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.

1. Introduction

It is usually well-known that there is radiation (acoustical or electromagnetic monochromatic field in the far zone) at the spatial frequency h₀ < k₀ (k₀ -wave number in the medium) of sources, and at the spatial frequency h₀ < k₀ of sources radiation is absent or very small [1-7]. 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 h₀ from h_{0 = 0} to h₀ > k₀ (immediately before radiation falling to zero at h₀ > k₀) can reach infinity when approaching h₀ to k₀ to . 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 ) by a traveling distribution (with a traveling phase [8])

of normal vibrational velocities on the surface S (plane surface S, z = 0,  r = (x,y,0 )), where U(w₀, h₀) is the complex amplitude of travelling wave on the time frequency w₀ and spatial frequency h₀ of normal vibrational velocities (Fig. 1-a). Below we will consider the radiation of various patterns (as modifications of (1)) with a traveling phase.

2. Plane (acoustical field)

Particle velocity v(x, z, t) and sound pressure p(x, z, t) are determined by potential ψ as

where  is the mass density of medium at z > 0.

Assuming the spatial two-dimensionality of the boundary value problem (i.e.), let us substitute the solution in the equation

( c is the speed of sound in a compressible medium) for the acoustic wave potential ψ(x, z, t) in the form

with functions X(x), Z(z), T(t), of separable variables x, z, t, in the absence of waves incident on the boundary z=0. In this case for wave potential ψ(x, z, t) (in the absence of incident waves and satisfying the boundary condition (1) or  we obtain the following expression

where

thus where  k_{0 =} ω₀/c .  Now we get the power flux density

at the boundary z = 0 . 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 2π /ω₀ ). This is due to an abrupt change in the eigenvalue of the boundary value problem from purely real to purely imaginary when going from h₀ < k₀ to h₀ > k₀.  For the reactive power flow, we obtain the expression

of the energy transfer of the boundary z = 0 (into the half-space ) . P Lines (Pointing vector) of power flow go out of the plane z = 0  at an angle of inclination

However, when the traveling pressure

is set at the boundary z = 0, there is no resonance in the following corresponding  functions (instead (13), (14))

(contrary to superficial judgment about the relationship between pressure and velocity through the impedance of the medium, see Figs. 2-c, 2-d).

Below, the value [W(h₀) / W(0)]_n  (n = 1 – 5, i.e. in various modifications) will also be called the transmission function (in terms of power) of a spatial low-frequency filter (due to boundary value problem, Fig. 1) of spatial frequencies h₀ or also the normalized (by W(0) ) radiation resistance of a radiating pattern (pattern number n = 1 – 5 ) with spatial frequency h₀ along the vector called as phasor h₀ . Below the quantity U₀( x,t ) we also will call “pattern” with surface or boundary .   

The essence of the effect of resonant radiation: the wave generated by the pattern (1) at |k₀ – |h₀|| / k_{0 << 1}, takes much more energy for radiation (from the devices that support the pattern (1)) than the wave with h₀ = 0 (at the same fixed amplitude module ∆ = |U(h_{0, ω0)|} of pattern (1), Fig. 2-e, 2-f). In turn, devices that support pattern (1) must have an infinite internal impedance Z_u = ∞  (or not depend on any wave pressure at z > 0 ) or an impedance Z_u >> pc that is many times greater than the impedance pc of the medium (at ), for example: air (z > 0)—piezoceramic (thickness )—steel .

Here is another explanation of the effect. When |k₀ – |h₀|| / k_{0 < 1,} the solution of equation (3) in the half-space z > 0  is a plane sound wave with a wave potential , where A is the magnitude. To fulfill condition (1) at the boundary z = 0, the amplitude A of the radiated wave must be increased at  |h₀|→ k₀ (or  α→0 ) so that the projection of particle velocities in the radiated wave ψ onto the axis “z” coincides with pattern (1), which magnitude has been given independently on α. But supporting a given pattern (1) at α→0 requires more and more energy (energy of radiation).

3. 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

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8. V.V. Arabadzhi, Solutions to Problems of Controlling Long Waves with the Help of Micro-Structure Tools, Bentham Science Publishers, 2011.
9. C.H. Walter, Traveling Wave Antennas, McGraw-Hill, 1965, Dover, 1970, reprinted by Peninsula Publishing, Los Altos, California, 1990.
10. Robert T. Beyer, “Radiation pressure – the history of a mislabeled tensor,” Journal of Acoustical Society of America, vol. 63, no. 4, pp. 1025-1030, 1978, doi:10.1121/1.381833.

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