\label{eq17}
1/\langle \mathrm{DF} \rangle = 1 + M
\langle \cos \theta\rangle
\end{equation}}\unskip
expression that extends  Equation~(\ref{eq2}) to an interval of any
duration.  Equation~(\ref{eq13}) also gives:
{\begin{equation}\label{eq18}
\langle\cos \theta\rangle  =
(\mathrm{BE} - \mathrm{AE}) / \mathrm{AB}
\end{equation}}\unskip
relationship which has the practical interest of calculating the
average cosine on a scale drawing. If we assume a real configuration
where the angles of  Figure~\ref{fig6} are respected and where a sound
source emitting a dominant frequency of 100~Hz passes from points A to
B at a speed of Mach 0.2, we obtain by simple measurement of the
lengths of  Figure~\ref{fig6}:  $\langle \cos
\theta\rangle\approx0.69$,  $\langle \mathrm{DF}\rangle \approx 0.88$,
and the frequency perceived at E, $\langle f\rangle \approx 88$~Hz.

\section{Generalization}\label{sec5}
To summarize the previous paragraph, we can say that in the classical
configuration where the trajectory is rectilinear, the speed of the
source constant and the propagation medium isotropic, the average
Doppler factor between two points of the trajectory and for a given
observation point is equal  to:
{\begin{equation}\label{eq19}
\langle\mathrm{DF} \rangle = 
\Delta t_{\mathrm{S}}/\Delta t_{\mathrm{R}}
\end{equation}}\unskip
where  $\Delta t_{\mathrm{S}}$ denotes the duration of the signal at
emission and  $\Delta t_{\mathrm{R}}$ the duration of the signal at
reception. This duration  $\Delta t_{\mathrm{R}}$ depending of many
factors, the relationship~(\ref{eq19}) leads us to considering the
Doppler effect as a distortion of the time between the emission and the
reception---distortion meaning in physics ``lack of proportionality
between the signal emitted and the signal received''. This phenomenon
obviously causes a shift in frequency as well as a variation in
acoustic intensity, by assuming that the acoustic energy is conserved,
apart from absorption phenomena, in a given emission volume.

This being admitted, we see that  Equation~(\ref{eq12}) is indifferent
to the phenomena that can occur between transmission and reception, at
least to the refraction of sound rays by the wind and to temperature
gradients, and that the same is true for the phenomena that determine
the transmission: trajectory, speed and acceleration of the sound
source, variation of its power over time. However, the calculation will
only be of interest if we consider a time step sufficiently short
compared to the scale of the variations of these parameters. The
frequencies $f_{\mathrm{R}}$ and the levels $L_{\mathrm{R}}$ of acoustic intensity calculated
at the reception point without taking, in a first step, the Doppler
effect into account, will be modified as follows by the Doppler factor
to give the final values:
{\begin{eqnarray}
f_{\mathrm{D}} &=&
f_{\mathrm{R}} (\Delta t_{\mathrm{S}}/\Delta t_{\mathrm{R}})    
\label{eq20}\Seqnsplit
L_{\mathrm{D}} &=&
L_{\mathrm{R}} + 10 \log (\Delta t_{\mathrm{S}}/\Delta t_{\mathrm{R}}) 
\label{eq21}
\end{eqnarray}}\unskip

Thus, frequency and acoustic intensity are governed by the same ratio
equal to the Doppler factor. This relationship, which seems us
fundamental, is neither used nor pointed out in the literature we
consulted, except in  Ref.~\cite{11} which indicates for supersonic
jets a shift in sound power level of 
$10\log 1/\mathrm{DF}$ instead 
$10\log \mathrm{DF}$
(page 17), unless there is an error in the sign.

Figures~\ref{fig9} and~\ref{fig10} summarize the issues to be
considered in real-world situations. Figure~\ref{fig9} does not show
the wind direction at successive altitudes, which generally needs to be
considered. In this case, the ray paths shown in  Figure~\ref{fig10}
would probably not be included in a plane. The process consists of
choosing a time step $\Delta t_{\mathrm{S}}= t_2\ndash t_1$ along the
sound source trajectory, and of calculating, using a sound ray computer
code, the corresponding signal duration $\Delta
t_{\mathrm{R}}=t_2^\prime\ndash t_1^\prime$ at the listening point E. 
Equations~(\ref{eq20})  and~(\ref{eq21}) then give the correction to be
applied to the frequencies and sound levels $f_{\mathrm{R}}$ and
$L_{\mathrm{R}}$ obtained in static mode, in order to compare with the
experimental data the final results $f_{\mathrm{D}}$ and
$L_{\mathrm{D}}$ integrating the Doppler effect.

\begin{figure}
\includegraphics{fig09}
\caption{\label{fig9}Example of a real atmosphere  compared to a
standard atmosphere without wind.}
\end{figure}

\begin{figure}
\includegraphics{fig10}
\caption{\label{fig10}General case of any sound source  trajectory and
non-straight sound ray 
\mbox{trajectories}.}
\end{figure}

\section{Application case}\label{sec6} 
We see that the approach described above allows us to calculate the
Doppler effect from signal durations, beyond any other considerations,
provided we are able to calculate, when they exist, the trajectories of
the acoustic rays that leave each chosen calculation point and pass
through the reception point, as well as their propagation times. This
requires knowing the trajectory of the sound source, the
characteristics of the acoustic emission and the atmospheric
conditions, as well as taking into account atmospheric absorption and
Earth's roundness when the distance requires it. This problem and its
converse, consisting of calculating the trajectories of fictitious rays
coming from the listening point to target the acoustic source, were
addressed at ONERA as part of a study conducted in another 
context~\cite{15}.

This model concerning the Doppler effect was developed in the framework
of the Ariane 5 program of the French National Center for Space Studies
(CNES). It involved to test in a dynamic context the jet noise model
developed by the ONERA, based on NASA studies and improved from static
firings of rocket engines of different sizes, up to a ballistic missile
first  stage~\cite{16}. More precisely, the proposed objective was to
reproduce the ``acoustic fallout'' recorded in Kourou by the ``Toucan''
listening station of ONERA, 4800~m from the launch pad, during the
initial phase of Flight 521 of the Ariane 5 space launcher.

In such a take-off context, the problem arises of the orientation of
the launcher speed in the opposite direction to that of the jets and
therefore of the noise sources emitted downstream at an initially
supersonic speed. Since no classical model seems to be able to adapt to
this particular configuration, we finally imagined the time approach
described previously and formulated the following hypothesis: the jets
being included in the aerodynamic wake and quickly protected by the
supersonic shock wake of the launcher (see  Figure~\ref{fig2}), their
emitted noise spectrum must remain fairly close to that of static
firings and there is no need, in a first analysis, to modify the jet
noise model used. Such a hypothesis is also adopted in  Ref.~\cite{11}
for supersonic jets, but a ``convection Mach number'' where the vehicle
speed is subtracted from jet speed is used in the subsonic case.

This being established, it remained to use the data provided by the
CNES, namely the position as a function of time of the launcher and
therefore of the acoustic emission zone, which makes it possible to
calculate the propagation times of the sound rays towards the listening
station and to determine the value of the Doppler factor in successive
time intervals. The length of the sound rays also gives the spatial
decrease in the acoustic level for the portion of the trajectory
considered, and their direction relative to the axis of the jets makes
it possible to calculate the acoustic power spectrum to be taken into
account. Since no sonic boom is perceived, it is concluded that the
listening station remains inside the sound cone and receives only the
``reverse'' Doppler frequencies (see  Section~\ref{sec2} and 
Figure~\ref{fig2}).

The particularly favorable atmospheric conditions (see 
Figure~\ref{fig9}) made it possible to use a simplified propagation
model mainly taking into account a temperature gradient very close to
that of the standard atmosphere [J], the influence of the wind being
considered as negligible. In addition to the uncertainties linked to
the simulation of the acoustic emission and to the aerodynamic
disturbances in flight, the position of the free-field microphones at
height (about 2.5~m over a low vegetation) raised the question of the
reflected sound field, which led us to compare the measurements with
the sound levels calculated in free field on the one hand, above a
fictitious reflective plane on the other hand (sound level increased by
3~dB). Note that the trajectory of the launcher is not straight, it
bends rapidly in an easterly direction.

We present two calculations corresponding to a case where the launcher
speed is subsonic, below an altitude of 4000~m, and to a case where it
is supersonic, around an altitude of 13,000~m.  Figures~\ref{fig11}
and~\ref{fig12} give the acoustic pressures recorded by station
``Toucan'' in each case at the acquisition frequency of 5 kHz, and in
red the calculated arrival times of two sound rays emitted in altitude
at one second interval: this allows a direct reading of the value of
the average Doppler factor of the framed period, i.e.\ $\langle
\mathrm{DF}\rangle = 1$~s/1.5~s ${=}$ 0.667 in the first case, $\langle
\mathrm{DF}\rangle =1$~s/2.6~s ${=}$ 0.385 in the second. This allows us
to calculate the ``apparent speed'' of the launcher using
formula~(\ref{eq2}), knowing that the angles of the sound rays with the
trajectory are $\theta\approx50\text{\textdegree}$ and
$\theta\approx22\text{\textdegree}$ respectively: we obtain 
$M\approx0.78$ in the first case,  $M\approx 1.72$ in the second.

\begin{figure}
\includegraphics{fig11} 
\caption{\label{fig11}Signal recorded by the ``Toucan'' station.
Launcher altitude  3100 to 4600~m, signal analysis window centered at
3750~m.}
\end{figure}

\begin{figure}
\includegraphics{fig12}
\caption{\label{fig12}Signal recorded by the ``Toucan'' station.
Launcher  altitude 11,700 to 13,700~m, signal analysis window centered at
13,000~m.}
\end{figure}

Figures~\ref{fig13} and~\ref{fig14} give on the one hand the sound
power spectral density from the signals of  Figures~\ref{fig11}
and~\ref{fig12} between the red time markers (dark blue curve), on the
other hand the raw result of the simulation of the jet noise from the
three rocket engines of the launcher at a distance equal to the length
of the calculated sound rays (sky blue curve), finally the framing
obtained by integrating the Doppler factor by means of 
Equations~(\ref{eq20})  and~(\ref{eq21}), in the presence or not of a
reflected field (purple and red curves).

\begin{figure}
\includegraphics{fig13}
\caption{\label{fig13}Sound power spectral density for the window in
Figure~\ref{fig11}.  Simulations without and with Doppler factor.}
\end{figure}

\begin{figure}
\includegraphics{fig14}
\caption{\label{fig14}Sound power spectral density for the window in
Figure~\ref{fig12}.  Simulations without and with Doppler factor.}
\end{figure}

Knowing that the jet noise spectrum is very regular in the case of
static firings of small rocket engines (boosters at scales 1/20 and
1/10), it appears that the measured noise spectrum in 
Figure~\ref{fig13}, which corresponds to the signal between the two
time marks in  Figure~\ref{fig11}, seems to be strongly disturbed by
the aerodynamic effects due to the vortex wake of the launcher at
subsonic speed, but that the envelope of the maxima given by the model
nevertheless appears to be correct. 

In Figure~\ref{fig14}, the noise spectrum corresponding to the period
indicated in  Figure~\ref{fig12} has the same general appearance but is
much more regular, the jets being apparently shielded by the shock wake
of the launcher at supersonic speed. The slope of the spectrum is
clearly accentuated compared to that of  Figure~\ref{fig13}, because of
the decreasing value of the Doppler factor. It can be said that the
``free field'' simulation (red curve) is very satisfactory up to 50~Hz,
the drop in the measured noise level below 3~Hz being probably due to
the cut-off frequency of the measurement devices. In contrast, the
minima at 28 and 36~Hz, visible in both  figures, probably have a
physical cause. 

We can admit that the results of the simulations constitute an apparent
validation of the models used and of the simplifying assumptions that
accompanied their implementation. It appears in particular that the
diminution of sound levels by the Doppler effect (${-}$2~dB in case of 
Figure~\ref{fig13}, ${-}$4~dB in case of  Figure~\ref{fig14}), sometimes
omitted in previous or contemporary studies, has improved the accuracy
of the  calculation.

\section{Conclusion}\label{sec7}
Despite the uncertainties mentioned, the application example given in 
Section~\ref{sec6} can be considered as a first validation by
experience of the method of calculating the Doppler effect applied to
aerial sound sources stemming from a supersonic vehicle. More
generally, the specificity of the model is that it can be applied to
any trajectory and to a real environment, unlike geometric models whose
field of application is restricted by the assumptions of the
rectilinear and uniform displacement of the sound source, and above all
of a homogeneous and isotropic propagation medium.

The approach described, which proposes calculating the Doppler factor
from the duration of the emitted and received signals, is equivalent to
attributing the cause to a distortion of time, distortion being
understood in the physical sense of a lack of proportionality between
the emitted and received signals.

It should be noted that the principle of conservation of acoustic
energy in a given emission volume, omitting the atmospheric absorption
effects, allows to apply the Doppler factor to both acoustic intensity
and frequency. In our opinion, this assertion represents a step forward
in the understanding of these phenomena, given that the variation of
sound levels due to the Doppler effect does not have a well-defined
formulation in the literature: its calculation is in fact either
clearly empiricist without obvious theoretical justification, or simply
omitted.

The model applies to sound sources linked up to both subsonic and
supersonic vehicles. In the latter case, regarding the Doppler effect,
the definition of the Mach cone created by a point harmonic source, and
upon passage of which a sonic boom is perceived, is often found in 
the~literature. This may be true theoretically, but it should be noted that
the sonic boom caused by the passage of a supersonic aircraft is only
related to the shock wake created in the atmosphere by the body of the
mobile, and is therefore neither caused nor modified by the Doppler
effect. We felt it was important to clarify this point. 

The major difficulty relating to the model, apart from the classical
cases in homogeneous and isotropic medium where it applies directly, is
that it must be coupled with an acoustic propagation computer code to
calculate, by integrating the wind and temperature data, the path of
the sound rays and their travel time between the emission points chosen
and the listening point. The main difficulty consists in correctly
targeting the listening point, so as to minimize the number of
iterative calculations. This problem has been solved at the ONERA, in
particular by making reversible the calculation of the sound rays,
which was theoretically impossible in the presence of wind gradients,
due to the inversion of the concavity of the ray 
trajectories~\cite{17}: as it is proposed in Ref.~\cite{15}, to make it
possible it was in fact sufficient to reverse, at all altitudes, the
direction of the wind given by the weather report.

\section*{Declaration of interests} 
The author does not work for, advise, own shares in, or receive funds
from any organization that could benefit from this article, and has
declared no affiliations other than his research institution.

\section*{Acknowledgements}
The author expresses his gratitude to Geraldine M\'{e}nexiadis, PhD,
Claude Bresson, Research Engineer, and Georges Elias, PhD, Research
Director at the National Office for Aerospace Studies and Research
(ONERA).

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