1. Introduction
Elastic surface waves that exist in solid waveguides seemingly have very little in common with surface plasmon polariton (SPP) electromagnetic waves propagating in metal-dielectric waveguides. However, with the advent of new elastic metamaterials, this assertion must be revisited.
Indeed, one can argue that the invention of metamaterials was one of the most significant events in physics at the turn of the XX and XXI centuries [
1,
2]. In fact, metamaterials challenged many tacit assumptions and beliefs accumulated over decades about the properties of matter and wave motion herein. Combining basic research with a judicious engineering design, researchers devised many new materials with unprecedented properties. In the domain of elastic media, we observed the emergence of elastic metamaterials with a negative mass density [
3,
4,
5], anisotropic mass density [
6], negative elastic constants [
7,
8], etc. Not surprisingly, these new properties opened possibilities for the existence of new types of acoustic waves, which were previously considered impossible.
To date, it has been commonly agreed that shear horizontal (SH) elastic surface waves cannot exist at the interface between two elastic half-spaces [
9]. In this study we challenge the above assertion, showing that SH acoustic (ultrasonic) surface waves can efficiently propagate at the interface between two elastic-half-spaces, providing that one of them is elastic metamaterial with special properties, i.e., with a negative shear elastic compliance.
Inspired by the newly developed elastic metamaterials, we propose in this paper a new type of shear horizontal (SH) elastic surface waves that were impossible in conventional elastic waveguides [
9]. The new SH elastic surface waves can propagate at the interface between two elastic half-spaces one of which is a metamaterial with a negative elastic compliance
. If, in addition, the compliance
changes with angular frequency
as the dielectric function
in Drude’s model of metals, the proposed SH elastic surface waves can be considered as direct elastic analogues of Surface Plasmon Polariton (SPP) electromagnetic waves propagating at a metal-dielectric interface.
As a result, special attention was paid in this paper to similarities between the newly proposed SH elastic surface waves and the electromagnetic surface waves of the surface plasmon polariton (SPP) type, propagating at a dielectric-metal interface [
10,
11,
12]. In fact, SPP surface waves are transverse magnetic (TM) electromagnetic modes with only one transverse component, namely the magnetic field
that is analogue of the SH particle velocity
of the new proposed SH elastic surface wave. It is noteworthy that both types of waves share one crucial property, i.e., very strong subwavelength decay in the transverse direction away from the guiding interface
, especially in the metal and elastic metamaterial half-spaces.
Due to strong formal similarities between the SPP electromagnetic surface waves and the new proposed SH elastic surface waves, most of the results obtained in this paper can be transferred verbatim into the SPP domain by mutual substitution of the appropriate symbols. However, a transition from the SPP domain into the SH elastic surface wave domain can be very beneficiary for the latter due to a very large number of interesting new phenomena observed already in the SPP domain, such as trapping of light (zero group velocity) [
13], transformational optics systems [
14] or nonreciprocal and topological waveguides [
15], just to name a few. Therefore, the proposed new SH elastic surface waves may open new fascinating possibilities to control wave phenomena occurring in elastic solids.
The new SH elastic waves have the character of surface waves since they decay exponentially in the direction of axis , perpendicular to the interface () and perpendicular simultaneously to the direction of propagation .
Another advantage of the proposed new SH elastic surface waves is the fact that they have only one component of the mechanical displacement
(along axis
), which is completely uncoupled with the remaining components of mechanical vibrations, such as longitudinal (L, along axis
) and shear vertical (SV, along axis
). Multimodal coupling may be a significant problem in conventional bulk ultrasonic devices [
16,
17].
The proposed new SH elastic surface waves can have deep subwavelength penetration depth, in both half-spaces of the waveguide, therefore they offer a potential for applications in subwavelength acoustic imaging, superlensing, and/or acoustic sensors with extremely large sensitivity, analogously to their SPP counterparts in electromagnetism. These are very attractive properties of the newly discovered SH elastic surface waves.
The frequency range, in which the new SH elastic surface wave can propagate, covers practically the range from several kHz to several MHz. The maximum wave frequency
depends on the resonant frequency of local resonators
and is given by Formula (24) in
Section 3.3. For example, when an exemplary waveguide structure depicted in
Section 2.1 consists of (1) the metamaterial half-space (
) composed of ST-Quartz with embedded local resonators with a selected resonant frequency
and (2) a conventional PMMA elastic half-space (
), the maximum frequency of the new SH elastic surface waves equals approximately
, according to the Formula (24) in
Section 3.3.
The proposed new SH elastic surface waves have a potential for very high resolution (of the order of micrometers) using relatively low ultrasonic frequencies (of the order of a few MHz). So far, using the conventional ultrasonic waves and imaging systems a comparable resolution could be achieved using frequencies of the order of 1 GHz. Needless to say, such a frequency range is still quite difficult to handle in ultrasonic practice.
The concentration of the elastic energy near the guiding interface can be of crucial importance in subwavelength acoustic imaging, acoustic energy harvesting as well as in miniaturized modern ultrasonic devices at the micro and nano-scale.
Several analytical equations developed in this paper are new and have not yet been published elsewhere. As a result, we hope that they can provide fresh physical insight into the wave phenomena occurring in both domains, namely SPP electromagnetic waves and SH elastic surface waves, proposed in this paper. For example, Equations (30), (33), (36) and (37) that relate complex power flow with penetration depths in both half-spaces of the waveguide, were to the best of our knowledge not yet published in the literature.
Due to their close similarity with the electromagnetic SPP waves the proposed new ultrasonic waves are characterized by a large confinement of acoustic energy near the surface. For this reason, these newly discovered SH acoustic waves can constitute the basis of a new generation of acoustic (ultrasonic) sensors with a giant mass sensitivity.
The layout of this paper is as follows.
Section 2.1 introduces the geometry and material parameters of two half-spaces forming the metamaterial waveguide.
Section 2.2 presents the metamaterial half-space with a negative elastic compliance
. In
Section 2.3 we derive a complete quantitative model of a metamaterial, whose elastic compliance
obeys the Drude relation. How to fabricate the elastic metamaterial with Drude-like elastic compliance is discussed in
Section 2.4. Mechanical displacement
and shear stresses
are subject to
Section 3.1. Boundary conditions and the dispersion equation of the new SH elastic surface waves are presented in
Section 3.2. The analytical formula for the wavenumber
was derived in
Section 3.3. The formulas for the phase
and group
velocities were developed, in
Section 3.4 and
Section 3.5, respectively. The equations for the penetration depth in both half-spaces of the waveguide are given in
Section 3.6. The net active power flow
, in the direction of propagation
, was determined in
Section 3.7. The average reactive power flow
, in the transverse direction
was analyzed in
Section 3.8. The correspondence between SPP electromagnetic surface waves and the proposed new SH elastic surface waves is outlined in
Section 4. The results of numerical calculations and the corresponding figures are presented in
Section 5. The discussion and conclusions are the subject of
Section 6 and
Section 7, respectively.
2. Physical Model
2.1. Geometry and Material Parameters of the Waveguide
The geometry of the waveguide supporting new SH elastic surface waves is sketched in
Figure 1. The waveguide consists of two semi-infinite elastic half-spaces, one of which is a conventional elastic material
and the second an elastic metamaterial
with a negative elastic compliance
, which is a function of angular frequency
. By contrast, the densities
in both half-spaces as well as the elastic compliance
in the conventional elastic material are positive and frequency independent (see
Figure 1).
Two elastic half-spaces, rigidly bonded at the interface , are uniform in the direction , therefore all field variables of the new SH elastic surface wave will vary only along the transverse direction , i.e., as a function of distance from the guiding interface . It is assumed that both half-spaces of the waveguide are linear and lossless.
2.2. Elastic Drude-like Compliance in the Metamaterial Half-Space ()
The important assumption made throughout this paper is about the elastic compliance
in the metamaterial half-space (
). Namely, it is assumed that
, as a function of angular frequency
, is given explicitly by the following formula:
where:
is the angular frequency of the local mechanical resonances of the metamaterial and
is its reference elastic compliance for
.
It is not difficult to notice that the elastic compliance
given by Equation (1), is formally identical to the dielectric function
in Drude’s model of metals [
18], in which the angular frequency
is named the angular frequency of bulk plasma resonance [
19].
Similarly, the density of the metamaterial half-space () corresponds to the magnetic permeability in Drude’s model of metals.
The second elastic half-space () is a conventional elastic material with a positive compliance and density that are both frequency independent.
In the following of this paper, it is assumed that the elastic compliance in the metamaterial half-space () is given by Equation (1), which is an exact analogue to the dielectric function in Drude’s model of metals. This assumption not only simplifies further analysis but also provides us with a full analogy with the SPP electromagnetic waves propagating at a metal–dielectric interface. Therefore, the results obtained in the SPP domain may be almost automatically transferred to the SH elastic domain and vice versa.
2.3. Quantitative Model of the Elastic Metamaterial with a Drude-like Elastic Compliance
To develop a quantitative model for elastic metamaterials with the Drude-like elastic compliance , described by Equation (1), we will consider a number of electromechanical analogies based on the close affinity between the new SH elastic surface waves and the SPP electromagnetic modes propagating at a metal–dielectric interface.
The correspondence between the new SH elastic surface waves and the SPP electromagnetic waves stems from the fact that they share formally identical mathematical models, derived from the first physical principles. Namely, from the equations of motion (second Newton’s law) governing the behavior of an elastic continuum with parameters and and Maxwell’s electromagnetic equations determining behavior of an electromagnetic continuum with parameters and .
The correspondence between the dielectric permeability and magnetic permeability and shear modulus and density can be expressed as follows:
and
. In
Section 4, we compare the properties of the new SH elastic surface waves and electromagnetic surface waves of the SPP type.
Consequently, the mathematical formulas that we can prove in the domain of the SPP electromagnetic waves using and can be automatically transferred to the domain of the new SH elastic surface waves, which employs and .
We begin our analysis by proposing a one-dimensional model of a mechanical resonator with the elastic properties described by the equation analogous to the dielectric function in Drude’s model of metals.
It is assumed that the one-dimensional mechanical resonator shown in
Figure 2 performs shear vibrations and consists of an elastic spring with a compliance
connected in series with mass
.
2.3.1. Equivalent Circuit Representation of the Mechanical Resonator Shown in Figure 2
The mechanical resonator given in
Figure 2 can be represented by equivalent mechanical and electrical circuits with lumped elements
and
(
Figure 3a) and
and
(
Figure 3b).
The mechanical equivalent circuit shown in
Figure 3a is governed by the equation of motion resulting from Newton’s second law of dynamics. However, the mechanical equivalent circuit shown in
Figure 3a has its electric counterpart in the domain of electric circuits (see
Figure 3b). Consequently, in the analysis of the mechanical equivalent circuit (
Figure 3a) we can employ the methods and notions already developed in the theory of electric circuits, such as e.g., impedance or admittance. In particular, the mechanical admittance of the mechanical equivalent circuit, defined in the frequency
domain as
, can be written as:
where
is the resonant frequency of the mechanical resonator.
Equation (2) shows that the overall behaviour of the mechanical resonator shown in
Figure 2 can be expressed in terms of a resulting shear compliance
represented by a lumped element (spring) in
Figure 4.
By virtue of Equation (2), the equivalent lumped elastic compliance
shown in
Figure 4 is given by the following formula:
The effective lumped (shear) elastic compliance is negative in the frequency range (, in which it grows monotonically from to 0. It means that the mechanical velocity lags in phase with respect to the driving mechanical force by 180°.
Comparing Equation (3) with Equation (1), it is clear that the effective shear elastic compliance
of the discrete representation of the mechanical resonator shown in
Figure 2 and the elastic compliance
of the metamaterial elastic continuum (
) given by Equation (1) (Drude’s model) share the same frequency dependence, if
is replaced by
. This is a very encouraging result since we are now in a position to propose an elementary cell (local oscillator) which constitutes the basis (microstructure) for the design of the elastic metamaterial continuum with a Drude-like elastic compliance
, described by Equation (1).
In the development of a quantitative model of the elastic continuum with a Drude-like elastic compliance , it is prerequisite to identify the elementary cell of local oscillators embedded in the considered elastic host continuum.
2.3.2. Unit Cell of Local Mechanical Resonators with SH Polarization
As a unit cell that can be used as a local resonator, we choose the following structure, see
Figure 5:
The proposed local resonator, embedded in a host elastic material, consists of a sphere of mass
connected to two microcantilevers, which act as a spring with an effective compliance
. It is assumed that the local resonator can vibrate only along the SH direction perpendicular to the line connecting the mass
with the cantilevers and perpendicular to the plane of
Figure 5. As a result, the proposed local resonator can interact only with an SH wave propagating in the host material.
The elastic compliance of the microcantilever is given by the following formula: , where stand, respectively, for the length, width, height, and Young’s modulus of the considered microcantilever. Consequently, the resonant frequency of the proposed local resonator equals .
2.3.3. Elastic Continuum with a Drude-like Elastic Compliance
The analytical formula for the average mechanical energy
stored in the mechanical resonator represented by the discrete mechanical circuit shown in
Figure 3a equals:
Up to now, we are still in the domain of the lumped element circuit theory. However, we are going now to perform the first crucial step by transferring the results obtained in the discrete 1-D circuit domain to the 3-D domain of the metamaterial continuum.
Indeed, in analogy to Equation (4) we are in a position to show that the average mechanical energy density
stored in the corresponding elastic continuum equals:
where:
is the elastic compliance of the corresponding elastic continuum,
is the shear stress equal to
and
is the shear force acting on the surface
of the local oscillator, see
Figure 6.
Therefore, the mechanical energy
stored in the reference volume
(shown in
Figure 7) in the elastic metamaterial equals:
where:
is the number of local shear resonators contained in the reference volume
(see
Figure 7). The coefficient
in Equation (6) represents the average value of the elastic compliance of the resulting 3-D elastic metamaterial continuum.
Now we are going to perform the second crucial step in our development of the quantitative model of the elastic continuum with a Drude-like elastic compliance. This time, we will use the equation developed in the electromagnetic domain by V.L. Ginzburg in [
20] for the energy density of the electromagnetic continuum, whose material parameters are dispersive, i.e., they change with the angular frequency
.
Indeed, using Equation (5) and transferring the electromagnetic equation B.2.5 from reference [
20] into the domain of elastodynamics we obtain:
In the derivation of Equation (7) we employed the correspondence between the dielectric function
and elastic compliance
, shown in
Section 4.
At this moment we are almost done. To obtain a quantitative model of the elastic continuum with a Drude-like elastic compliance we need to perform only a few technical steps. At first, we will integrate Equation (7) over
arriving at the following formula:
It is not difficult to note that Equation (8) is exactly Drude’s relation describing the elastic compliance
of the resulting elastic metamaterial continuum as a function of angular frequency
(see Equation (1) in
Section 2.1).
In the last technical step, we must relate the averaged value of the effective elastic compliance of the resultant elastic metamaterial continuum with the parameters of embedded elementary resonators in an elastic host material.
In fact, since the shear stress
, acting on an elementary resonator with the surface
(see
Figure 6), equals
, by virtue of Equation (6) we can write the following:
Now we have all the necessary elements to present our final model of an elastic metamaterial with a Drude-like elastic compliance
, see
Figure 7 below.
2.4. Fabrication of the Elastic Metamaterial with a Drude-like Elastic Compliance
Elastic metamaterial with a Drude-like elastic compliance in a certain frequency range was already proposed in [
21]. The unit cell of the proposed metamaterial was composed of four tungsten rods with four adjacent vacuum cavities embedded in a host foam. A circular vacuum cavity was placed in the center of the unit cell. Negative elastic compliance was due to the quadrupolar resonance occurring in the unit cell The negativity of the elastic compliance
was confirmed by the corresponding FEM calculations. The elastic compliance of the metamaterial had some characteristics of Drude’s model but was by no means described by the analytical formula given by Equation (1).
In the following, we have included numerical data for material parameters of the elementary mechanical oscillator shown in
Figure 8 as well as the resulting resonant frequency
and effective mechanical compliance
.
Numerical example:
As a unit cell that can be used as a local resonator, we can choose the following structure, see
Figure 8:
Figure 8.
Practical realization of the proposed local mechanical resonator with SH polarization embedded in an elastic host material.
Figure 8.
Practical realization of the proposed local mechanical resonator with SH polarization embedded in an elastic host material.
Effective elastic compliance
of the cantilever shown in
Figure 8, treated as a spring, can be expressed as:
: where:
= length,
= width,
= height and
= Young’s modulus.
Material parameters of the cantilever shown in
Figure 8 were chosen as follows:
, , and : (Bronze).
Reference Volume was assumed as: .
Surface
of the elementary shear resonator from
Figure 6 equals
.
The number of local resonators in the reference volume is equal to .
The mass of the sphere is: : (Tin-lead alloy).
Employing the above set of parameters, we get: ; .
The resonant frequency of the local resonator amounts to . As a host material, we can choose one of the plastics, for example: Nylon PA-6.
Finally, the effective elastic compliance equals: .
Ultrasonic waves in the considered frequency range (e.g., 50 KHz) can be generated and received using standard ultrasonic transducers operating in a conventional experimental setup consisting of a pulser-receiver, a measuring head with ultrasonic transducers, and a control electronic unit (PC computer).
The velocity of ultrasonic waves can be determined, using the above experimental setup, from measurements of the time-of-flight (TOF) between the selected ultrasonic impulses. In the precise determination of the time of flight and therefore the velocity of ultrasonic waves, we can employ the cross-correlation method, which can be effectively implemented digitally within the controlling PC computer.
It should be noted that the new SH elastic surface waves can also propagate in another class of elastic waveguides, in which the elastic compliance of the metamaterial half-space is described by an analytical formula different that the Drude’s formula, given by Equation (1). Namely, the analysis performed in the submitted manuscript will be also valid (after some modifications) when the elastic compliance fulfils the following 2 conditions:
and
- 2.
elastic compliance equals zero for the frequency .
As an example of the elastic compliance
that satisfies the above two conditions we can invoke a Lorentz-like function implying the following formula:
. All analytical equations developed in the submitted manuscript will be valid (after some modifications) for the Lorentz-like elastic compliance
. Similar can be said about figures presented in
Section 5 which will be different, but they will preserve anyway their qualitative properties. However, besides some complications the Lorentz-like elastic compliance does not bring important new phenomena, which are not already present in the Drude-like model.
Therefore, for the sake of simplicity and possible comparison with the SPP electromagnetic waves, which are commonly analyzed with the dielectric function of the Drude type, in the submitted manuscript we assumed that the elastic compliance in the metamaterial half-space is described by the Drude-like Equation (1).
The elastic metamaterial with a Drude-like elastic compliance, described by Equation (1) may be fabricated using 3-D printers and dip-in direct-laser-writing optical lithography [
22]. This activity will be the subject of the author’s future works.
3. Mathematical Model
3.1. Mechanical Displacement and Stresses
Since new SH elastic surface waves are time-harmonic, propagate in the direction
and are uniform along the transverse direction
, their mechanical displacement
, in both half-spaces (
) shown in
Figure 1, will be sought in the following generic form:
where
expresses variations of the mechanical displacement in the transverse direction
,
is the wavenumber of the new SH elastic surface wave and
its angular frequency.
The mechanical displacement
in both half-spaces of the waveguide is governed by the wave equation, resulting from the second Newton’s law, which with the help of Equation (10) reduces to the second order ordinary differential equation of the Helmholtz type [
23]:
where
is the wavenumber of SH bulk waves in both elastic half-spaces number
. In the conventional elastic half-space (
) the wavenumber
is positive and in the metamaterial half-space (
) the wavenumber
is always negative in the angular frequency range
.
Since the mechanical displacement
of the new SH elastic surface wave must vanish at large distances from the guiding interface
, namely for
, the solution of the Helmholtz Equation (11) will be sought in the following form:
where
(
) are arbitrary amplitude coefficients and the transverse wave numbers
are real (waveguide is lossless) and according to the Helmholtz Equation (11) is given by
, where
are wavenumbers of bulk SH waves in the metamaterial half-space
(
) and conventional elastic half-space
(
).
In the following of this paper, we will use two shear stresses of the new SH elastic surface wave, namely
and
that are defined, respectively, as:
Consequently, we can write the following formulas:
where the index
.
To provide an exponential decay of and the transverse wavenumber in Equations (13)–(16) have to be preceded by sign in the convention elastic half-space () and by sign in the metamaterial half-space (), since () in Equations (13)–(16) are real and positive.
3.2. Boundary Conditions and Dispersion Equation
From physical considerations it is obvious that the mechanical displacement
and the shear stress
must be continuous at the interface
, namely:
Substituting Equations (13) and (14) into boundary conditions, Equations (17) and (18), one obtains two linear homogeneous algebraic equations for two unknown amplitude coefficients
and
, namely:
Combining Equations (19) and (20), we get the following dispersion equation for the new SH elastic surface waves:
The sign “
” before the compliance
plays a crucial role in the analysis of new SH elastic surface waves, since it implies that if the transverse wavenumbers
and
are positive, the elastic compliances
,
must be of the opposite sign
. Consequently, if the elastic compliance
(see Equation (1)) in the metamaterial half-space is negative for
, the compliance
have to be positive (see
Figure 1).
Since (see Equation (19)) in the following of this paper we will use only one amplitude coefficient, denoted as .
3.3. Wavenumber
Substituting Equation (16), for transverse wavenumbers
and
, in the dispersion relation Equation (21), one obtains the following formula for the wavenumber
of the new SH elastic surface wave:
where the wavenumber of bulk SH waves in the conventional elastic half-space
.
Since the wavenumber
of the new SH elastic surface wave must be real and positive, Equation (22) imposes the following two necessary conditions on
and
:
The first condition requires that
and the second gives rise to
, where the cut-off angular frequency
and the angular frequency of local resonances
are related by:
Since is always higher than (), the two conditions given by Equation (23) imply that the frequency of the new SH elastic surface wave must be limited to the range .
In the context of the SPP electromagnetic surface waves, the angular frequency
is called the surface plasmon resonance frequency [
19].
3.4. Phase Velocity
Since by definition
, the analytical formula for the phase velocity
of new SH elastic surface waves results immediately from Equation (22):
where
is the phase velocity of bulk SH waves in the conventional elastic half-space.
3.5. Group Velocity
Differentiation of Equation (22) for the wavenumber
, with respect to the angular frequency
, leads to the following formula for the group velocity
of the new SH surface wave:
Despite its relative complexity, Equation (26) is quite elementary and can be easily implemented in numerical calculations.
3.6. Penetration Depths in Both Half-Spaces of the Waveguide
The penetration depth in the metamaterial half-space is defined as , where the transverse wave number (see Equation (16)) and . Similarly, in the conventional elastic half-space we have , where the transverse wavenumber (see Equation (16)) and .
Consequently, substituting Equation (22) for the wavenumber
into Equation (16) for the transverse wavenumbers
and
and noting that
, one obtains:
where
is the wavelength of the new SH elastic surface wave.
In general, the ratio of the penetration depths
,
is expressed by the dispersion equation (Equation (21)), i.e.,
that is independent on
. On the other hand, by virtue of Equations (27) and (28), the product of the normalized penetration depths equals:
However, if the density in both half-spaces of the waveguide is the same (
) then Equation (29) reduces to:
Thus, if the density in both half-spaces of the waveguide is identical () the product of the normalized penetration depths is independent of angular frequency and material constants of the waveguide and equals . In other words, if both normalized penetration depths , are inversely proportional. As a result, if increases then decreases accordingly to Equation (30) and vice versa. Simultaneously, if the angular frequency then both and are subwavelength and tend to the same value .
3.7. Net Active Power Flow in the Direction of Propagation
The complex Poynting vector, in the direction of propagation , of new SH elastic surface waves can be expressed as , where is the mechanical displacement (Equation (5)) and is the mechanical stress (Equation (15)), where .
Similarly, the net complex power flow (per unit length along the axis
) in the metamaterial half-space (
) is defined as
(see
Figure 1) and in the conventional elastic half-space (
) by
.
Consequently, using Equations (13) and (15), it can be shown that the net complex power flows
and
in both half-spaces of the waveguide are given by:
where
is an arbitrary amplitude coefficient.
It should be noticed that all field variables entering Equations (31) and (32) are real. Therefore, the power flows and in both half-spaces of the waveguide are active. In other words, new SH elastic surface waves can effectively transfer the active power along the guiding interface in the direction of propagation .
Employing the dispersion Equation (21) in conjunction with Equations (31) and (32), the ratio of the net active powers flows
in both half-spaces of the waveguide is given by the following:
Note that the ratio of the net active power flows in both half-spaces is always negative, since and are of the opposite sign and the transverse wavenumbers are real and positive . Consequently, and propagate in opposite directions along axis .
3.8. Average Reactive Power Flow in the Transverse Direction
The complex Poynting vector, in the transverse direction , of new SH elastic surface waves can be expressed as , where is the mechanical displacement (Equation (13)) and is the mechanical stress (Equation (14)), where .
Similarly, the average complex power flow (per unit length along the axis
) in the metamaterial half-space (
) is defined as
(see
Figure 1) and in the conventional elastic half-space (
) by
.
Consequently, using Equations (13) and (14) it can be shown that the average complex power flow
and
in both half-spaces are given by:
Thus, if then and both tend to zero. On the other hand, if then and tend to the same value, namely .
Since the elastic compliance is negative, in the frequency range , the average reactive power flows , in both half-spaces, are both positive (+) and correspond to the inductive type of the reactive power, in analogy to SPP electromagnetic waves.
Using Equation (1) together with Equations (34) and (35), the ratio of the average reactive power flows in both half-spaces can be written as:
Comparing Equations (33) and (36), one obtains a rather unexpected relation between the net active power flows
in the direction of propagation
and the average reactive power flows
in the transverse direction
, namely:
Thus, if the ratio of the net active power flows increases, say times, the ratio of the average reactive power flow grows only times, etc. In other words, repartition of the net active power flow () between two half-spaces of the waveguide is much more sensitive to changes in the penetration depths and than that of the average reactive power flow () in the transverse direction .
4. Correspondence between the SPP Electromagnetic Waves and the Proposed New SH Elastic Surface Waves
As it was stated before, the proposed new SH elastic surface waves can be considered an elastic analogue of the SPP electromagnetic surface waves propagating at a metal–dielectric interface. In fact, the mathematical models of both types of waves are formally identical. Therefore, it will be advantageous to identify explicitly the corresponding field variables in both domains, since the results obtained in one domain can be directly transferred to the other domain, alleviating thereby tedious from scratch derivations of the resulting analytical formulas (see
Table 1).
As a result, the analytical formulas for all field variables analyzed in this paper, such as mechanical displacement , shear stresses , transverse wavenumbers , wavenumber , phase velocity , group velocity , penetration depths , net active power flows , average reactive power flows , as well as the dispersion relation can be readily transferred to the SPP domain by a simple substitution of the corresponding symbols.
For example, phase velocity
of the SPP electromagnetic waves (see row 9 in
Table 1) is expressed by the same formula as phase velocity
of the new SH elastic surface waves, providing that
and
are substituted by
and
, respectively. The symbol
corresponds to phase velocity of bulk SH waves in the conventional elastic material (
) and to bulk transverse electromagnetic waves in the dielectric (
).
Interestingly, the crucial step in development of the quantitative model of the elastic metamaterial with a Drude-like elastic compliance (see
Section 2.3) was the reverse transfer of an analytical equation developed in the electromagnetic domain (Equation B.2.5 in [
20]) into the domain of the SH elastic waves (see Equation (7) in
Section 2.3.3 and the accompanying discussion).
6. Discussion
Elastic surface waves propagating in metamaterial waveguides were subject of a number of papers that analyzed the Rayleigh surface waves at the solid-vacuum interface [
24], Scholte interfacial waves at the solid-liquid interface [
25], shear horizontal waves on a semi-infinite half-space loaded with a metasurface [
26,
27] or Love surface waves in waveguides loaded with a resonant metasurface [
28].
The possibility of the existence of elastic SH waves propagating at the interface of two elastic half-spaces, one of which is an elastic metamaterial, was briefly announced in one of the author’s previous works [
29]. However, the present paper differs significantly from the former paper presented in [
29]. In particular, in the present study:
- (1)
A general theory of the new SH elastic surface waves propagating at an elastic interface has been developed from first physical principles;
- (2)
All considered field variables are normalized, e.g., we use the normalized angular frequency , normalized wavenumber etc.;
- (3)
The influence of the density of both half-spaces on the characteristics of the new elastic SH wave is taken into consideration;
- (4)
New analytical formulas for the penetration depths and were established. The newly developed formulas can be of significant practical importance in design of devices in the domain of SPP and in the domain of new SH elastic surface waves;
- (5)
A new quantitative model of the elastic metamaterial with a Drude-like elastic compliance has been developed.
It should be emphasized that all the five developments mentioned above have been included in the present paper and were not yet published elsewhere.
Our former research [
30] on elastic surface waves propagating in conventional elastic waveguides showed that SH surface waves, such as Love surface waves [
31], share many common properties with waves in other domains of physics, such as TM (Transverse Magnetic) modes in optical planar waveguides or wave function of quantum particles in a potential well. However, the present paper was mostly influenced by recent developments in the domain of elastic metamaterials and SPP electromagnetic surface waves propagating at the metal-dielectric interface [
32].
In this paper, we demonstrated that the ultrasonic analogue of SPP electromagnetic waves can exist in elastic waveguides consisting of two elastic half-spaces, providing that one of the elastic half-spaces is an elastic metamaterial with a negative elastic compliance that corresponds to the dielectric function in Drude’s model of metals. These two types of waves are described by formally identical mathematical models and, therefore, have similar (1) distribution field variables and (2) dispersion equation.
The dispersion curves of the new SH elastic surface wave, shown in
Figure 9, have the characteristic property that the wavenumber
tends to infinity
, when the wave angular frequency
approaches the cuff-of frequency
. Since
, the wavelength
of the new SH elastic surface wave tends to zero
when
. This phenomenon can be exploited in the subwavelength near field ultrasonic imaging.
Another very intriguing property of the new SH elastic surface waves is that their phase
and group
velocities tend to zero when the wave frequency approaches the cut-off frequency
(see
Figure 10 and
Figure 11). This property is of key importance in the potential applications of the new SH elastic surface wave in ultrasonic sensors with extremely large mass sensitivity, which can give rise to a new generation of biosensors and chemosensors with unprecedented sensitivity.
This paper contains several new original formulas which to the best of our knowledge were not yet published in the literature, namely:
- -
Relation for the product of penetration depths , in two half-spaces of the waveguide (Equation (30));
- -
Relation between net active power flows , in the direction of propagation and penetration depths , in two half-spaces of the waveguide (Equation (33));
- -
Relation between average reactive power flows , in the transverse direction and penetration depths , in two half-spaces of the waveguide (Equation (36));
- -
Relation between net active power flows , in the direction of propagation and average reactive power flows , in the transverse direction of the waveguide (Equation (37)).
All new equations mentioned above, which were developed in the elastic domain, can be directly transferred into the domain of SPP electromagnetic surface waves, using to this end
Table 1 presented in
Section 4. In particular, the relation between the penetration depths
,
in two half-spaces of the waveguide (Equation (30)) can be useful for designers of SPP electromagnetic sensors, in selection of proper wave frequency providing high subwavelength concentration of energy in the dielectric material of the waveguide leading to long range propagation of SPP waves.
Similarly, the new relations between the power flows and the penetration depths in two half-spaces of the waveguide (Equations (33) and (36)) indicate that the proper control of the net active power flow in the direction of propagation may be very important in achieving high sensitivity of long range SPP sensors with low losses.
The results presented in
Figure 9,
Figure 10,
Figure 11,
Figure 12,
Figure 13,
Figure 14 and
Figure 15 reveal that the densities
, in both half-spaces of the waveguide, have a profound impact on all parameters of the proposed new SH elastic surface waves. For example, if
, the penetration depth in the metamaterial half-space
is ~43 times smaller than the wavelength
of the wave, at
(see green curve
Figure 12). By contrast, if
the penetration depth
decreases significantly and is ~167 times smaller than the wavelength
(see red curve in
Figure 12).
Therefore, since the densities
correspond to magnetic permeabilities
in SPP electromagnetic waveguides (see rows 6 and 7 in
Table 1 in
Section 4) it implies that we can also effectively shape the characteristics of SPP electromagnetic waves by analogous adjustment of
and
.
On the other hand, due to strong formal similarities between the new SH elastic surface waves and SPP electromagnetic surface waves it may be possible in future to transfer many fascinating newly discovered SPP phenomena, such as cloaking [
14], trapping (zero group velocity) [
13] and topological protection [
15] into the domain of elastic metamaterials using to this end the new SH elastic surface waves, proposed in this paper.
As a result, the proposed new SH elastic surface waves can open new possibilities to control wave phenomena in elastic solids and can constitute the basis for a new generation of modern devices in the domain of sensors, acoustic imaging, and signal processing.
Using recently discovered elastic hyperbolic metamaterials [
33] we can achieve subwavelength imaging by amplification of the evanescent waves scattered from the object, which contain information about fine details of the object. The evanescent waves are not only amplified but also are converted to propagation modes, which can be focused in a far zone of the hyperbolic superlens. However, the same amplification of the evanescent waves and subwavelength imaging can be achieved with the proposed new SH elastic surface waves, but in a simpler way. In fact, the elastic hyperbolic metamaterials are quite complicated since they require that the mass density of the hyperbolic metamaterial must be simultaneously anisotropic and negative [
34]. By contrast, using the new SH elastic surface waves we can also achieve subwavelength imaging and amplification of the evanescent waves but in a much simpler way. In fact, two half-spaces of the waveguide supporting the new SH waves are always isotropic and only one metamaterial half-space must exhibit a negative Drude-like elastic compliance.
Finally, we must address the issue of losses that will inevitably occur in waveguides of the proposed new SH acoustic surface waves. Interestingly, the problem of losses was solved in SPP devices by the introduction of a multilayer waveguide structure. For example, a very thin layer (25 nm) of lossy metal (Au) was sandwiched between two low loss dielectrics (SU-8 polymer) provided a 5 mm long sensor [
35]. The presence of losses may also affect efficiency of specific wave phenomena occurring in metamaterial waveguides, such as zero group velocity. In fact, in reference [
36] it was shown that the minimal group velocity that can be achieved in waveguides with losses is always higher than zero.
Moreover, the presence of losses can limit the maximum value of the wavenumber of the SH surface wave propagating at the boundary of the elastic half-space and the metamaterial half-space with Drude-like elastic compliance. This may limit the resolution of Drude-type metamaterial superlenses used in near-field acoustic imaging.
This paper is a clear example of the multidisciplinary research that can bring new valuable and sometimes unexpected physical insight on the physical phenomena occurring in two domains of physics, i.e., theory of elasticity and electromagnetism.
It will be advantageous in future research to extend the analysis of the new SH elastic surface waves on waveguides with losses as well as to design a model of a biosensor based on the analogy with SPP electromagnetic devices [
35,
37].