Resonant Radiation of Boundary with a Travelling Distribution of the Field

: The problem of acoustic monochromatic radiation by boundary with a traveling distribution of phases of normal vibrational velocities is considered. It is shown that when the spatial frequency of the traveling phase of normal velocities approaches the wave number in the medium, the energy transfer from boundary into a “sliding” (with respect to the boundary) sound wave can resonantly increase to a value many times greater than the energy transfer from of the in-phase boundary, correspondingly, into the normal one (with respect to the boundary) sound wave at the same modules of amplitudes of vibrational velocities of boundary. In addition, the resonant energy transfer of the boundary into a "sliding" wave is the greater, the larger the wave dimensions of the radiating pattern on boundary. It is shown that when a similar traveling distribution of sound pressure (instead normal velocity) is specified at the boundary, there is no resonance. The influence of the curvature of the radiating boundary on the above phenomenon of resonant radiation was studied. It is shown that the resonant radiation of the boundary with given running phases of normal velocities generates a tangential (with respect to the boundary) constant in time radiation reaction force. It is shown that for the case of a linear chain of equidistant monopoles (or pulsing spheres separated from each other by medium) with a traveling phase (a traveling wave antenna) of their oscillatory velocities, the resonance does not appear.


Introduction
It is usually well-known that there is radiation (acoustical or electromagnetic monochromatic field in the far zone) at the spatial frequency k -wave number in the medium) of sources, and at the spatial frequency 0 0 k h  of sources radiation is absent or very small [1][2][3][4][5][6][7].This is probably why the researchers did not consider this area in sufficient detail.Below, using several examples of very simple boundary value problems [8], it is shown that radiation power with an increase in the spatial frequency k .This means the phenomenon of resonance, which is of particular interest to any physicist, especially since we are talking about such an important physical quantity as the surface density of the radiated power.On the other hand, it is known that the traveling amplitude distribution of radiating elements (separated from each other by the medium) in traveling wave antennas does not lead to resonant radiation [9].In addition, many highly educated researchers, without delving into details (on the basis of the hastily applied relationship between pressure and velocity through the impedance of the medium), are inclined to declare that there is no fundamental difference between boundary radiation with a given pressure and boundary radiation with a given normal velocity.Thus the purpose of this work is to fill the above-mentioned small, but very common (as experience shows) gaps in understanding the process of wave radiation.
First, let's consider a sound field excited in a compressible nonviscous linear medium (in a half-space 0  z ) by a traveling distribution (with a traveling phase [8]) h of normal vibrational velocities (Fig. 1-a).Below we will consider the radiation of various patterns (as modifications of (1)) with a traveling phase.

Plane (acoustical field)
Particle velocity where  is the mass density of medium at where ) , ( exp ) , , ( . This is the work performed by a section of a strip of unit width (along the axis " " x ) and unit length (along the axis y ) and averaged in time over a period ).This is due to an abrupt change in the eigenvalue of the boundary value problem from purely real to purely imaginary when going from For the reactive power flow, we obtain the expression Normalizing to the value of the energy transfer of the boundary ) .Lines P (Pointing vector) of power flow go out of the plane 0 = z at an angle of inclination However, when the traveling pressure , there is no resonance in the following corresponding functions (instead (13), ( 14)) (Fig. 2-c (contrary to superficial judgment about the relationship between pressure and velocity through the impedance of the medium, see Figs.
Here is another explanation of the effect.When

Plane (electromagnetic waves).
The resonance characteristics (13), ( 14) described above are a property of the wave equation.Therefore, it is natural to assume the possibility of their appearance in the boundary value problem for an electromagnetic field.And indeed, for instance, the tangential electric field   ) is equal to the following expression Due to this, the local pressure force of the medium on the boundary 0 = z at the point x at t the moment has a tangential component The average (for the time period where is the phase velocity of the pattern ) , ( 0 t x U (or the speed with which it would be necessary to translate the rigid profile along the axis " " Obviously: (a) On the other hand, the pattern acts on the liquid with a force ) that can generate a flow in liquid.
Thus, we have obtained the radiation reaction force for a perfectly linear compressible nonviscous medium (unlike [10]).Now consider the case when the pattern (1) of normal vibrational velocities (at the boundary where, using ( 10) and ( 23), we can write

Cylinder (azimuthal phasor)
Next, we will continue to consider examples of resonant radiation by various patterns.Let's consider patterns with a finite curvature of a bearing surface -an infinite cylindrical surface S of pattern with a cross-sectional radius R , on which a certain distribution ) , ( 0 t U r of normal vibrational velocities is given (Fig. 4-a).Using the cylindrical wave equation (instead (3)) (where r , J are the corresponding cylindrical coordinates of point r ) for pattern ) we obtain the normalized radiation resistance (Fig. 4-b) ).Looking at the evolution of the graphs of the magnitude   However, the convergence of these values significantly slows down (this can be judged by the slowdown in the growth of the maximums (varying .In book [4], for example, the author could get the resonance dependence (Fig. 2-a) if he would normalized the ordinate by ) as interpolation nodes on the continuous axis of spatial frequencies.Note that the integral flow of the Poynting vector P through an arbitrary closed cylindrical surface S (which embraces S ) is equal to the integral power flow on the surface S of the cylindrical radiating pattern (Fig. 4-a).

Cylinder (axial phasor)
Now for the cylindrical wave equation (instead (29)) we consider the pattern (boundary condition) where the phase runs along the axis of the cylinder S ( +    − x , Fig. 5-a).So the normalized radiation resistance of such a pattern looks like varying 0 h , in Fig. 5-c).Power flux lines P exit the cylinder surface in a taper angle , where n is the sphere number ( is the phasor module 0 h .Thus this is a wellknown 3D travelling wave antenna [9].Then, for the normalized radiation resistance of the pattern (chain of monopoles), we obtain the expression where The function   6-b) has no resonance, and The difference between the boundary value problem and the previous ones lies in the absence of diffraction (small scattering on rigid fixed spheres) of the fields of pulsating spheres on each other due to their small wave sizes (

 ka
) and relatively large distances ( a   ) between their centers.Surface S becomes lumped into a set of 1 N 2 + small spheres (see Fig. 6-a).In other words, we summarize in the far zone the fields of the single pulsating spheres that each of them creates in a space free from other spheres.In the same way, there is no resonance when noninteracting spheres are placed on a ring (if we try to formulate the problem in Section 5 in a similar way).
For ( to make given velocity or pressure on the spheres.This is probably the reason for the absence of resonance.The ) in the following form: where L -space scale of localization.We need to estimate the ratio of the total power emitted by the pattern (41) at 0 0 = h (Fig. 7-a) and the total power emitted by the pattern (41) at 0 0  h (Fig. 7-b).Below we will consider the boundary-value problem of radiation as a low-frequency filter of spatial frequencies with a transmission coefficient (13) (in terms of power (Fig. 7-c) where ] [ I -Heaviside step function (see (8)), x h , y h are the components of the vector h ( ) of the spatial frequency along the axes "x" and "y" , respectively (Fig. 7-d).Let's write the spatial Fourier spectrum of pattern (41) as , where

Conclusion
In this work, the phenomenon of resonant sound emission by a boundary is analyzed analytically when a pattern is set on it in the form of a running sinusoidal spatial distribution of normal vibrational velocities at a certain temporal frequency (Section 1).
The initial problem is the resonant emission of sound by a spatially infinite harmonic distribution of normal velocities on an infinite plane (Section 2).
It is shown that setting a sinusoidal distribution of acoustic pressure on the plane does not generate resonant radiation (Section 2).In addition, it is shown that a linear chain of monopoles with a running phase of vibrational velocity amplitudes does not generate a radiation resonance (Section 7).
Further, the effect of the following factors on the magnitude of the resonant peak is estimated: surface curvature along the phasor (Section 5); surface curvature transversal to the phasor (Section 6); limitedness of the pattern in space (localization, Section 8).
A complete analogy of the phenomenon of resonant radiation in two problems is pointed out: (a) in the acoustic problem of Section 2; in the problem of electromagnetic radiation of a flat boundary, on which a traveling distribution of a tangential electric field with a polarization parallel to the phasor is specified (Section 3).
In practice, if we need to obtain the highest radiation power of a certain pattern with a limited dynamic range of actuators, then it would be most expedient to excite a sliding (with respect to the pattern) wave with a distribution of normal vibrational velocities with a traveling phase, and not excite a normal (with respect to the pattern) wave with an in-phase distribution normal vibrational velocities.
It has been shown that a resonantly radiating boundary with a traveling phase is capable of generating a timeconstant tangential radiation reaction force and,

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www.jenrs.comJournal of Engineering Research and Sciences, 1(4): 01-08, 2022 2 of normal vibrational velocities on the surface S (plane surface S , travelling wave on the time frequency 0 and spatial frequency 0

.
Now we get the power flux density a onedimensional radiation problem), we obtain a simple expression that means the resonant dependence for relative active

Figure 2 .Fig. 2 -
Figure 2. Resonant radiation of the infinite plane boundary S with: (a) normalized real power (13) for velocity pattern (1); (b) normalized imaginary power (14) for velocity pattern (1); (c) normalized real power (17) for given pressure pattern (16); (d) normalized imaginary power (18) for given pressure pattern (16), left white area corresponds to real part of power (radiation), right gray area means reactive imaginary power (no radiation); (e) radiation of normal wave by pattern with amplitude modulus D , phasor 0 0 = h A is the magnitude.To fulfill condition (1) at the boundary 0 = z , the amplitude A of the radiated wave must be increased at in the radiated wave  onto the axis " " z coincides with pattern (1), which magnitude has been given independently on a .But supporting a given pattern (1) at 0 → a requires more and more energy (energy of radiation).

Figure 3 .
Figure 3. On the appearance of tangential acoustical radiation pressure on an infinite flat pattern.

(
spatial harmonics running to the left, i.e. caused by spatial harmonics running to the right, i.e.
26)-(28) it is easy to see that the value (26) is the balance of the power of waves radiated to the right (

Figure 4 .
Figure 4.The phase runs along the azimuth J of the cylinder S : (a) geometry of the boundary value problem; (b) normalized radiation resistance   2 ) 0 ( / ) ( 0 W h W

)
Power flux lines P exit the cylinder surface at an sliding angle .4-c) and become straight radial in the far zone at the distance 5-b).Note that here the argument  of the Hankel function turns from purely real to purely imaginary.The quantity

2 Figure 5 . 7 .
Figure 5.The phase runs along the axis " " x of cylinder S : (a) geom- etry of the boundary value problem; (b) normalized radiation resistance   3

Figure 6 .
Figure 6.Linear chain of acoustic monopoles: (a) geometry of the boundary value problem; (b) normalized radiation resistance of the chain of monopoles at different wave sizes maximum radiation power of the line of monopoles corresponds to the direction of the Poynting vector P in a cone with an inclination angle radiation pressure is present as in above sections.8.Spatially Localized Pattern on a Plane.For a rough estimate of the effect of amplification of acoustic radiation

Figure 7 . 5 M
Figure 7. On the spatially localized pattern (41) on the infinite plane: (a) " " x 0 0 = h in the absence of waves incident on the boundary