Utilizing the phenomenon of diffraction for noise protection of roadside objects

3.1 Theoretical concepts

In the process of designing sound-absorbing barriers, it’s crucial to take into consideration that the noise protection effect manifests only within the acoustic shadow zone. Noise originating from a transportation highway can reach any point in space in the form of sound transmitted directly through the body of the screen and in the form of diffracted sound enveloping the upper and side edges of the screen.

Implementing diffraction effects in the Fraunhofer zones and using Fresnel lenses allows for the concentration of vibrations.

The laws of reflection and refraction of sound waves are explained using the Huygens principle to determine the position of the wavefront at any given time as the envelope of all secondary waves created by elementary sources. The sources of secondary waves are the points reached by the wavefront at the preceding time.

To explain the phenomena of diffraction, it’s necessary to apply the Huygens-Fresnel principle, according to which the excited wave is represented as the result of a superposition of coherent secondary waves. In this case, there is a departure from the laws of geometric optics. When sound is scattered, the sound wave from the source is transformed into multiple waves caused by numerous obstacles encountered in its path.

The propagation and absorption of sound waves in a homogeneous medium are accompanied by energy dissipation, i.e., a decrease over time due to conversion into other forms of energy. In acoustics, energy dissipation occurs due to internal friction and heat conduction.

The dependence of the frequency of sound waves ν perceived by the receiver of vibrations on the speed of the source of vibrations, which is a moving vehicle, is expressed by the Doppler effect.

where c is the speed of sound waves in a stationary medium, ν ₀ is the frequency of oscillations emitted by the source, and θ is the angle formed between the vector u and a vector connecting the sound source and the receiver.

The legitimacy of using this formula for the problem under consideration is expressed through the correspondence of its dimensions to the units of measurement of the parameters included in it. Units of measurement of a physical quantity are conventionally chosen quantities having the same physical meaning as the measured quantity. A system of units of measurement is considered to be a set of units of measurement defined in a certain way for all quantities considered in a given field. The formulas of the dimensions of a physical quantity are the ratio defining the relationship between the unit of measurement of this quantity and the basic units. Dimensionality of the frequency of oscillation of the source [ν ₀] = [s⁻¹]. Dimensionality of the speed of sound in a dynamically moving extended medium [u] = [m s⁻¹]. Dimensionality of the speed of sound in a stationary medium [c] = [m s⁻¹], cos θ – dimensionless value. Using this information, after performing the appropriate calculations, according to the right part of this formula, we obtain the value of this dimensionality according to the right part of this formula [s⁻¹]. The left part of this formula also has the same dimensionality value [ν] = [s⁻¹]. Equality of values of dimensions in both parts of this equation testifies to the correctness of using this formula in the considered problem.

The issue of sound wave scattering by individual scatterers of a noise screen can be examined based on the wave equation of the potential ψ with unit amplitude.

where Δ is the Laplace operator.

The solution of the wave equation can be represented as

where i = − 1 , k is the wave vector, equal modulo the wave number 2π/λ and having the direction of normal to the wave surface, r is the radius-vector of a point on the wave surface.

Function (3) describes some wave, the form of which is determined by additional conditions. The boundary condition for scattering can be interpreted as the Neumann condition, which states that the normal component of the sound wave’s velocity becomes zero when it encounters the surface of the noise screen.

In this context, ψ and ψ ₀ represent the field potentials in the external environment and within the noise-absorbing screen, while ρ and ρ ₀ refer to the density of the material in the environment and the screen, respectively, n is the external unit normal.

The acoustic efficiency of the screen is determined by its geometric dimensions, encompassing its height, length, and the distance between the screen and the traffic artery.

Fraunhofer diffraction arises when plane waves diffract under conditions where the source of oscillation and the observation point are situated far beyond the obstacle causing the diffraction, significantly exceeding the radiation’s wavelength. In this phenomenon, rays directed toward the observation point and striking the obstacle align to form nearly parallel rays.

To practically implement Fraunhofer diffraction, the oscillation source is positioned at the focal point of a converging lens, while the diffraction pattern is observed in the focal plane of a second converging lens positioned behind the obstacle (Figure 1).

Figure 1

The principle of active sound absorption control based on Fraunhofer diffraction.

In this scenario, the placement of the converging lens in front of the screen should be chosen so that the moving extended source of traffic noise aligns with its focus. In these circumstances, the wave surfaces of the incident wave, responsible for generating acoustic noise, the plane of the noise shield, and the focal plane of the second collecting lens will become mutually parallel. The secondary waves generated within the elementary regions of the portion of the open slot for sound, emitted at an angle φ relative to the optical axis of the lens, gather at a specific point M. The oscillation at point M with the coordinate x is given by:

where A ₀ represents the oscillation amplitude at the center of the diffraction pattern, φ denotes the diffraction angle measured from the outward normal to the incident wavefront, b stands for the gap width, λ is the wavelength, and ωt − 2π/λ xsin φ corresponds to the oscillation phase.

When φ equals 0, the fluctuations converging at point M will be in phase, resulting in an amplitude equal to the algebraic sum of the amplitudes of oscillations from the elementary zones. The resulting oscillation at point M is

where A 0 sin [ ( π / λ ) b sin φ ] ( π / λ ) b sin φ is the amplitude at the observation point.

The intensity of oscillation I is directly proportional to the square of the amplitude. The graph depicting its relationship with the diffraction angle φ at the characteristic points of the wavelength is presented in Figure 2.

Figure 2

Intensity distribution in Fraunhofer diffraction.

The majority of the noise that has passed through the slit is concentrated within the central maximum. The angular width of the central maximum φ _m is determined by the expression

The positioning of the intensity minima is determined by the ratio of the slit width b to the wavelength λ. Given that |sin φ| ≤ 1, this implies that k ≤ b/λ. Under these circumstances, the presence of the first converging lens becomes essential, as the incident wave on the slit needs to be planar. If the slit width is considerably smaller than the focal length of the second converging lens, the rays extending to point M from the slot’s edges will remain parallel. Consequently, within the Fraunhofer diffraction observation setup, when generating noise from moving vehicles, the use of a second converging lens can be omitted.

The diffraction pattern maintains symmetry concerning the center of the lens. When the slit is shifted within the screen’s plane, the diffracted pattern remains stationary. Alternatively, with a fixed slit, adjusting the lens’s position results in a shift of the diffraction pattern. The configuration of the diffraction pattern is defined by the value of the wave parameter:

where L represents the distance from the plane of the slit to the parallel observation plane.

If P < 1, then the wave’s amplitude at the observation point situated opposite the center of the slot remains the same as in the absence of the slot. However, if P > 1, the primary intensity maximum will be positioned opposite the middle of the slit.

In scenarios involving two slits, the diffraction maxima will exhibit greater brightness compared to the case of single-slit diffraction. Within a diffraction grating composed of N identical slits uniformly spaced, the intensity of the principal maxima grows proportionally to N².

The consideration of boundary conditions, contingent on the nature of the obstacles, necessitates approximate methods due to the intricacy of solutions. One such method is based on Fresnel Zones of Focused Zones (FZF). FZF transforms a diverging spherical wavefront into a converging one. The wavefront emanating from a point source of oscillations S ₀ is partitioned into transparent and opaque annular zones, where the difference in distance from the outer radii of adjacent zones to the point of wave concentration is half the wavelength, λ/2. Subsequently, these zones on the spherical surface are projected onto a plane. The diagram depicting the amplitude of the acoustic flat Fresnel zone, showcasing wave concentration due to maximum interference of secondary waves, is illustrated in Figure 3.

Figure 3

The principle of dividing Fresnel zones on a spherical surface.

Oscillations from even and odd Fresnel zones are in antiphase, causing mutual attenuation; conversely, placing a plate in the path of a sound wave that would encompass all even and odd zones results in a substantial increase in oscillation amplitude. The outcome of diffraction hinges on the extent of zone overlap. For considerable values, diffraction effects hold minor significance, and the laws of geometric optics aptly describe intensity distribution. Situations where approximately 10 zones remain open correspond to Fresnel diffraction, yielding intricate intensity distribution. Within the central section of the image, one can observe intensity minima and maxima. Graphical computation of diffraction can be accomplished using the Cornu spiral (Figure 4).

Figure 4

Diffraction calculation from a slit using the Cornu spiral.

The Cornu spiral consists of two branches symmetrically arranged around the origin, which asymptotically approach the poles. The parametric equation representing the spiral is given by Fresnel integrals and takes the following form:

where parameter υ = 2 λ L ( x − x 0 ) , x ₀ is the observation point coordinate, and x is the observation point coordinate.

The meaning of the parameter υ is that | υ | gives the arc length of the curve Cornu, counted from the origin.

Points F ₁ and F ₂ to which the Cornu spiral asymptotically approaches at υ → ± ∞ are the focal points or poles of the Cornu spiral.

The resultant oscillation generated by the section of the wavefront confined between the straight lines x = x ₁ and x = x ₂ is determined by the formula

were A ₀ is the oscillation amplitude at a fully open wavefront, and B ₁ B ₂ is the length of the segment connecting two points on the spiral corresponding to the values.

Utilizing the Cornu spiral enables the determination of oscillation amplitudes for points situated at various distances from the boundary of the geometric shadow.

The zone plate also possesses imaginary focal points, enabling it to function concurrently as a combination of converging and diverging lenses.

The conventional demarcation between Fresnel and Fraunhofer diffraction is known as the Rayleigh distance. At this distance, when a plane monochromatic wave is incident on the central point of a round hole, only the first zone opens. In the Fraunhofer region, the diffraction patterns increase linearly with the distance from the slit. The angular size of the central diffraction maximum in the far zone is determined by the ratio of the wavelength to the diameter of the hole or slit. At the Rayleigh distance, this angular sector possesses a linear dimension equivalent to the width of the slit.

3.2 Rationale for the experiment

A fundamental requirement for practically harnessing the phenomena of diffraction and interference for noise mitigation alongside road structures is the establishment of a planar incident wavefront. When situating a point source at infinity, an alternative to achieving a plane wave at the FZP’s focus yields minimal energy concentration.

The attainment of a plane incident wavefront can be enhanced through the utilization of a Pictet mirror. The Pictet mirror, a concave spherical metal mirror equipped with supports, facilitates adjustments in height and inclination relative to an object positioned before it. The focal point lies at a distance of half the mirror’s radius from its midpoint. If the sound source is positioned at the mirror’s focal point, the reflected rays emerge as parallel beams, ensuring that the entire mirror reflects and amplifies the acoustic signal when observed along the optical axis. Incrementally rotating the mirror alongside the vibrations source positioned at its focus, to direct reflected sound rays towards different spatial regions, allows for a pronounced amplification of sound vibrations when the observer aligns with the mirror’s axis. Progressing along the oscillation source’s positional axis leads to the identification of the Pictet mirror’s focal point. By introducing a second concave mirror, similar to the Pictet mirror, within the path of rays emanating from the latter, the position of the second mirror’s focus can be determined using a sound-reflecting screen. The point of sound amplification will coincide with the Pictet mirror’s focus.

The experiment proposes fabricating a Pictet mirror shaped as a paraboloid, with a FZP affixed to its open section. The selection of a paraboloid aims to leverage its inherent property: a parallel beam of rays to the parabola’s axis, upon reflection, converges at its focus, and conversely, sound vibrations from a focal point source on the paraboloid are reflected by its inner surface, transforming into parallel rays. The acoustic signal also sustains a uniform phase. In this context, the distance between the Pictet mirror’s position and its focus aligns with the focal length of the FZP, and the paraboloid’s focus coincides with that of the FZP.

Of particular interest is ascertaining the most efficient acoustic energy conversion processes within the Pictet mirror. When the Pictet mirror serves as an absorber, only a fraction of the sound wave’s energy impinges on the FZP within the solid angle where the plate rests, with its apex positioned on the paraboloid’s axis. Conversely, when utilized as a reflector, the process occurs beyond the solid angle within the remaining volume of the sphere. When employing a point source, the vibrational energy is uniformly disseminated, generating a spherical wavefront.

It is recognized that sound vibration energy is directly proportional to the area of its dispersion. Hence, to assess the efficacy of absorption and reflection processes within the proposed paraboloid-FZP configuration, evaluating the ratio of the energy section areas involved in these processes is necessary.

where W _ref1 is the energy of the reflected wave, Wabs is the energy of the absorbed wave, S represents the reflective area, and S _segm symbolizes the area of a spherical segment that characterizes the energy of sound vibrations during absorption. ρ denotes the outer radius of the largest zone within the FZP, while R corresponds to the focal length of the FZP.

In this form of representation, the value of K is a dimensionless quantity; i.e., the dimensionality of the numerator and denominator of the fraction must coincide.

Hence, the coefficient K signifies the proportion of reflection energy to absorption. Given that S > S _segm, when utilizing a paraboloid as an absorber, a substantial portion of energy dissipates, whereas when employed as a reflector, all the energy converges onto the FZP. In the configuration involving the paraboloid as a reflector, all energy is channeled toward the FZP, resulting in an amplification of the incident sound wave’s energy by a factor of K.

The amplitude A of a plane wave, propagating along the x-axis, undergoes alteration in accordance with the exponential law:

where A ₀ signifies the amplitude of sound at the vibration source, and γ represents the absorption coefficient.

The material’s absorption coefficient is ascertained by the quotient of absorbed sound wave energy to incident wave energy. In accordance with Bouguer’s law, the intensity of incident oscillations I ₀ diminishes exponentially within the absorbent material, as follows:

If the surface dimensions are significantly larger than the incident wavelength and possess considerable thickness, the absorption coefficient is determined via the reflection coefficient. Sound-absorbing materials encompass substances with an absorption coefficient exceeding 0.3. This absorption coefficient exhibits dependence on the frequency of the sound wave.

The quantitative representation of the reflection coefficient for the noise component is attainable through analyzing the pressures of the incident p ₀, reflected p _refl, and transmitted p _p waves, along with their respective velocities υ ₀, υ _refl, and υ _p.

where ρ ₁ and ρ ₂ denote the density of the medium through which sound propagates and the material composing the noise barrier, respectively. Similarly, c ₁ and c ₂ represent the velocities of sound vibration propagation in air and the soundproofing material.

Considering these relationships, the reflection coefficient α is determined as the ratio of acoustic impedances ρc between air and the environment.

To mitigate the influence of the noise component resulting from multiple reflections of the incident wave from secondary elementary sources, the model experiment employs a combination of a reflector and an absorber within the paraboloid structure, accompanied by the installation of a FZP on the exposed section of the paraboloid. The parameters and fundamental geometric relationships of this revolutionized paraboloid are selected based on the analogy to an open antenna with a mirror in radar systems. In such apparatus, shielding serves as a constructive method for localizing the electromagnetic field, designed to safeguard against external interference and radiation localization. The shielding effect is determined by the cumulative attenuation of energy due to material thickness absorption and the differential reflection caused by distinct acoustic characteristics of the medium.

For determining the geometric dimensions of the expanding segment of the paraboloid when coupled with the FZP, one should employ a definition of a parabola stating that the set of points equidistant to the focus and the directrix should be utilized. A coordinate system is established at the vertex of the parabola. The line extending through the focus and parallel to the directrix intersects the parabola at two distinct points. Half of this focal chord is equivalent to the parabola’s parameter, denoted as “P,” and its abscissa is P/2.

Let “f” denote the distance between the parabola’s focus and its apex. In terms of “F,” the focal length of the parabola, and “r,” the radius of the last Fresnel zone, which also serves as the focal radius of the point belonging to the parabola, the relationship can be expressed as follows:

where

Since x = r, considering the notation introduced earlier, it becomes feasible to determine the distance at which the FZP is positioned within the paraboloid.

By solving equations (21) and (22) simultaneously, we can derive the corresponding parabola equation.

The scientific rationale behind conducting the experiment to observe diffraction effects within the Fresnel and Fraunhofer zones was based on the notion that if the source of oscillations is positioned at the focal point of the FZP, and the plate itself would function as an acoustic collecting lens with its distinct focus “F”. This focus can be defined in a manner similar to that of refractive lenses.

where d stands for the object distance, and f denotes the image distance.

Substituting d = F ₁ yields f = ∞, signifying that a plane wave should emerge from the FZP and propagate along its principal axis.

By substituting the components of this formula with the parameters of the FZP, f ₁ = r ₀ and d = R, we obtain

The radius ρ _n of the zone in the Fresnel plate is determined as:

where λ signifies the wavelength of the sound wave.

In this expression, the value r ₀ signifies the distance between the point of maximum amplitude and the zone plate, while R represents the distance from the vibration source to the zone plate.

The area of the Fresnel zones is determined by the following expression:

where a is the distance from the source to the outer edge of the zone, b is the distance to the focus area.

The zone plate multiplies the intensity of the vibrations, acting like a converging lens.

4.1 Used equipment

During the simulation experiment, the following equipment was utilized:

• Benetech GM1357 (AR824) sound level meter;

• Mini portable speaker FM/mp3;

• TD-V26 speaker functioning as a point sound source;

• Computer;

• Amplitude annular FZP employed as the source of oscillations; and

• Pictet mirror.

The parameters of the FZP (as detailed in Table 1) were computed for a frequency of 6,000 Hz and a focal length of 1 m.

The research employed methods for studying the phenomena of diffraction and interference on sound gratings as research techniques. The spatial resolution of this particular FZP was established via the Rayleigh criterion, using the width of the outermost opaque ring denoted as Δ.

According to Table 1, we have r ₇ = 0.6298 m, r ₈ = 0.6733 m, and Δ ₇₈ = 0.0435 m. Taking into account that the wavelength for 6,000 Hz is 5.7 cm, the spatial resolution in terms of the wavelength becomes Re = 0.76λ.

4.2 Design features of the installation

In order to achieve a flat front and concentrate oscillations, a parabolic absorber was crafted from multiple screens of acoustic foam rubber. At the apex of the paraboloid, there was an aperture through which wires were affixed to the speaker. The speaker was capable of moving freely along the symmetry axis of the parabolic absorber. This enabled adjustments to the sound source’s position, thus extending the measurement range. Throughout the experiment, efforts were made to maintain a distance from the edge of the zone plate located on the exposed part of the paraboloid that equaled the plate’s focal length. This paraboloid configuration allowed for dual functionality, serving as both an absorber and a reflector of sound energy, particularly when considering the primary focusing, frequency, and shaping characteristics of the diffractive elements within the FZP.

The FZF represents the simplest diffractive focusing element, and exploring its information properties, along with investigating its limiting focusing and frequency attributes, facilitates a deeper comprehension of its nature and practical applications when utilizing a Pictet mirror (Figures 5 and 6).

Figure 5

Photograph of the FZP utilized in the experiment.

Figure 6

Appearance of the experimental setup.