RESULTS OF A STUDY OF PROPELLER NOISE

E. A. Nepomnyashchii

INTRODUCTION

The struggle against noise is a task of enormous practical importance and very often a serious and difficult scientific-technical problem. One such task is the study and attenuation of aircraft noise.

Over the past decade, scientific research on the study of aircraft noise has been conducted very intensively; it has led to the clarification of the basic properties of the sounds emitted by aircraft. Naturally, the first question to be answered was: what are the sources of aircraft noise, how great is the contribution of each of them to the overall noise, and which of them, consequently, should be subjected first of all to careful and systematic investigation.

When one of two equally loud noise sources is completely silenced, the overall loudness level decreases by only approximately 3 db¹); when the difference in noisiness between two sources is 5–10 db, the less loud component will be practically completely masked by the louder one.

These remarks evidently determine the order and the required degree of attenuation of each of the sources of aircraft noise. It has been established that these sources are: the propeller, the engine, the lifting surfaces, various kinds of struts, bracing wires, etc.; the loudest are the first two sources.

Without dwelling here on the properties of aircraft-engine noise, we shall point out that its sources are: intake, exhaust, the impeller of the supercharger, the reduction-gear pinions, valves, and other mechanical elements; moreover, in most cases the noise level of the engine exhaust is comparable with the noise of the propeller.

In view of the successes achieved in the application of exhaust-noise mufflers and the ever-increasing noisiness of propellers, it may be asserted that the study of the noise produced by the propeller is at present the most urgent problem in the general complex of questions concerning aircraft noise suppression.

¹) Roughly speaking, 1 db represents a difference in the loudness of two sounds that is barely perceptible to the ear.

The present article acquaints the reader with questions of propeller sound generation, the basic properties of this sound, and methods for calculating it.

Let us note that the study of all these questions was begun abroad more than 20 years ago, but work in this direction has been carried out especially intensively in recent years. In the USSR these studies were initiated by N. N. Andreev, in whose laboratory they were developed.

1. CAUSES OF PROPELLER SOUND GENERATION

In terms of the complexity of the sound it emits, the propeller occupies a special place among known noise sources. This complexity is due to the fact that the sound generation of a propeller may be associated with the most diverse physical processes occurring in the space surrounding it.

The various sources of propeller noise generate sounds that differ greatly in their properties, which considerably confuses the overall picture and makes investigation difficult.

One of these sources of sound is the periodic disturbance of the medium by the propeller blades. The process of formation of this sound is as follows.

As a result of the flow of air around a propeller blade, a certain pressure distribution is established at its surface; namely, on one (convex) side of the blade there is a decrease, and on the other (concave) side an increase, of pressure relative to atmospheric pressure; the difference of these pressures determines the thrust force of the propeller. This pressure distribution is carried along with the propeller blade, and if we imagine that a microphone is placed on the propeller, then the latter will not perceive any sound, since the pressure relative to it is unchanged.

The situation is otherwise for a stationary microphone or the ear of a stationary observer; the pressure carried by the rotating propeller blade alternately approaches and recedes from the observer; consequently, it is alternately intensified and weakened depending on the variable distance, i.e. the microphone and the ear will perceive sound. The exception will be the case in which the observer is located on a straight line coinciding with the axis of rotation of the propeller: in this case the distance from the observer to the pressure carried by the propeller blade remains unchanged. Experiment, provided only that it is carried out under clean conditions without any reflections, clearly confirms this conclusion.

The same process of sound formation may be conceived somewhat differently.

Let us mentally cut out, in the plane of rotation, an element of the medium. At each passage of a blade through this element, forces from the blade will act upon it (thrust force, resistance force to rotation), moving this element approximately perpendicularly …

to the plane of rotation; moreover, at the place where a blade of finite thickness passes, the air will be pushed apart in both directions.

Acoustically, this element of the medium may be likened to a certain body which at the same time both vibrates, like a solid body, and pulsates, periodically expanding and contracting.

Each such element of air is thus a source of acoustic oscillations throughout space.

It is obvious that the spectrum of the sound caused by this periodic process consists of sinusoidal sounds whose frequencies are multiples of the fundamental frequency, equal to the product of the number of revolutions of the propeller by the number of its blades.

This sound is customarily called “rotation sound.” But in addition to the rotation sound, whose formation is possible even in an ideal fluid, in a real fluid, owing to vortex formation near the propeller blades, there arises the so-called “vortex sound.”

The first separate observation of rotation sound and vortex sound was carried out in the laboratory of N. N. Andreev.

In doing so, a very simple but extremely ingenious method was used: the microphone was placed near the plane of rotation, not far from the end of a blade; in this position the rotation sound is very strong in comparison with the vortex sound.

Fig. 1. Spectrogram of the rotation sound.

In Fig. 1 is shown the spectrum of the rotation sound, taken with Rodman’s analyzer. The individual peaks, situated to the right of the peak marked with the digit zero, represent the separate harmonics of the rotation sound; the small irregularities on the peaks are due to traces of the vortex sound. The oscillogram (Fig. 2), pertaining to the same experiment, also shows a quite definite periodicity; here too the small irregularities are due to the presence of a weak vortex sound.

If the microphone is placed on the axis of rotation of the propeller, then, owing to the symmetry indicated above, the rotation sound will not act on the microphone at all, and the latter will record a pure vortex sound; Fig. 3 gives an oscillogram of the latter; the irregular character of the oscillations in the vortex sound appears on it very clearly. In accordance with this, the spectrum of the vortex sound, shown in Fig. 4, has the form of a continuous spectrum containing a continuous set of different frequencies.

There are indications in the literature of the possibility of the appearance of sound as a result of elastic oscillations of the blade and the shaft. However, in no case, under ordinary operating conditions of the propeller, have we observed sound from vibrations.

Fig. 2. Oscillogram of vortex sound.

Labels in the figure:

• Alternating current 50 Hz
• Time of one complete revolution of the propeller
• Alternating current 50 Hz

Fig. 3. Oscillogram of the vortex sound of a cylindrical rod.

Label in the figure:

• Time of one complete revolution of the propeller

In N. N. Andreev’s laboratory some work was carried out to clarify this question. A theoretical study was made to establish the possibility of the occurrence of self-oscillations of a propeller blade due to the presence of a falling part in the function expressing the dependence of the resistance to rotation on the rotational speed of the propeller. It turned out that oscillations of this kind cannot arise. Further, experiments were set up in which rather large vibrations of the blades were artificially excited. Under these conditions a certain increase in the noise of the propeller model was found; however, in practical operation of the propeller these conditions are absent.

In any case, an oscillating propeller constitutes an acoustic antenna of order higher than zero. When the blade widths are small in comparison with the wavelengths possible here, the very conditions of radiation, as is known from acoustics, will be unfavorable. In addition, the frequencies of propeller vibrations observed in practice are very low. For example, Carter1 gives, for vibration frequencies, values of 10–20 Hz, which lie below the sound frequencies perceived by the ear.

Fig. 4. Spectrogram of the vortex sound of a plate.

Therefore, sound production due to oscillations of the blade, in solving the general problem of propeller noise, need not be considered—at any rate until the principal sources of this noise, the sound of rotation and vortex sound, have been studied and sufficiently attenuated.

At supersonic speeds of motion of individual elements of the blade, exceeding the local speed of sound, the formation is possible of surfaces of strong discontinuity or, as they are otherwise called, shock waves. The process of formation and propagation of these waves is connected with new, still unstudied acoustic phenomena in the space surrounding the propeller.

In connection with the increasing tip speeds of propellers, the study of this source of sound is highly desirable; for the present, however, this sound is absent in the noise of modern propellers.

Frankl recently solved the problem of the aerodynamics of a propeller in the presence of supersonic tip speeds. It is very probable that this solution will also make it possible to understand the acoustic phenomena of propellers with such tip speeds.

The majority of investigations concern the study of the sound of rotation and vortex sound, which are always present in the spectrum of the total noise of a propeller. Below, the basic properties and regularities of these sounds will be set forth.

2. RESULTS OF THE INVESTIGATION OF VORTEX SOUND

When a solid body is flowed around by a real fluid, one may speak of a certain “boundary layer” adjacent to the body; in this layer the action of viscosity is concentrated. Flowing off the body, the boundary layer, as is said, breaks up, and vortices are thereby formed; this process occurs rather irregularly. The intensity of the vortices formed depends on the character of the flow, i.e., on the shape of the blade and on the velocity of the incident stream.

The impulses associated with vortex formation are sources of sound. This fact was pointed out already by Strouhal^1 in investigating the acoustic vibrations of strings. He found that the periodic separation of air and the vortex formation associated with it are the cause of the appearance of sound, and that the frequency of a single sound is related to the velocity and diameter of the string by the following dimensionless relation:

\[ \frac{N d}{V} = \text{const.} = 0.185 \]

\(N\)—frequency of the sound, \(d\)—diameter of the string, \(V\)—velocity of motion.

A great many works^2–12 have been devoted to the investigation of the frequencies of vortex formation behind a circular cylinder. As a result of these investigations, a dependence has been established of the dimensionless frequency \(\frac{N d}{V}\) on the Reynolds number

\[ Re = \frac{V d}{\nu} \]

(\(\nu\)—kinematic coefficient of viscosity). For \(Re > 1000\) the frequency \(\frac{N d}{V}\) is practically independent of \(Re\) and is equal to 0.2.

The question of the frequencies of vortex formation behind smooth plates has been subjected to careful investigation; the same question has been investigated somewhat less for airfoil profiles at various angles of attack^1).

Various authors^11–13 propose a value of the universal dimensionless frequency \(\frac{N L}{V}\), equal to from 0.165 to 0.180, valid for angles of attack in the range \(20^\circ < \alpha < 90^\circ\). Here \(L = b \sin \alpha + \delta \cos \alpha\) (\(b\)—width of the plate, \(\delta\)—thickness of the plate, \(\alpha\)—angle of attack).

In the region of small angles of attack there is a strong dependence of the frequency on the number \(Re\), and the measurements made are still not complete. Nevertheless, all these works considerably facilitate the task of studying the vortex sound of a propeller. It may be assumed that the vortex sound of a propeller owes its origin to the combined process of vortex formation behind its individual elements, situated at different distances from the axis of rotation. Therefore, unlike the line spectrum obtained in vortex formation behind bodies flowed around—

^1) The angle of attack is the angle between the direction of the flow velocity and the straight line connecting the leading and trailing edges of the blade profile.

plane flow of liquid with constant velocity, the spectrum of the vortex sound of a propeller proves to be continuous, since, with a known approximation, it may be assumed that each element will form vortices with its own frequency, corresponding to its velocity and geometrical features.

The character of the spectrum will be determined by the magnitudes of the sound pressure, depending on the intensity of vortex formation for the individual frequency bands of this spectrum.

The first investigation of vortex sound was carried out by Stowell and Deming[^14].

In this work the greatest interest is presented by changes in the spectra. The object of Stowell and Deming’s investigation was the sound emitted by a round cylindrical rod of diameter \(\sim 12\) mm, length \(\sim 60\) cm, rotating at various speeds within the range 1080–1450 rev/min. The authors used a NACA analyzer; the spectra obtained by them are continuous, with a maximum of intensity in a certain frequency band. Assuming that the maximum radiation corresponds to the end of the rod, Stowell and Deming obtained agreement of this frequency with Strouhal’s dimensionless relation

\[ \frac{Nd}{V}=0.185. \]

Fig. 5. Spectrogram of vortex sound.

A more complete investigation of this question was carried out by the author[^15] in the laboratory of N. N. Andreev. Here measurements were made of the spectra of the vortex sound of round and elliptical cylindrical rods, plates, and a propeller model. The sound receiver was a piezoquartz microphone; the analysis was performed by means of Rodman’s analyzer[^16], developed in the laboratory. A spectrogram characteristic of all the measurements is shown in Fig. 5. It refers to a round cylindrical rod, 2 cm in diameter and about 60 cm long, rotating at a speed of 27 rev/sec. The measurement method used in the experiments made it possible, with sufficient accuracy, to determine the frequency of the spectrum at which the sound intensity is maximal.

In Fig. 6 a graph is plotted of the dependence of frequency on the number of revolutions \(n_s\) for rods of different diameter. As a result of these measurements it may be considered that, for cylindrical rods,

\[ \frac{Nd}{V} \simeq 0.20, \]

where \(N\) is the frequency of maximum intensity in the spectrum, corresponding to the element located at \(0.8\)–\(1.0\) of the length of the rod.

Exactly the same character of the spectrum was obtained in measurements of the sound of a rotating plate and of a propeller model. In Fig. 7 is shown...

a spectrogram of the vortex sound of the propeller model. In Fig. 8 the values of the dimensionless frequency \(\dfrac{NL}{V}\) are plotted for various angles of attack. In this case the phenomenon pertains to a certain blade-tip element (\(R = 0.9 R_0\), for the propeller), and as the characteristic dimension the value

\[ L = b \sin \alpha + \delta \cos \alpha . \]

is adopted.

Fig. 6. Dependence of the frequency of the vortex sound of a cylinder on the number of revolutions.
1 — circular cylinder, 2 — elliptical cylinder.

Fig. 7. Spectrogram of the vortex sound of a propeller model.

From the graphs it is evident that, in the region of large angles of attack, the relation \(\dfrac{NL}{V} = \text{const.}\) is satisfied for different angles of attack.

The formula \(N = \dfrac{\text{number } V}{b \sin \alpha + \delta \cos \alpha}\) explains the large difference in the spectrum of the vortex sound of propeller models, axial fans, and full-scale propellers operating on the ground and in the air.

For a fan, for example, the most pronounced frequencies of the vortex sound lie in the region of 200–400 Hz. For propellers operating on the ground at ordinary angles of attack, these frequencies likewise lie in the region of 200–400 Hz. When, however, a propeller on a flying airplane is considered, the frequencies of the vortex sound will occupy the region 2000–4000 Hz, since the angle of attack of this propeller is greatly reduced in flight in comparison with rotation on a stationary airplane.

Fig. 8. Dependence of the dimensionless frequency \(\dfrac{NL}{V}\) of a propeller model on the angle of attack.

An interesting property of the vortex sound of a propeller is the directivity pattern, or its distribution around the propeller.

Stowell and Deming investigated the directional properties of the vortex sound of a rotating cylindrical rod. They found that the maximum total intensity is radiated in the direction of the axis of rotation; in the plane of rotation the sound intensity falls to zero; then in this direction only the sound of rotation remains.

Practically the same result was obtained by Ernsthausen^17 for the case of radiation from a symmetrically profiled rod rotating with zero angle of attack. Figure 9 shows the directivity characteristics measured by Ernsthausen for various rotational speeds over a wide range of speed variation. In his experiments the tip speeds \(U\) reach values corresponding to full-scale propellers.

Fig. 9. Directivity characteristics of vortex sound.

Studies of the directivity of the vortex sound of propeller models in the laboratory of N. N. Andreev led to the same result; the directivity curve has the form of a figure eight. Applying the results of studies on models to larger propellers operating on the ground does not lead to contradictions. Here, in the laboratory, measurements were also made of the intensity level^1) of the vortex sound of a propeller model for various setting angles^18. Figure 10 presents the dependence of the intensity level on the rotational speed of a screw of diameter \(0.8\ \text{m}\), for various setting angles.

In addition to the curves for various setting angles, a curve \(40\lg V\) is plotted here, corresponding to a quadratic dependence of the sound pressure on the speed.

Fig. 10. Dependence of the intensity level of the vortex sound of a propeller model on rotational speed.

Such a dependence was derived by N. N. Andreev from simple dimensional considerations and is justified fairly well in various cases, in particular for a propeller with a noise level of \(90—100\ \mathrm{db}\). The fact that the curves for small angles of attack lie parallel to the curve \(40\lg V\) indicates the independence of the dimensionless pressure quantity

^1) The sound intensity level is \(l = 20\lg \dfrac{p}{p_0}\), where \(p\) is the sound pressure, \(p_0\) is the pressure at the threshold of audibility, taken equal to \(2.04\cdot 10^{-4}\) bar (for normal values of temperature and pressure).

\(\dfrac{p}{V^2}\) on the velocity, or the Euler number on the Reynolds number. At large angles this is no longer observed. At such angles separation of the jets from the blade elements already occurs, and it is natural to suppose that the stronger increase in sound intensity with velocity is connected precisely with this intense vortex formation. To verify this supposition, the same dependences were obtained for the case of rotating cylindrical rods, which are a particularly convenient object of investigation owing to the intense vortex formation. It turned out that the course of the curve corresponds to the course of the curves for large angles of the propeller model.

Investigations have shown that the intensity level of the vortex sound near the propeller and of the rotation sound in those listening directions where both are present is approximately of the same order. In the case of a propeller on a flying airplane, when the angles of attack are very small, the intensity level of the vortex sound is somewhat less than that of the rotation sound. In addition, it was indicated above that the frequencies of the vortex sound of a propeller on a flying airplane will be of the order of 2000–4000 Hz.

The latter circumstance is very important for estimating the role of vortex sound in comparison with the rotation sound of the propeller when listening to a flying airplane from the ground.

It is known that the atmosphere has the property of absorbing sound, and as the sound frequency increases this absorption increases. Thus, for example, for our estimate one may take the absorption for sound with a frequency of 3000 Hz to be of the order of 15–20 db per 1 km for ordinary values of the relative humidity of the air. If one bears in mind that rotation sound with frequencies of 50–200 Hz is practically not absorbed by the atmosphere, then the predominating importance of the rotation sound in the overall propeller noise at large distances from a flying airplane becomes clear. Stowell and Deming\({}^{19}\) came to the same conclusion in their very interesting investigations of the noise of a two-bladed propeller. The regularities of the vortex sound of a propeller obtained as a result of all the investigations set forth above may be used in considering the vortex noises of other similar sources. These may include the vortex noises of ventilation installations, all kinds of rotating objects, lifting surfaces, various struts, and braces on an airplane.

Incidentally, we note that airplane vortex noises will continue to increase as flight speeds increase and may have some significance in the overall noise of the airplane.

3. SPECTRUM OF THE ROTATION SOUND NEAR A PROPELLER BLADE

If a sound-pressure meter is placed near the blade of a rotating propeller, the sound recorded with its aid will be determined by the pressure of only the nearest elements of the blade.

The recording instruments may be an oscillograph and various kinds of analyzers. The first instrument will show the change of pressure with time; with the aid of the second this pressure is represented as a function of frequency, i.e., a spectrogram is obtained. In order to understand the physical processes connected with sound generation, it is desirable first of all to study the variable pressure field near the propeller.

The first experimental works in this direction were apparently carried out by the Japanese Obata, I. Iosida, and U. Iosida[^20], who accumulated a large number of oscillograms of the sound of various types of screw models. These measurements gave a clear idea of the nature of the pressure oscillations near the blade, which in fact are the sources of the sound of rotation. A further step in the investigation of this question could be made after the theoretical work of B. P. Konstantinov, carried out in 1934 in N. N. Andreev’s laboratory. Here the question was posed of the possibility of calculating the sound impulse for the case of a rotating circular cylindrical rod by applying the method known in aerodynamics for calculating the pressure near an infinite circular cylinder moving in the potential flow of an incompressible fluid. For this purpose, at small distances from the surface of the cylinder (1–2 cylinder diameters), the pressure distribution was calculated; as a function of time, connected with the rotational speed of an element of the rod, it represents a periodically varying sound impulse, which can be resolved into a line spectrum by Fourier’s method.

The experiments performed showed very good agreement with the calculation both in the form of the impulse and in the spectrum. Later, Ernsthausen[^17] calculated the sound impulse for an element of a symmetrically profiled rod. In doing so he also calculated the pressure distribution near a symmetric profile, using a plane potential flow of an incompressible fluid.

To calculate the pressure field, the profile was replaced by a dipole, on which a parallel flow was superposed. The form of the calculated impulse proved to be in agreement with experiment. An approximation to real profiles was made in N. N. Andreev’s laboratory. Here systematic calculations of sound impulses and spectra were carried out for propellers with Zhukovsky profiles[^21]. In calculating the pressure, the Prandtl–Glauert correction for compressibility was introduced1. In Fig. 11 are shown impulses for various distances \(r\) from the blade element—

part of the propeller model. One can see the smoothing of the pulse with increasing distance from the profile.

The maximum rarefaction with increasing distance from the profile varies inversely with the distance; this agrees well with experiment. At large distances (more than the diameter of the screw) it is impossible to carry out the calculation because of the influence of the pressure field of the neighboring blade1.

Decomposing the pulse into a Fourier series, we obtain the sound spectrum, i.e., the sound pressure for each of the discrete frequencies. It turns out that at small distances from the blade, when moving away from it, the spectrum is considerably impoverished in overtones.

In addition, dependences of the sound pressure on various parameters of the blade elements were established. These dependences are characterized by the following figures: when the angle of attack is increased by \(1^\circ\), the pressure increases by \(7\%\); an increase of the relative thickness of the profile by \(1\%\) (with respect to the chord) increases the pressure by \(16\%\). These dependences are satisfactorily confirmed by experiments on models. We must recall here that all these results refer to the pressure produced by the action of an individual blade element, and not of the whole propeller.

Fig. 11. Pressure pulses at various distances from an element of a propeller blade.

4. THE SOUND FIELD OF A PROPELLER

The study of the sound field at distances from the propeller that are large in comparison with its diameter is of the greatest and most direct interest.

In 1919, Lynam and Webb²², investigating the dependence of sound on the speed of rotation of a propeller, found that the sound of this source possesses directivity, i.e., is distributed around the propeller according to certain laws, but by no means uniformly, as may be observed when considering a point radiator.

To determine the character of the directivity of the sound, they identified the acoustic action of the propeller with the action of an aggregate of acoustic sources and sinks. In this case the strength of such sources, distributed over disks in front of and behind the propeller, varies periodically with a frequency equal to the fundamental frequency of the rotational sound. The phase relations of the sources and sinks were determined by the rotation of the propeller. The aggregate of sources and sinks in the given case constitutes dipoles with axes parallel to the axis of rotation of the propeller.

In this case the amplitude of the sound pressure is proportional to the expression determining the character of the directivity:

\[ J_{mz}\left(\frac{3mz\,\omega R_0}{2c}\sin\vartheta\right) \sin\left(\frac{3mz\,\omega R_0}{4c}\cos\vartheta\right), \]

where \(J_{mz}\) is the Bessel function of the first kind of order \(mz\), \(m\) is the number of the harmonic, \(z\) is the number of blades, \(c\) is the speed of sound, \(R_0\) is the propeller radius, \(\omega\) is the rotational frequency, and \(\vartheta\) is the angle between the radius vector from the center of rotation to the point of observation and the direction of flight.

From this expression it is evident that the directivity characteristic must be symmetric with respect to the plane of rotation, and the sound pressure must vanish in the directions of the plane of rotation \((\vartheta = 90^\circ)\). Neither of these is confirmed by experiment.

Another assumption of Lynam and Webb was to represent the propeller only as an aggregate of sources. In this case the directivity characteristic is determined by the expression

\[ J_{mz}\left(\frac{3mz\,\omega R_0}{2c}\sin\vartheta\right). \]

The directivity characteristic will in this case again be symmetric with respect to the plane of rotation. The directivity characteristics computed according to the theories mentioned are shown in Fig. 12.

Garrick²³ combined both of these theories, replacing the propeller by an aggregate of dipoles and sources. After a whole series of very artificial assumptions, he arrived for the directivity characteristic at the expression

\[ J_{mz}\left(\frac{3mz\,\omega R_0}{2c}\sin\vartheta\right) \{\,2.6 - 2\sin(6mz\,\omega R_0\cos\vartheta)\,\}, \]

which satisfactorily describes the experimental directivity characteristic obtained by him for the fundamental tone. As is evident, the present theories do not make it possible to calculate the absolute

significance of the sound pressure and do not establish a connection between the acoustic characteristics of the propeller and its aerodynamic properties.

The artificiality of the assumptions is also a major shortcoming of the theories set forth. It is expedient to begin the exposition of the quantitative theories with the later one, as the more complete, taking into account the features of sound generation by a propeller. The earlier existing theories may be obtained as special cases of this general theory, which, however, is not independent, but is in a definite relation to those that existed earlier.

Fig. 12. Directional characteristics according to the theories of Linn and Webb.
I—the first theory, II—the second theory.

This quantitative theory was developed by E. A. Nepomnyashchii[^18] in 1941. The basic hypotheses that form the basis of the calculation are as follows.

1. To compute the sound field around a propeller with sufficient accuracy, one may use the linear wave equation:

\[ \frac{\partial^2 \varphi}{dt^2}-c^2 \Delta \varphi = 0, \tag{1} \]

where the sound potential \(\varphi\) is related to the pressure and velocity by the relations:

\[ p=\rho_0 \frac{\partial \varphi}{dt}; \quad V=\operatorname{grad}\varphi, \tag{2} \]

In this case, for the \(m\)-th harmonic of the sound potential at a certain point of space \(M\), the following expression is valid:

\[ \varphi_{mM} = -\frac{1}{4\pi} \iint_S (V_n)_S \frac{e^{-jkr}}{r}\,dS + \frac{1}{4\pi \omega \rho} \iint_S (p)_S \frac{\partial}{\partial n} \left( \frac{e^{-jkr}}{r} \right) dS, \tag{3} \]

which makes it possible to compute \(\varphi_{mM}\), if the values of the normal velocities \(V_n\) and the pressure \(p\) are known on a certain surface \(S\) enclosing the propeller at such a distance from it that outside it the propagation of sound may be regarded as occurring according to (1).

In the first approximation, the surface of the propeller was taken as the boundary surface \(S\).

1. The boundary disturbances \(V_n\) and \(p\) on the blade surface may be computed on the basis of the plane motion of a compressible ideal fluid, or else the corresponding experimental values may be assigned.

Expression (3) shows that the action of an airscrew is acoustically equivalent to the combined action of sources (the first term) and dipoles (the second term), the intensity and moment of which are determined by the properties of the propeller. Accordingly, ...

we divide the total radiation into radiation of zeroth order (sources—normal velocities) and radiation of first order (dipoles—pressure).

Expression (3) may be given another form:

\[ \varphi_{nm} = -\frac{1}{4\pi}\iint\limits_S V_{nm}\frac{e^{-jkr}}{r}\,dS + \frac{1}{4\pi\omega}\iint\limits_S \left[ dX_m\cos(r,x)+ dY_m\cos(r,y)+dZ_m\cos(r,z) \right] \frac{\partial}{\partial r}\left(\frac{e^{-jkr}}{r}\right), \tag{4} \]

where \(dX\), \(dY\), \(dZ\) are the components of the pressure forces acting from the side of the propeller on the element of the medium surface adjacent to it. Periodically acting forces and normal velocities may be represented in the form:

\[ \left. \begin{aligned} dX_m&=-\frac{dQ(R)}{dR}\frac{R}{a}K_{Qn}e^{jmz\omega t-jmz\theta-j\eta_m}\sin\theta\,dR\,d\theta,\\ dY_m&=\phantom{-}\frac{dQ(R)}{dR}\frac{R}{a}K_{Qn}e^{jmz\omega t-jmz\theta-j\eta_m}\cos\theta\,dR\,d\theta,\\ dZ_m&=-\frac{dP(R)}{dR}\frac{R}{a}K_{Pn}e^{jmz\omega t-jmz\theta-j\varepsilon_m}\,dR\,d\theta,\\ V_{nm}&=VK_{Vnm}e^{jmz\omega t-jmz\theta-j\chi_m}. \end{aligned} \right\} \tag{5} \]

Here \(Q(R)\) and \(P(R)\) are the resistance force to rotation and the thrust force of a blade element at a distance \(R\) from the axis of rotation; \(a\) is the projection of the blade width on the plane of rotation; \(m\) is the number of the harmonic; \(z\) is the number of blades; \(\omega\) is the frequency of rotation of the propeller; \(\theta\) is the polar angle in the plane of rotation; \(V=\sqrt{(\omega R)^2+V_0^2}\) is the total velocity of the blade element (\(V_0\) is the flight speed of the airplane); \(K_{Qm}\), \(K_{Pm}\), \(K_{Vam}\) are harmonic coefficients representing the moduli of the Fourier-series expansion of the distribution curves along the projection \(a\) of the forces \(Q(R)\), \(P(R)\) and the normal velocities \(V_n\); \(\eta_m\), \(\varepsilon_m\), \(\chi_m\) are the initial phases of these expansions. The period with respect to which these boundary impulses are expanded is equal to \(\dfrac{2\pi}{z\omega}\).

As a result of some calculations, for the amplitude of a certain harmonic of the sound pressure at a point of space \(M(r_0,\vartheta)\), the following expression is obtained:

\[ |\bar p_m|= \frac{mz\bar U_c^{\,2}}{r_0} \left| \int\limits_{\bar r} \left\{ K_{Vnm}\cos\chi_m\sqrt{\bar V_0^{\,2}+\bar R^2} + j\frac{4K_{Qm}dT(\bar R)}{\pi^3 a\bar R^2\,d\bar R}\cos\eta_m - j\frac{4K_{Pm}\bar U_c\cos\vartheta}{\pi^2 a} \times \frac{dP(\bar R)}{d\bar R}\cos\varepsilon_m \right\} J_{mz}\!\left(mz\bar U_c\bar R\sin\vartheta\right)\bar R\,d\bar R \right|, \tag{6} \]

where relative quantities have been introduced: \(\overline{p}_m=\dfrac{p_m}{\rho \dfrac{c^2}{2}}\), \(\overline{U}_c=\dfrac{\omega R_0}{c}\)—relative tip speed; \(\overline{r}_0=\dfrac{r_0}{R_0}\)—relative distance to the center of the propeller; \(\overline{R}=\dfrac{R}{R_0}\); \(\overline{V}=\dfrac{V}{\omega R_0}\sqrt{V_0^2+R^2}\), \(\overline{p}=\dfrac{p}{R_0}\)—radius of the nonworking part of the propeller: \(\overline{a}=\dfrac{a}{R_0}\), \(dP(\overline{R})=\dfrac{dP(\overline{R})}{\rho n_s^2D_0^4}\), \(dT(\overline{R})=\dfrac{dT(\overline{R})}{\rho n_s^3D_0^5}\)—elementary power; \(n_s\)—number of revolutions per second; \(D_0\)—propeller diameter.

In addition, here \(J_{mz}\) is a Bessel function of the first kind, \(\vartheta\) is the listening angle, i.e., the angle between the radius vector \(r_0\) from the center of rotation to the receiving point and the direction of flight of the aircraft. Formula (6) is a computational formula for sound pressure. It can be considerably simplified if one uses the mean theorem instead of laborious integration along the propeller radius.

Then the expression for the intensity equation will take the form:

\[ I_m=S(mz,\overline{U}_c,\vartheta)+10\lg\left[(0.1785+0.1347\lambda)K^2V_{nm}+ \right. \]

\[ \left. +\left(0.3055\frac{K_{Qm}\cos\eta_m}{za}\beta -0.4053\frac{K_{pm}\cos\varepsilon_m\overline{U}_c\cos\vartheta}{za}\alpha\right)^2\right] -20\lg\overline{r}_{\Delta}, \tag{7} \]

where \(\alpha=\dfrac{P}{\rho n_s^2D_0^4}\)—thrust coefficient; \(\beta=\dfrac{T}{\rho n_s^2D^5}\)—power coefficient; \(\lambda=\dfrac{V_0}{n_sD_0}\)—relative advance of the propeller; \(\overline{r}_{\Delta}=\dfrac{r_0}{R_0}\cdot\dfrac{1}{\Delta}\)—“reduced distance” \(\left(\Delta=\dfrac{\rho}{\rho_0}\)—relative density of the air\(\right)\); \(S(mz,U_c,\vartheta)\)—“radiation function,” determining the principal dependence of the sound-intensity level on speed and on the number of blades.

It must be pointed out that formula (7) was obtained for a value of the mean radius \(\overline{R}_{\mathrm{cp}}=0.65\), which for large values of the numbers \(mz\) must be increased, and formula (7) will change accordingly.

With a certain degree of accuracy one may adopt the following values of the mean radius for various numbers: \(mz=2\), \(\overline{R}_{\mathrm{cp}}=0.65\); \(mz=3\), \(\overline{R}_{\mathrm{cp}}=07\); \(mz=4\), \(\overline{R}_{\mathrm{cp}}=0.75\); \(mz\ge 6\), \(\overline{R}_{\mathrm{cp}}=08\).

For greater accuracy of calculation it is, of course, necessary to use formula (6), since the magnitude of the sound pressure is very sensitive to changes in \(\overline{R}_{\mathrm{cp}}\); however, for practical calculations in estimating the noisiness of propellers, formula (7) can be successfully applied. The computation is considerably facilitated by the fact that

graphs have been compiled for the radiation function \(S\) and for the harmonic coefficients, as functions of the parameters of the propeller arc.

Before proceeding to a further exposition of the regularities of rotational sound and to a discussion of their agreement with experimental data, let us pause briefly on the theoretical works that preceded the exposition given above.

In 1936 L. Ya. Gutin\(^{24}\) proposed a very ingenious and simple theory for calculating the sound field of a propeller, which was the first serious attempt at a rational study of this problem. The author proceeded from the following premises. It is known that each element of a propeller is acted upon by a thrust force and by a force of resistance to rotation. On an element of the medium adjacent to the propeller blade there act periodic disturbing forces equal in magnitude to the thrust and resistance-to-rotation forces and opposite to them in direction.

In determining the sound field Gutin uses the expression for the sound potential of concentrated forces:

\[ \psi=\frac{i}{4\pi\rho kc}\left(X\frac{\partial}{\partial x}+Y\frac{\partial}{\partial y}+Z\frac{\partial}{\partial z}\right)\frac{e^{-jkr}}{r}, \tag{8} \]

where the force components \(X, Y, Z\) are calculated as functions of the thrust and of the resistance moment. If the observation point is far from the plane of rotation, the following expression is obtained for the amplitude of a certain harmonic of the sound pressure:

\[ |p_m|=\frac{mz\omega}{2\pi c r_0} \left\{ \left[ -P\cos\vartheta+\frac{c}{\omega R_1^{2}}M \right] J_{mz}\left(kR_1\sin\vartheta\right) \right\}, \tag{9} \]

where \(P\) is the propeller thrust force; \(M\) is the moment of resistance; \(R_1\) is a certain mean radius, taken by Gutin to be equal to \(0.7\) of the screw radius and dependent on the distribution of the aerodynamic forces along the blade.

Calculations of the sound field by this very elegant formula gave satisfactory agreement with Kemp’s measurements for the fundamental tone; for the second and higher harmonics the theory disagrees with experiment, and this discrepancy increases appreciably with increasing harmonic number. This disagreement follows from the imperfection of the theory and of the calculation.

The calculation of the sound field is carried out here by formula (8), which represents only one part of the solution of the wave equation, namely the part determined by the forces acting on the medium [compare with formula (4)]. The second part of the solution, i.e. radiation of zero order, is not taken into account here. An analysis of the radiation of zero order shows that it must be taken into account both for the fundamental tone and, especially, for the harmonics. The significance of zero radiation will be clearly shown below. In addition to this imperfection of the theory, the calculation of the higher harmonics suffers from inaccuracy owing to

arbitrariness in specifying the aerodynamic forces over the blade width. In practically encountered cases these errors may already be substantial for the first and second harmonics. From the fundamental point of view these shortcomings of Gutin’s theory were overcome in 1937–1938 in the laboratory of N. N. Andreev^21.

However, the cumbersomeness of the calculation and the absence, in the theory constructed here, of a direct connection between the acoustic characteristics of the propeller and its aerodynamic parameters made it impossible to use this theory for practical calculations. On the other hand, however, it played a certain role in the creation of the theory set forth above, in which the space surrounding the propeller was divided into two parts: the part immediately adjacent to the propeller and the part remote from it. The calculation of the pressure and velocities near the propeller was carried out according to the equations of plane motion of an ideal incompressible fluid, with the Prandtl–Glauert correction for compressibility introduced. The sources of disturbances thus computed on a certain boundary surface served as boundary conditions in solving the wave equation for the space remote from the propeller. In a more exact calculation the boundary surface was chosen at such a distance from the plane of rotation of the propeller that, on the one hand, calculation by the equations of plane motion was possible and, on the other hand, the disturbances on it would be so small that the linear acoustic equations would be valid. The calculation of the sound field led to satisfactory agreement with experiment.

Somewhat later, in 1938, Ernsthausen and Wilms^25 proposed a calculation of rotation sound based on analogous assumptions. They made use of the complete solution of the wave equation, i.e. formula (3). Expressing the normal velocity through the blade thickness, and the pressure difference through the circulation around the corresponding blade elements, they obtained the following formula for the potential at a certain point of space:

\[ \left|\varphi_{mM}\right| = -mz\omega K_m \left[ \int_{0}^{R} \delta f_1(R) J_{mz} \left( mz\frac{U}{c}\sin\vartheta \right) R\,dR + j\frac{2\pi}{mc} \int_{0}^{R_0} \Gamma\cos\vartheta\, f_2(R) J_{mz} \left( mz\frac{U}{c}\sin\vartheta \right) R\,dR \right], \]

where \(K\) is the coefficient of expansion in a Fourier series of a rectangular impulse; \(f_1(R)\) and \(f_2(R)\) are certain distribution functions of the normal velocities and circulation along the blade; \(\delta\) is the blade thickness; \(U\) is the circumferential velocity.

The directional characteristic calculated by this formula turns out to be symmetric with respect to the plane of rotation. The reason for this discrepancy between theory and experiment lies in the assumptions made.

by the authors, the principal one of which consists in an incorrect conception of first-order radiation. Having quite correctly expressed the discontinuity of the potential in terms of the circulation of an element, or its lift force, they did not decompose it into the thrust force and the rotational-resistance force, but directed it along the direction of the propeller thrust force. This circumstance was what chiefly caused both the qualitative and the quantitative error.

In 1938 Deming\(^{26}\) considered the problem of the sound of a propeller with elements without lift, i.e., the problem of zero radiation. He proceeded from Rayleigh’s relation

\[ d^2\varphi=\frac{1}{2\pi}\frac{d\varphi}{dn}\frac{e^{-jkr}}{z}\,dS \]

and arrived at the following expression for the amplitude of the sound pressure

\[ |p_m|=\frac{\rho_0 mzHR_0 A_{mz}}{z_0}\Pi K^2U^2J_{mz}(kR_0\sin\vartheta), \]

where \(H\) and \(K\) are empirical coefficients determined from experiment; \(\Pi\) is a certain function of the shape and dimensions of the profile, which Deming approximately assumes equal to \(\Pi=\dfrac{a}{b}\left(a-\dfrac{\delta}{2}-\right.\) half-thickness, \(b\)—chord); \(A_{mz}=\dfrac{mz^2b^2}{4\pi R^2}\) is the coefficient of expansion of a rectangular impulse in a Fourier series.

Experimental verification of this theory showed good agreement between calculation and experiment for the first three harmonics. For higher harmonics Deming introduces additional empirical factors in order to obtain agreement with experiment.

It will not be superfluous to note that the possibility of comparing the theory of zero radiation with measurements of the sound from a propeller with symmetrical profiles is hampered by the circumstance that this propeller, absorbing power for its rotation in a real medium, will also produce sound due to first-order (force) radiation. Deming did not pay attention to this circumstance.

Later Deming\(^{27}\) made an attempt to improve Gutin’s theory, which amounted to the fact that, having expanded the Bessel function in a series and introduced the distribution of aerodynamic forces along the propeller radius, he integrated the expression for the amplitude of the sound pressure over the radius. The difference thereby obtained from Gutin’s theory gives slight quantitative discrepancies, but, of course, does not change the calculation in principle and does not eliminate the known shortcomings of Gutin’s calculation. Of great interest are Deming’s measurements, made for comparison with theory, which we shall also use below for the same purpose.

Deming carried out his measurements on a model of a two-bladed propeller, 2.9 m in diameter, whose blade had in its sections

of the \(RAF\) profile and was set at an angle of \(5^\circ\). Measurements were made at a distance of \(25\ \text{m}\) from the propeller axis; when directivity diagrams were taken, the number of revolutions was kept constant, \(n = 1700\) rpm, and measurements were made around the circumference at intervals of \(15^\circ\). The sound was received by an electrodynamic microphone and, after appropriate amplification, was recorded by an automatic tone analyzer. The apparatus was carefully calibrated and permitted measurement of absolute values of sound pressure with an accuracy of up to \(5\%\). The measurement conditions were specially selected. In Fig. 13 a polar diagram is plotted from Deming’s measurements for one of the three

Fig. 13. Polar diagram for the fundamental tone of rotational sound.
\(1\) — experiment, \(2\) — Gutin’s theory, \(3\) — Nepomnyashchii’s theory.

harmonics of rotational sound (the first harmonic), together with the corresponding calculated curve computed according to the theories of Gutin and Nepomnyashchii, with the sound pressure recalculated in terms of intensity levels and plotted in decibels. For the second harmonic an analogous curve is obtained; the directivity of the third harmonic looks somewhat different (Fig. 14).

Consideration of these curves leads to the conclusion that the theory satisfactorily describes the details of the directivity characteristics and gives good quantitative agreement between calculation and experiment, especially in the directions of maximum sound radiation.

It also follows from these same curves that radiation of zero order must be taken into account even for the fundamental tone; it explains the actual absence of a zero value of the sound-pressure amplitude in directions other than \(\vartheta = 0^\circ\) and \(\vartheta = 180^\circ\), whereas according to Gutin’s theory a secondary drop of the sound to zero is obtained in the direction \(0^\circ < \vartheta < 180^\circ\).

In addition to the polar diagrams, Deming made measurements of the sound pressure for four harmonics at various rotational speeds.

It is of interest to compare the relative course of the dependence of the sound pressure on the circumferential velocity, as obtained from calculation and from experiment.

Fig. 14. Polar diagram for the second overtone.
1—experiment, 2—Gutin’s theory, 3—Nepomnyashchii’s theory.

It was indicated above that the dependence of sound on the circumferential velocity and on the number of blades is determined by the radiation function \(S(mz,\overline{U}_c,\vartheta)\). In Fig. 15, for illustration, the radiation function \(S\) is given for the value of the listening angle \(\vartheta = 90^\circ\) (plane of rotation).

From the graph it is evident that, for the given direction \(\vartheta\), the dependence on the circumferential velocity is different for different values of \(mz\), irrespective of why \(mz\) changed—because of the number of blades or because another harmonic was considered. Deming measured different harmonics of the spectrum at different angles \(\vartheta\). In Fig. 16 are plotted Deming’s experimental dependences and the corresponding theoretical curves of the dependence of sound on velocity, determined for the angles \(\vartheta\) and numbers \(mz\). The agreement is obtained very good—

Fig. 15. Graph of the radiation function \(S(mz,\overline{U}_c,\vartheta)\) for the listening direction \(=90^\circ\).

…except at large values of the velocities, where the experimental curves run somewhat above the theoretical ones. This deviation may be explained by the influence of compressibility at high speeds, which is not taken into account by the theoretical curves presented, but, generally speaking, can be included in the present theory. In the region of low speeds, the dependence of the intensity level on speed is stronger than at high speeds (Fig. 16). It is therefore desirable to compare the theory with measurements at low speeds.

Fig. 16. Dependence of the level of intensity of rotational noise on the circumferential tip speed for different numbers \(mz\) and angles \(\vartheta\).
\(1\) — experiment, \(2\) — theory

In Fig. 17 are given the experimental dependences of the intensity level on the circumferential speed, obtained on models of two-bladed propellers at various blade setting angles \(\gamma\). The intensity levels presented can be referred only to the fundamental tone, i.e. to the number \(mz=2\) and the direction \(\vartheta=90^\circ\). The curves in Fig. 17 were obtained in N. N. Andreev’s laboratory on a metal model, \(0.8\,m\) in diameter; there are analogous determinations by Obata, Kawada, U. Iozida and I. Iozida on a meter-scale propeller model. The graph includes a segment of the theoretical curve, which, as is seen, is sufficiently parallel to the experimental curves; this indicates agreement between theory and experiment also at low rotational speeds.

Fig. 17. Dependence of the intensity level of the rotational noise of a propeller model on the circumferential tip speed.

These comparisons of experimental dependences with the radiation functions \(S(mz,\overline{U}_c,\vartheta)\) indicate agreement between theory and experiment for the dependence of the sound on the circumferential velocity \(\overline{U}_c\), the listening angle \(\vartheta\), and the number \(mz\). The dependence of the sound intensity on the pitch, thickness, and width of the blade is determined mainly through the change, due to these parameters, in the power coefficient \(\beta\) and thrust coefficient \(\alpha\), which enter linearly into the expression for the sound pressure.

The blade width, moreover, has a certain influence on the distribution of harmonics in the spectrum. This influence manifests itself through the ratio of the width of the boundary impulse \(\tau\) to the period of the expansion \(T\). Roughly speaking, for the first harmonics the sound pressure changes proportionally to the change in

\[ \frac{\tau}{T}, \]

i.e., it increases approximately in proportion to the width of the blade.

Let us return to the curves in Fig. 15.

From them one can draw a very curious conclusion: at small circumferential velocities only the fundamental tone predominates in the sound spectrum; at large velocities the relative weight of the higher harmonics increases considerably, and they may predominate in the propeller sound spectrum. This property of the sound of rotation is confirmed by all known measurements. Ernsthausen and Wilms\(^{25}\) confirmed this fact by their measurements of propeller sound on a flying airplane. The measurements were made with the aid of a Siemens and Halske rapid-action spectrometer. Two-, three-, and four-bladed propellers with different circumferential velocities were tested.

Fig. 18. Directional characteristic of the sound of a four-bladed propeller, measured from the ground during airplane flight.

The directional characteristics were recorded while the airplane was moving in a straight line, with the angle indicator being automatically linked to the spectrometer. Subsequently it was necessary to reduce the readings to the same distance. As an example we give the directional characteristic for a four-bladed propeller (Fig. 18). The authors do not give the rotational speed, but characterize it as “medium.” It is probable that the tip speed was of the order of \(0.8\) of the speed of sound.

From Fig. 18 it is seen that the directional characteristics are unusually (for the case of propeller operation on the ground) turned in the direction of the airplane flight. This displacement of the directional characteristic can be explained by a certain lag of the sound relative to the moment

fixing the aircraft from the ground and characterized by the angle \(\Delta \vartheta = \dfrac{V_0}{c}\) (\(V_0\) is the flight speed).

To construct the true directivity pattern it is therefore necessary to introduce this angular correction, which at ordinary flight speeds may reach \(15\text{–}20^\circ\).

In the case of a moving aircraft it is also necessary to introduce a correction for the change in frequency due to the Doppler effect, which may reach \(\pm 20\text{–}30\%\) in cases of practical interest.

In conclusion it may be said that, although the basic properties of propeller sound and its regularities have been studied quite comprehensively, there are still many important and interesting questions requiring the attention of both theoreticians and experimenters. Among these questions one may perhaps include: the study of the intensity of vortex sound, acoustic phenomena at supersonic speeds of individual blade elements, a more systematic study of propeller sound on an aircraft in flight, questions of various kinds of interference, and, of course, within the field of view of every investigation there must lie questions of the propagation of sound in the real atmosphere.

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1. This phenomenon is known in the literature as the “grid effect.” ↩↩↩