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Using the Partial Wave Method for Wave Structure Calculation and the Conceptual Interpretation of Elastodynamic Guided Waves Christopher Hakoda and Cliff J. Lissenden * Department of Engineering Science and Mechanics, Pennsylvania State University, University Park, PA 16802, USA; [email protected] * Correspondence: [email protected] Received: 8 May 2018; Accepted: 8 June 2018; Published: 12 June 2018







Featured Application: Shortcut to wave-structure calculation for elastodynamic guided waves and prediction of acoustic leakage behavior. Abstract: The partial-wave method takes advantage of the Christoffel equation’s generality to represent waves within a waveguide. More specifically, the partial-wave method is well known for its usefulness when calculating dispersion curves for multilayered and/or anisotropic plates. That is, it is a vital component of the transfer-matrix method and the global-matrix method, which are used for dispersion curve calculation. The literature suggests that the method is also exceptionally useful for conceptual interpretation, but gives very few examples or instruction on how this can be done. In this paper, we expand on this topic of conceptual interpretation by addressing Rayleigh waves, Stoneley waves, shear horizontal waves, and Lamb waves. We demonstrate that all of these guided waves can be described using the partial-wave method, which establishes a common foundation on which many elastodynamic guided waves can be compared, translated, and interpreted. For Lamb waves specifically, we identify the characteristics of guided wave modes that have not been formally discussed in the literature. Additionally, we use what is demonstrated in the body of the paper to investigate the leaky characteristics of Lamb waves, which eventually leads to finding a correlation between oblique bulk wave propagation in the waveguide and the transmission amplitude ratios found in the literature. Keywords: partial wave method; slowness curves; lamb wave; stoneley wave; mode sorting; acoustic leakage; rayleigh wave; surface waves; elastodynamics

1. Introduction In this paper, we demonstrate how the partial-wave method, as first introduced by Solie and Auld [1], can be used to gain a new perspective on guided waves. This new perspective adds to our understanding of the structure of guided waves, and provides valuable insight into the leaky characteristics of Lamb waves (which are not to be confused with the study of leaky Lamb waves and their corresponding dispersion curve solutions). At present, our understanding of guided waves is primarily informed by the derivation of characteristic equations and dispersion curve solutions. The dispersion curve solutions in particular have become significantly more accessible with the widespread use of the semi-analytical finite element (SAFE) method. However, the SAFE method does not provide the same level of understanding that would be possible by analyzing a characteristic equation. The most well-known example of this is that the two-part Lamb wave characteristic equation shows that all of the Lamb waves can be divided into two types: symmetric and anti-symmetric. Hence, the importance of the present work. However, this paper will not

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cover the calculation of dispersion curves, attenuation, energy loss, or anisotropic materials. Neither will it cover the derivation of characteristic equations for various guided waves. For the purpose of clarity, in this paper, bulk waves are plane waves that travel through the bulk of a media, and surface waves are plane waves that travel along the surface of a media. It is emphasized that all of the waves are considered to have a planar wavefront. Sections 2 and 3 will provide a brief history and the fundamentals of the partial-wave method and the Christoffel equation, respectively. These sections give special attention to the diction used and the assumptions made, as they are pertinent to how the interpretation presented in this paper deviates from the literature. Section 4 describes this new method of interpretation. It discusses in detail how the slowness curve solutions of the Christoffel equation can be used to interpret the characteristics of most guided waves, and the Lamb wave solutions in particular. Sections 5 and 6 apply the new method of interpretation to a variety of guided waves, which include Rayleigh waves, Stoneley waves, Lamb waves, and shear horizontal waves. Section 7 focuses on Lamb waves, and a method developed in Section 6 is used to investigate the contributions of each partial wave to Lamb waves from 50 kHz to 20 MHz for a 1-mm thick aluminum plate. This section also identifies characteristics of Lamb waves that have not been formally discussed in the literature. Section 8 uses the results from the previous sections to develop an understanding of the leaky characteristics of Lamb waves. Various characteristics of the acoustic leakage from the waveguide can be easily determined, including the angle of propagation and the amplitude of emission based on the interface conditions. A correlation between the amount of acoustic leakage and the refraction qualities of obliquely travelling bulk waves in the guided wave is also identified. 2. The Partial Wave Method The term ‘partial-wave method’ first appeared in a paper by Solie and Auld in 1973 [1]; specifically, it was called “the method of partial waves (or transverse resonance)”. This method was described as follows: “In this method, the plate wave solutions are constructed from simple exponential-type waves (partial waves), which reflect back and forth between the boundaries of the plates . . . ” They go on to write, “A basic principle of the partial wave method is that the partial waves are coupled to each other by reflections at the plate boundaries”. In Auld’s 1990 monograph [2], he discusses the method further: “[the superposition of partial waves method] can be further simplified by making use of the transverse resonance principle, which has been used to great advantage in electromagnetism” [emphasis added by Auld]. That is, what is known colloquially as the partial-wave method can be broken down into two parts. First, there is the superposition of the partial-waves method, which uses the principle of superposition with the Christoffel equation’s solutions (i.e., partial waves) [2]. Second, there is the transverse resonance principle, which assumes that the standing wave perpendicular to wave propagation can only exist if the partial waves satisfy the relevant boundary conditions [2,3]. Modern application of the partial-wave method includes the calculation of dispersion curves using the global-matrix method or the transfer-matrix method for anisotropic plates and/or multilayered plates [4,5]. 3. The Christoffel Equation and the Use of Slowness Curves The Christoffel equation is critical to the flexibility and strength of the partial-wave method. The Christoffel equation can be derived by assuming a plane wave (Equation (4)) propagating in a solid elastic media (Equations (1)–(3)). σij,j = ρui,tt (1) σij = Cijkl ekl eij =


1 ui,j + u j,i 2

(2) (3)

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ui ( x, y, z, t) = Ui ei(−k j r j +ωt)

(4)

Here σij , eij , and ui are the stress, strain, and displacement; ρ and Cijkl are the mass density and elastic stiffness; k j and ω are the wave vector and angular frequency; i, j, k, l = x, y, z, and t is the time variable. When combined, Equations (1)–(4) yield the Christoffel equation: Cijkl k k Ul k j − ρω 2 Ui = 0

(5)

When Equation (5) is solved, a slowness curve/surface can be calculated and plotted in an inverse phase velocity space. 3.1. Deriving the Expression for the Slowness Curves of an Isotropic Solid For an isotropic linear elastic solid, Equation (2) can be replaced with: σij = λekk δij + 2µeij

(6)

where λ and µ are the Lamé parameters and δ is the Kronecker delta. As a side note, in addition to replacing Equation (2) with Equation (6), if Equation (4) is replaced with: ˆ i(−k x x+ωt) uiL ( x, t) = Uo xe

(7)

ˆ i(−k x x+ωt) uiS ( x, t) = Uo ze

(8)

or: then, the Christoffel equation can be used to derive the relationship between the Lamé parameters, the longitudinal wave speed, cL , and the shear wave speed, cS : λ + 2µ = ρc2L

(9)

µ = ρc2S

(10)

where xˆ and zˆ are the unit vectors along the x-axis and the z-axis. Using Equation (6), assuming ky = kz = 0 (i.e., wave propagation in the x–z plane), and dividing through by ω2 , Equation (5) can be rearranged into matrix form: Aui = 0 (11) where: 

2 −2 (λ + 2µ)c− x + µcz − ρ  A= 0 1 c −1 ( λ + µ ) c− x z

0
2 + c −2 − ρ µ c− x z 0

 1 −1 c− x cz (λ + µ)  0  − 2 − 2 µc x + (λ + 2µ)cz − ρ

(12)

1 Equations (11) and (12) represent a system of linear homogeneous equations, where c− x = k x /ω − 1 and cz = kz /ω. The non-trivial solution to Equation (11) only exists if:

det( A) = 0

(13)

Expanding Equation (13) based on Equation (12) results in: 

   2 2 −2 2 −2 ρ − (λ + 2µ) c− ρ − µ c− =0 x + cz z + cx

(14)

The solutions to the characteristic equation (Equation (14)) are: 2 −2 c− x + cz =

ρ (λ + 2µ)

(15)

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2 −2 c− x + cz =

ρ µ

(16)

If Equations (9) and (10) are considered, the right-hand side of Equations (15) and (16) become 2 and c− S , respectively. These represent the slowness curve solutions to the Christoffel equation of an isotropic material. It should be noted that in Equation (14), the squared term represents the multiplicity in the shear wave’s slowness curve solutions. 2 c− L

3.2. Interpretation of Slowness Curve Solutions From the previous section, it is clear that the solutions to the Christoffel equation do not take into consideration the boundary conditions (BCs) of the problem. That is, the solution only depends on the material properties of the elastic solid in which the plane waves propagate. As a result, in the literature, it is shown that the Christoffel equation can be used to determine the reflection and refraction characteristics of plane waves at oblique incidence to an interface between anisotropic elastic solids [6–8]. Additionally, the energy velocity vector and the bulk wave’s skew angle can be calculated [4]. Slowness curves are often plotted in terms of the ‘inverse phase velocity’. However, since phase velocity is not a vector, the use of the term ‘inverse phase velocity’ to describe the slowness vector can be confusing. A different notation that may be easier to conceptually understand is plotting slowness 1 −1 curves on the kz /ω and k x /ω axis instead of the c− z versus cx axis. Therefore, if a ray is drawn from the origin to a point on the slowness curve, it is understood that its unit vector describes the wave propagation direction in the x–z plane. Since the focus of the partial-wave method is on guided waves, for the purposes of this paper, it is assumed that the wave is propagating parallel to the boundary/interface, which we take to be in the x-direction. The z-axis is through the thickness of the elastic solid. This coordinate system will be maintained throughout the entirety of the paper, with additional details depending on the problem being considered. Further, continuing to assume that ky = 0, and using the frequency–wavenumber (ω-k x ) pairs obtained from any one of the many methods for calculating dispersion curves (i.e., enforcing the BC and solving the eigenvalue problem), the Christoffel equation becomes a complex valued quadratic eigenvalue problem in three dimensions. The solution, should it exist, will always be composed of six complex valued eigensolutions. These six solutions are the partial waves of the partial-wave method. The partial waves can be paired up into three pairs, and each pair is composed of an upward and a downward propagating wave (i.e., the magnitudes of the eigenvalues are the same, but the signs are opposite) [1,4]. In Auld [2], there is a discussion of regions for an increasing k x , in which the partial waves have specific characteristics. Also, in Auld [2], it is recognized that at an interface, k x is the same for all of the partial waves in order to satisfy Snell’s law, and that a given guided wave must theoretically have a single k x value for a given ω. Auld gives an example for an isotropic material, which has two slowness curves and three regions (see Equations (15) and (16)):

• • •

Region 1: 0 < c p < cS All six partial waves are surface waves (imaginary-valued kz ) (see Figure 1a). Region 2: cS < c p < c L : Two partial waves are surface waves (imaginary-valued kz ), and four are bulk waves (real-valued kz ) (see Figure 1b). Region 3: c L < c p : All six partial waves are bulk waves (real-valued kz ) (see Figure 1c).

1 −1 The inverses of the shear and longitudinal group velocities (c− S and c L ) are also the radii of the two slowness curves, which are circles for an isotropic material. By Crandall’s paper [6], it is also understood that outside of these circles, the value for k Z becomes imaginary, and the magnitude traces a hyperbola. Auld also gives examples of slowness curve representations for a shear horizontal (SH) mode composed of obliquely travelling bulk waves, and a piezoelectric plate with leaky electromagnetic radiation [2]. This concludes the state-of-the-art section of the paper. How these theories and concepts allow us to reinterpret our understanding of guided waves in a novel way is discussed from here onward.

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Figure 1. Depiction of the characteristics of (a) region 1, (b) region 2, and (c) region 3 with respect to

Figure 1. Depiction of the characteristics of (a) region 1, (b) region 2, and (c) region 3 with respect to the the slowness curves of an isotropic solid. Slowness curves with solid lines denote real-valued 𝑘 slowness curves of and an isotropic solid. curves with solid lines denote solutions, dotted lines denoteSlowness imaginary-valued 𝑘 solutions. This figure is areal-valued compilation kofz solutions, and dottedelements lines denote imaginary-valued kz solutions. from the figures in Auld [2] and Crandall [6].This figure is a compilation of elements from the figures in Auld [2] and Crandall [6]. 4. Reinterpreting the Dispersion Curves

As mentioned previously, the Christoffel equation is only dependent upon the material 4. Reinterpreting the Dispersion Curves properties of the elastic solid in which the plane waves propagate. The BCs are not used until the

partial-wave method is applied. Not only that, but theisboundary conditionsupon are notthe limited to theproperties As mentioned previously, the Christoffel equation only dependent material conventional traction-free BCs (e.g., Lamb waves). As a result, the partial-wave method is not limited of the elastic solid in which the plane waves propagate. The BCs are not used until the partial-wave to any specific type of waveguide. In fact, most types of guided waves should be representable using method isslowness applied.curves. Not For onlysome, that,such butasthe conditions notthis limited to the conventional theboundary shear horizontal guided are waves, approach is more traction-free BCs (e.g.,than Lamb waves).derivation. As a result, the partial-wave is not alimited to any specific cumbersome the standard However, its value lies inmethod how it provides fundamental foundation In for fact, all of most the guided waves, and a common and comparing each type of waveguide. types of guided waves method shouldfor beinterpreting representable using slowness curves. one. When considering reflection/transmission, mode conversion, or multi-mode propagation, this For some, such as the shear horizontal guided waves, this approach is more cumbersome than the perspective is quite beneficial. standard derivation. However, valuethelies incommonly how it provides a fundamental In modern literature, its perhaps most used dispersion curves are offoundation Lamb wavesfor in all of the guided waves, and a common method for interpreting and comparing each one. When considering an aluminum plate in terms of phase velocity versus the frequency–thickness product. Using the three regions depicted in Figure 1, we can divide the phase velocity dispersion curves into three reflection/transmission, mode conversion, or multi-mode propagation, this perspective is quite beneficial. regions, as shown in Figure 2. By doing this, it is possible to view how the dispersion curve solutions In modern literature, perhaps the most commonly used dispersion curves are of Lamb waves match up with the partial waves’ characteristics. For example, this is a good way to dispel the in an aluminum in termsthat ofguided phasewaves velocity versus the frequency–thickness product. potentialplate misconception are composed of multiple reflecting bulk waves. That is, Using the three regions in Figureof1,plane we can divide the phase velocity curves into the depicted original assumption wave propagation (Equation (4)),dispersion which is fundamental to three the regions, does notitexclude the surface wave (i.e., imaginary-valued 𝑘 ) as 𝑘 as shown inChristoffel Figure 2.equation, By doing this, is possible to view howsolution the dispersion curve solutions match up with increases. It would be more accurate to say that guided waves are a superposition of reflecting bulk the partial waves’ characteristics. For example, this is a good way to dispel the potential misconception waves and surface waves with elliptical particle motion (this will be shown in the later calculations that guided waves are composed of multiple reflecting bulk waves. That is, the original assumption of and examples). plane wave propagation (Equation (4)), which isoffundamental the Christoffel equation, not exclude In fact, guided waves composed solely reflecting bulkto waves exist only in Region 3 (c),does where they are highly dispersive. However, there is one exception to this rule: there exists a range of phase the surface wave solution (i.e., imaginary-valued kz ) as k x increases. It would be more accurate to say velocities in Region 2 (Figure 1b), where only bulk waves propagate, which will be discussed in a that guided waves are a superposition of reflecting bulk waves and surface waves with elliptical particle later section. motion (this will in Rayleigh the laterwaves calculations and examples). Thebe A0shown mode and exist in Region 1 (Figure 1a), where each guided wave can be In fact, guided waves composed solely of reflecting bulk waves existisonly in Region (c),iswhere they represented in terms of six surface waves. The low-frequency S0 mode in Region 2 (b), 3and of two surface waves twoexception bulk waves to (asthis will rule: be shown later in theapaper), are highly composed dispersive. However, thereand is one there exists range excluding of phase velocities two shear horizontal partial waves. in Region 2the (Figure 1b), where only bulk waves propagate, which will be discussed in a later section. The line separating Region 1 (Figure 1a) and Region 2 (Figure 1b) has great significance, since The A0 mode Rayleigh waves exist in Region near 1 (Figure each Lamb guided wave can be many Lamband wave modes appear nearly non-dispersive it. That 1a), is, thewhere higher-order waves representedapproach in terms of six surface waves. The low-frequency S0 mode is in Region 2 (b), and is composed the shear wave speed, and the A0 and S0 modes approach the Rayleigh wave speed for large 𝜔 values. The shear horizontal modes also converge to this line for large 𝜔 values. The line of two surface waves and two bulk waves (as will be shown later in the paper), excluding the two shear horizontal partial waves. The line separating Region 1 (Figure 1a) and Region 2 (Figure 1b) has great significance, since many Lamb wave modes appear nearly non-dispersive near it. That is, the higher-order Lamb waves approach the shear wave speed, and the A0 and S0 modes approach the Rayleigh wave speed for large ω values. The shear horizontal modes also converge to this line for large ω values. The line separating Region 2 (Figure 1b) and Region 3 (Figure 1c) has many higher-order Lamb modes, which appear nearly non-dispersive close to it as well.

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separating Region 2 (Figure 1b) and Region 3 (Figure 1c) has many higher-order Lamb modes, which 6 of 22 appear nearly non-dispersive close to it as well.

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Figure 2. Lamb wave dispersioncurves curves for plate divided according to theto three Figure 2. Lamb wave dispersion foran analuminum aluminum plate divided according theregions three regions shown in Figure shown in Figure 1. 1.

The A0 and S0 Lamb wave modes’ tendency to approach the Rayleigh wave phase velocity can

The A0explained and S0 Lamb wave modes’ tendency to approach wave phase of velocity can also be using this reinterpretation, since increasing 𝑘 the will Rayleigh increase the magnitude the also be explained using this Figure reinterpretation, increasing k x will the magnitude imaginary-valued 𝑘 (see 1a). Once 𝑘 since increases sufficiently, the increase surface waves decay with of the the depth fromkzthe interacting with thesufficiently, opposite traction-free boundary, thereby imaginary-valued (seesurface Figurebefore 1a). Once kz increases the surface waves decay with the Rayleighbefore wave. interacting This also explains why quasi-Rayleigh wavesboundary, [9] have similar wave depthimitating from thea surface with the opposite traction-free thereby imitating structures to Rayleigh waves. That why is, thequasi-Rayleigh quasi-Rayleigh waves wave and Rayleigh arestructures both a Rayleigh wave. This also explains [9] the have similarwave wave to composed of partial waves (with slight differences in amplitude and exponential decay). Rayleigh waves. That is, the quasi-Rayleigh wave and the Rayleigh wave are both composed of partial Many higher-order Lamb modes have wave structures that are sinusoidal through the thickness, waves (with slight differences in amplitude and exponential decay). but have slightly larger or smaller amplitudes near the traction-free boundaries. In Region 2 (Figure Many higher-order Lamb modes have waveaffecting structures sinusoidal the thickness, 1b), this could be explained by surface waves the that waveare structure near through the traction-free but have slightly larger or smaller amplitudes near the traction-free boundaries. In Region 2 (Figure 1b), boundaries, while the reflecting bulk waves create the sinusoidal appearance. this couldThis be explained by surface waves affecting the wave thecalculation traction-free boundaries, reinterpretation of guided waves provides for thestructure relativelynear simple of wave basedbulk on the superposition the partial-wave eigensolutions. Based on the preceding whilestructures the reflecting waves create the of sinusoidal appearance. concepts and with wave in the 𝑥-direction, = 𝜔/𝑘 =simple 𝑐 . Solving Equations (15) and This reinterpretation of propagation guided waves provides for the𝑐relatively calculation of wave structures (16) for 𝑐 results in: based on the superposition of the partial-wave eigensolutions. Based on the preceding concepts and with 1 results in: wave propagation in the x-direction, c p = ω/k c . Solving Equations (15) and (16) for c− x = z (17) ± 𝑐x − 𝑐 𝑐 = 1 c− z =±

and:

𝑐

and:

q

2 −2 c− L − cx

=± 𝑐

(17)

−𝑐

(18)

q

−1 2 2 ± bec− − c− x respectively. Equations (9), (10) and (17)czcan=then into Equation (11) to get matrix 𝐴 Ssubstituted as a function of 𝑐 and 𝑐 :

(18)

respectively. Equations (9), (10) and (17) can then be substituted into Equation (11) to get matrix A as −1 1 (𝜆 + 𝜇)(𝑐 − 𝑐 ) a function of c− x and c L : 0 ± 𝑐 − 𝑐 𝑐 (𝜆 + 𝜇) 𝐴

( )

0

≔

−(𝜆 + 𝜇)𝑐  2𝑐 𝑐 −2(𝜆 + 𝜇) 0 − c (λ +± µ𝑐) c− x 0 L

(19)

q0  −2 2 −1 (𝜆 )− + 𝜇)(−𝑐 c − c c ± λ + µ ( ) x x L   −2 A( L)Equations :=  −(λ +into µ)cEquation  And (9), (10) and0 (18) can be substituted () to get matrix 𝐴 as a function 0 L  q
− 2 − 2 − 2 0 of and 𝑐 and 𝑐 ±: c − c x c−1 (λ + µ) (λ + µ) −c x x L 

(19)

And Equations (9), (10) and (18) can be substituted into Equation (11) to get matrix A as a function of −1 1 and c− x and cS :   A(S) : =  

  2 + µ c−2 − c−2 − µc−2 (λ + 2µ)c− x x S S 0

±

q

2 c− S

2 −1 − c− x c x (λ + µ)

0 0 0

±

q

2 −2 −1 c− S − c x c x (λ + µ)



    0 −2 2 −2 − µc−2 µc− x + ( λ + 2µ ) cS − c x S

(20)

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Finding the reduced row echelon form of A(L) and A(S) gives the solution to the eigenvectors, which contain, at most, two free variables (i.e., m and p). Considering A(L) and A(S) , and also the “±” terms in them, which correspond with the sign of kz , results in four possible eigenvectors (not including the multiplicity). For the positive kz eigenvalue of the longitudinal wave, the eigenvector is:  (+ L)

 = 

ui

−1 q cx m 2 −2 c L −c− x

0 m

   

(21)

While for the negative k z eigenvalue of the longitudinal wave, the eigenvector is:  (− L)

ui

−q

 = 

1 c− x m 2 −2 c L −c− x

   

0 m

(22)

For the positive k z eigenvalue of the shear wave, the eigenvector is:  (+S)

−

q

1 c− x

 = 

ui

2 −2 c− S −c x

p m

 m   

(23)

While for the shear wave’s negative k z eigenvalue, the eigenvector is:  (−S)

ui

q

2 −2 c− S −c x 1 c− x

 = 

p m

 m   

(24)

The superscripts L and S correspond to the longitudinal and the shear waves, respectively. The S can be made more specific by using SV and SH to denote the shear vertical and shear horizontal polarizations. 1 > c−1 , the square root becomes Examining Equations (21) and (22), it is clear that for c− x L −1 1 imaginary-valued, and for c− x < c L , the square-root becomes real-valued. These two scenarios correspond with the surface wave and bulk wave solutions, respectively. The same holds true for Equations (23) and −1 −1 1 −1 (24) with cS instead of cL , with p = 0. For c− x  c L or cx  cS and positive kz , the corresponding eigenvector is:   −im   (+) ui =  0  (25) m and for negative k z it is: 

(−)

ui

 im   = 0  m

(26)

1 Hence, for very large c− x values (relative to their respective wave speeds), the eigenvectors become circularly polarized, regardless of whether it is a shear wave or longitudinal wave. Lastly, for a given k x and ω, the kz values of the four partial waves (not including the multiplicity of the shear waves) can be calculated using Equations (15) and (16):

(± L)

kz

= ±ω

q

2 −2 c− L − cx

(27)

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(±S)

kz

= ±ω

q

2 −2 c− S − cx

(28)

As implied previously, since the solution of the Christoffel equation does not depend on the boundary/interface conditions, a linear combination of the partial waves should be able to represent most elastodynamic guided waves, regardless of the boundary/interface conditions. We demonstrate this in the following sections. The material properties shown in Table 1 will be used. For each partial-wave eigenvector calculated in the following examples, the free variable is set equal to unity; then, the resulting eigenvector is normalized into a unit vector by dividing by its magnitude. Table 1. Material properties used for the calculations shown in this paper. Aluminum [4]

Tungsten [4]

Air [10]

2700 56.98 25.95 6350 3100

19,250 199.4 158.6 5180 2870

1.2 343 -

ρB , kg/m3 λB , GPa µ B , GPa cL , m/s cS , m/s

5. Rayleigh Waves and Generalized Rayleigh Waves 5.1. Rayleigh Waves The Rayleigh wave, which travels at the surface of a half-space, is a good starting example, since its solution is well-known, and this different approach will mirror the conventional derivation in the literature. The term ‘generalized Rayleigh waves’ will be used to refer to a class of wave propagation solutions developed following the original Rayleigh wave derivation, as was done by Stoneley [11]. This will be considered later in section V.B. Assuming an aluminum half-space with the boundary at z = 0, an example Rayleigh wave calculation is completed using the previously mentioned methods. The eigensolutions for the six partial waves at a frequency of 3.188 × 106 rad/s and a wavenumber of 1103.4 rad/m are shown in Table 2. The frequency and wavenumber were chosen to match the non-dispersive phase velocity solution to the standard Rayleigh wave problem. Table 2. Eigenvalues and normalized eigenvectors from the Christoffel equation for aluminum at ω = 3.188 × 106 rad/s and k x = 1103.4 rad/m. Partial Waves

kz , rad/m

ux

uy

uz

(+L) (−L) (+SV) (−SV) (+SH) (−SH)

982.6i −982.6i 400.0i −400.0i 400.0i −400.0i

−0.747i 0.747i −0.341i 0.341i 0 0

0 0 0 0 1 −1

0.665 0.665 0.940 0.940 0 0

There are several notable results in Table 2:

•

All of the k z values are imaginary, since Rayleigh waves exist in Region 1 (Figure 1a);

•

is larger in magnitude than k z kz imaginary-valued slowness curves;

• •

kz , kz , kz , and k z have a multiplicity due to the isotropic material properties; The eigenvectors are complex, which suggests elliptical particle motion [9];

•

ui

(± L)

(±SV )

(+SV )

(−SV )

(+SH )

and ui

(+SH )

(−SH )

(±SH )

and k z

because of the hyperbolic shape of the

(−SH )

are not complex-valued.

As shown in Figure 3, only the exponentially decaying solutions are necessary for the Rayleigh wave solution, because the domain is a half-space.

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3. Conceptual depiction of the partial waves, which are necessary for the Rayleigh-wave FigureFigure 3. Conceptual depiction of the partial waves, which are necessary for the Rayleigh-wave problem, overlaid on a traction-free boundary of an aluminum half-space. problem, overlaid on a traction-free boundary of an aluminum half-space.

Although the derivation method is different, closer inspection of the 𝑘 solutions shows that

Although is different, closer of the kderivation, shows that they they matchthe thederivation values of 𝑞 method and 𝑠 given by Viktorov [9]. inspection In the conventional the Rayleigh z solutions solution superposition two exponentially decaying terms that are dependent upon the wave matchwave the values ofisqaand s given byofViktorov [9]. In the conventional derivation, the Rayleigh phase and material properties, so this close agreement is that expected. A superposition thephase solution is avelocity superposition of two exponentially decaying terms are dependent uponofthe ) (𝑥, 𝑧, 𝑡) = 𝑢 (𝑧)𝑒 ( two partial waves, when 𝑢 , can be represented as: velocity and material properties, so this close agreement is expected. A superposition of the two partial ( ) ( ) i (−k(x x + ωt)( ) ( ) waves, when ui ( x, z, t) = u𝑢i ((𝑧) z)e= (29) 𝐵 ) 𝑢 , can 𝑒 be represented + 𝐵( ) 𝑢as: 𝑒 ( ) ( ) The amplitudes of each(−partial wave, 𝐵(− , are solved bySVsubstituting into one L) and 𝐵 (− ) (− L) −ik (−SV ) for L) z (−SV ) z z u z = B u e + B ui e−ikz ( ) i of the BCs: i





𝜎



= 0 at 𝑧 = 0,



(29)

(30)

The amplitudes of each partial wave, B(− L) and B(−SV ) , are solved for by substituting into one of 𝜎 = 0 at 𝑧 = 0, (31) the BCs: and assuming 𝐵( ) = 1 . This approachσ is = taken 0 at in z =lieu 0, of satisfying both BCs (by setting the (30) zz

determinant of the coefficients matrix equal to zero), since 𝑘 is an approximate value, and the σxz attempts = 0 at z at=getting 0, determinant is never exactly zero, so most a reduced row echelon form of the (31) matrix result in the trivial solution of the eigenvector. To avoid this dilemma, we set 𝐵( ) = 1, and (− L) = 1. This approach is taken in lieu of satisfying both BCs (by setting the and assuming normalizedBas necessary. 𝜎 = 0 could be used in place of the other BC, but only one is necessary. ) ) determinant of the matrix to zero), since k x is (29) an approximate value, and the Solving for 𝐵( coefficients , it is found that 𝐵( equal = −1.154. Plotting Equation with the calculated values ( ) ( ) determinant isand never zero, sowave moststructure attempts getting wave a reduced row echelon form of 𝐵 𝐵 exactly gives the same for at a Rayleigh as that found in Viktorov [9].of the

matrix result in the trivial solution of the eigenvector. To avoid this dilemma, we set B(− L) = 1, 5.2. Stoneleyas Waves and normalized necessary. σxz = 0 could be used in place of the other BC, but only one is necessary. (− SV ) , approach can be applied Rayleigh waves. Generalized Rayleigh wavesvalues Solving forThe B same it is found that B(−SV ) to =generalized −1.154. Plotting Equation (29) with the calculated (− L ) (− SV ) refer to guided waves that demonstrate Rayleigh wave-like characteristics, such as for example, of B and B gives the same wave structure for a Rayleigh wave as that found in Viktorov [9]. Stoneley waves, Scholte waves, and Love waves [11–13]. For this example, Stoneley waves will be the focus, but a similar approach can be applied to the others as well. 5.2. Stoneley Waves Stoneley waves, which propagate along the interface between two half-spaces, will be analyzed

The same approach canwaves be applied generalized Rayleigh waves. Generalized Rayleigh as an example. Stoneley typicallytohave a non-dispersive phase velocity that is slightly less thanwaves refer to waves that Rayleigh wave-like characteristics, as for theguided denser material’s bulkdemonstrate shear wave velocity, which means that it is in Regionsuch 1 (Figure 1a).example, A Rayleigh wave-like structure should exist[11–13]. in the denser half-space [2].Stoneley A tungsten half-space Stoneley waves, Scholte wave waves, and Love waves For this example, waves will be the connected to an aluminum half-space will others be usedasfor the example. Using Auld’s example focus, rigidly but a similar approach can be applied to the well. (i.e., on pages 104–105 in [2]), the six eigensolutions are calculated frequency of will 3.188be × analyzed 10 Stoneley waves, which propagate along the interface between for twoa half-spaces, rad/s and a wavenumber of 1172.1 rad/m. The frequency and wavenumber are chosen to meet the as an example. Stoneley waves typically have a non-dispersive phase velocity that is slightly less non-dispersive phase velocity solution to the Stoneley wave problem [2]. The eigensolutions at this than the denser material’s bulk shear wave velocity, which means that it is in Region 1 (Figure 1a). frequency and wavenumber are shown in tables 3 and 4 for tungsten and aluminum, respectively. A Rayleigh wave-like wave structure should exist in the denser half-space [2]. A tungsten half-space rigidly connected to an aluminum half-space will be used for the example. Using Auld’s example (i.e., on pages 104–105 in [2]), the six eigensolutions are calculated for a frequency of 3.188 × 106 rad/s and a wavenumber of 1172.1 rad/m. The frequency and wavenumber are chosen to meet the non-dispersive phase velocity solution to the Stoneley wave problem [2]. The eigensolutions at this frequency and wavenumber are shown in Tables 3 and 4 for tungsten and aluminum, respectively.

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Table Table 3. Eigenvalues and eigenvectors from the Christoffel equation for at tungsten at 3. Eigenvalues andnormalized normalized eigenvectors from the Christoffel equation for tungsten 𝜔= ω = 3.188 and𝑘k x==1172.1 1172.1rad/m. rad/m. 3.188××10 106 rad/s rad/s and Partial Waves (+𝑳) (+L) L) (−(−𝑳) SV) (+ (+𝑺𝑽) (−SV) (−𝑺𝑽) (+SH) (+𝑺𝑯) (−SH) (−𝑺𝑯)

Partial Waves

𝒖𝒚 uy 0 0 00 00 0 0 1 −11 −1

𝒖 𝒌 , rad/m ux 𝒙 997.6𝒊 −0.762𝒊 997.6i −0.762i −997.6𝒊 0.762i 0.762𝒊 −997.6i 374.5i 374.5𝒊 −0.304i −0.304𝒊 −374.5i −374.5𝒊 0.304i 0.304𝒊 374.5i 0 374.5𝒊 −374.5i 0 0 −374.5𝒊 0

kz , 𝒛rad/m

𝒖𝒛 uz 0.648 0.648 0.648 0.648 0.953 0.953 0.953 0.953 0 00 0

Table 4. Eigenvalues and normalized eigenvectors from the Christoffel equation for aluminum at Table 4. Eigenvalues and normalized eigenvectors from the Christoffel equation for aluminum at ω = 3.188 × 106 rad/s and k x = 1172.1 rad/m. 𝜔 = 3.188 × 10 rad/s and 𝑘 = 1172.1 rad/m.

Partial Waves Partial Waves kz𝒌, 𝒛rad/m , rad/m L) 1058.1i (+(+𝑳) 1058.1𝒊 L) −1059.1i (−(−𝑳) −1059.1𝒊 562.5i (+SV) (+𝑺𝑽) 562.5𝒊 SV) −562.5i (− SH) 562.5i (+(−𝑺𝑽) −562.5𝒊 (−SH) −562.5i

(+𝑺𝑯) (−𝑺𝑯)

562.5𝒊 −562.5𝒊

uy𝒚 𝒖 00 0 0 0 00 10 −1 1 −1

ux 𝒖 𝒙

−0.742i −0.742𝒊 0.742i 0.742𝒊 −0.433i −0.433𝒊 0.433i 0 0.433𝒊 0

0 0

𝒖𝒛uz 0.671 0.671 0.671 0.671 0.902 0.902 0.902 0 0.902 00 0

For both tungsten and aluminum, the Stoneley wave is in Region 1 (Figure 1a). To construct the For both tungsten andsolution, aluminum,the thefollowing Stoneley wave is inwaves Region are 1 (Figure 1a). To construct(− theL) and Stoneley wave displacement partial needed: solutions Stoneley wave displacement solution, the following partial waves are needed: solutions (−𝐿) and (−SV ) from aluminum’s Christoffel equation, and solutions (+ L) and (+SV ) from the tungsten’s (−𝑆𝑉) from aluminum’s Christoffel equation, and solutions (+𝐿) and (+𝑆𝑉) from the tungsten’s Christoffel equation. Solutions (− L) and (−SV ) from the aluminum are considered to be exponentially Christoffel equation. Solutions (−𝐿) and (−𝑆𝑉) from the aluminum are considered to be decaying solutionsdecaying for +z values, while solutions and (+SV the tungsten considered (+ L)solutions ) from exponentially solutions for +𝑧 values, while (+𝐿) and (+𝑆𝑉) from theare tungsten to be exponentially decaying solutions for −z solutions values. for These are shown Figure with respect to are considered to be exponentially decaying −𝑧 values. Theseinare shown4in Figure 4 the interface. with respect to the interface.

Figure 4. Conceptual depiction of the partial waves from tungsten and aluminum, which are

Figure 4. Conceptual depiction of the partial waves from tungsten and aluminum, which are necessary necessary for the Stoneley wave problem, overlaid on the two half-spaces. for the Stoneley wave problem, overlaid on the two half-spaces. ) The superposition of partial waves for each material, when 𝑢 (𝑥, 𝑧, 𝑡) = 𝑢 (𝑧)𝑒 ( , is i (− k x + ωt ) x shown in Equationsof(32) and (33). Forfor tungsten (using values Tables 3): ui (z)e The superposition partial waves each material, whenfrom ui (x, z, t) = , is shown in ( from ) Equations (32) and (33). For tungsten Table ( values ) ( (using ) ( ) 3):( (𝑧)

𝑢

=𝐵

𝑢

𝑒



(+ L) (+ L)Table while for aluminum W (using values (+ L) from −ik z 4):z

ui ( z ) = B

ui

+𝐵



e

)

𝑢



𝑒

(

)

(32)

SV ) (+SV ) −ik(+ z z



SV ) (−SV ) −ik(− z ui e z



+ B(+SV ) ui

e

(32)

while for aluminum (using values from Table 4): uiAl (z)

=B

(− L)



L) (− L) −ik(− ui e z z



+B

(−SV )



(33)

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𝑢 (𝑧) = 𝐵(

)

𝑢

(

)

𝑒

(

)

+ 𝐵(

)

𝑢

(

)

𝑒

(

)

11 of 22

(33)

Since are assumed assumed to to be Since the the two two half-spaces half-spaces are be welded welded at at the the interface, interface, the the three three of of the the four four available available interface conditions: interface conditions: Al W 𝜎σzz ==𝜎σzz atz at 𝑧==0,0,

(34) (34)

Al W 𝜎σxz ==𝜎σxz atz at 𝑧==0,0,

(35) (35)

Al W 𝑢uz ==𝑢uz atz at 𝑧==0,0,

(36) (36)

Al W 𝑢u x ==𝑢u x atz at 𝑧==0,0,

(37) (37)

Coordinate Along the Z-axis (m)

( (+) SV ) (, and ) B (−SV )( (i.e., ) assuming B (+ L) = ) can be used to solve for the B(− L)𝐵 ,B can be used to solve foramplitudes the amplitudes , 𝐵 , and 𝐵 (i.e., assuming 𝐵( 1). = 1). (− L ) (+ SV ) ( )= −0.3465, B ( ) = −1.2181, Using these interface conditions, it was found that B these interface conditions, it was found that 𝐵 = −0.3465 , 𝐵 = −1.2181 , B((−SV) )==0.3108 0.3108.. Equations (33) cancan nownow be plotted to get to theget Stoneley wave’s wave structure, 𝐵 Equations(32) (32)and and (33) be plotted the Stoneley wave’s wave as shown in structure, as Figure shown5.in Figure 5.

Figure structure a Stoneley wave calculated using the the partial-wave method at an Figure 5. 5. Wave Wave structureof of a Stoneley wave calculated using partial-wave method at aluminum/tungsten interface. BlueBlue lineslines are the axis.axis. −𝑧 is−inz the half-space, and an aluminum/tungsten interface. are coordinate the coordinate is intungsten the tungsten half-space, +𝑧 inisthe aluminum half-space. andis+z in the aluminum half-space.

6. and Shear Shear Horizontal Horizontal Waves Waves 6. Lamb Lamb Waves Waves and Consider BCs, and and the as was Consider aa 1-mm 1-mm thick thick aluminum aluminum plate plate with with traction-free traction-free BCs, the same same frequency frequency as was analyzed in the previous section. The wavenumbers are calculated based on the Lamb wave analyzed in the previous section. The wavenumbers are calculated based on the Lamb wave dispersion dispersion curve at that frequency. curve solutions at solutions that frequency. 6.1. A0 Lamb Wave Mode The six eigensolutions × 10 rad/s and rad/m eigensolutions at ataafrequency frequencyof of 3.188 3.188 × 106 rad/s and aa wavenumber wavenumberof of 1688.7 1688.7 rad/m are shown in Table Table 5. 5. This frequency–wavenumber pair corresponds with the red-triangle symbol plotted in the dispersion-curve plot shown in Figure 2. Similar to the Christoffel equation solutions used for the Rayleigh -wave case, the solutions in Table 5 are typical of Region 1 (Figure 1a), in which the A0 A0mode mode exists. Unlike the Rayleigh wavethere case,is now therea is now surface, a bottom surface, so the exists. Unlike the Rayleigh wave case, bottom so the exponentially exponentially increasingare eigensolutions now necessary, will be written as increasing eigensolutions now necessary,are although they will bealthough written asthey exponentially decreasing − ik ( z − H ) ( ) z from the bottom surfacefrom upward as e surface. upward Using the system in Figure 6, exponentially decreasing the bottom as 𝑒coordinate. Using theshown coordinate system (z) assignment to the6,top or bottomto ofthe thetop plate the plate partialmeans wavesthat willthe be partial definedwaves using will e−ikzbe shown in Figure assignment ormeans bottomthat of the − H ) , respectively. ( ) ( ) or e−ikz (zusing the rule used this paper the to categorize waves defined 𝑒 or For 𝑒 simplicity, , respectively. Forinsimplicity, rule usedthe in partial this paper to between the top or bottom of the plate is shown in Table 6. categorize the partial waves between the top or bottom of the plate is shown in Table 6. Table 5. Eigenvalues and normalized eigenvectors from the Christoffel equation for aluminum at 𝜔 = 3.188 × 10 rad/s and 𝑘 = 1688.7 rad/m.

Partial Waves

𝒌𝒛 , rad/m

𝒖𝒙

𝒖𝒚

𝒖𝒛

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Table 5. Eigenvalues and normalized eigenvectors from the Christoffel equation for aluminum at ω = 3.188 × 106 rad/s and k x = 1688.7 rad/m. Appl. Sci. 2018, 8, x

12 of 22

Partial Waves

(+L)(+𝑳) (−L)(−𝑳) (+SV ) (+𝑺𝑽) (−SV) (−𝑺𝑽) (+SH ) (−SH) (+𝑺𝑯)

kz , rad/m

ux

uy

1612.4𝒊−0.723i −0.723𝒊 1612.4i −1612.4i −1612.4𝒊0.723i0.723𝒊 1339.5i 1339.5𝒊−0.621i −0.621𝒊 −1339.5i 0.621i −1339.5𝒊 1339.5i 0 0.621𝒊 −1339.5i 1339.5𝒊 0 0

(−𝑺𝑯)

−1339.5𝒊

uz

00 00 0 0 0 10 −11

0

−1

0.691 0.691 0.691 0.691 0.784 0.784 0.784 0.7840 0 0 0

Table 6. Method for assigning partial waves to the top or bottom of the plate using the coordinate 6. Method for 6assigning partial waves systemTable shown in Figure and assuming e−ikz z .to the top or bottom of the plate using the coordinate system shown in Figure 6 and assuming 𝑒

.

TopTop of Plate of Plate (+L) ) k(+SV) ( Real-valued (bulk waves) kz 𝑘 (and Real-valued (bulk waves) z and 𝑘 (− L) Imaginary-valued (surface waves) (and ) kz(−SV) ( k z Imaginary-valued (surface waves) 𝑘 and 𝑘

Bottom of of Plate Bottom Plate ) )

( k𝑘 z

(−L) )

(−SV) and kz 𝑘 ( and (+L) (+SV) kz ( )and kz ( 𝑘 and 𝑘

) )

Figure 6. Conceptual depiction of the partial waves, which are necessary for the A0 Lamb wave

Figure 6. Conceptual depiction of the partial waves, which are necessary for the A0 Lamb wave problem, overlaid on a plate in vacuum. problem, overlaid on a plate in vacuum. The superposed displacement profile in the z-direction is constructed by considering the symmetry of thedisplacement problem: The superposed profile in the z-direction is constructed by considering the symmetry of ( ) ( ) ( ) the problem: 𝑢 (𝑧) = 𝐵( ) 𝑢( ) 𝑒 ( ) ( ) ( ) ( ) ( ) ( ) +𝐵 𝑢 𝑒 +𝐵 𝑢 𝑒 + (

ui ( z ) =

) (

)

(

(38)

)

𝑢 (− L𝑒) 𝐵 L) (− L), (+ L) −ik(+ B(+ L) ui e z (z− H ) + B(− L) ui e−ikz z ( )

+

SV ) (+SV ) −ik(+ (z− H ) B(+SV ) ui e z +

where 𝑢 (𝑥, 𝑧, 𝑡) = 𝑢 (𝑧)𝑒 . For partial waves existing on the bottom surface of the plate, it is SV ) (−SV ) −ik(− z z shown in Equation (38) that the 𝑧-position is )offset thickness B(−SV ui byethe , of the plate. Just as in the Rayleigh ( ) ( ) ( ) , 𝐵 , and 𝐵 can be solved by using three of the four BC available wave problem, 𝐵

(38)

( ) i(−k x x+ωt) where(assuming ui (x, z, t) that = u𝐵 . For partial waves existing on the bottom surface of the plate, it is i (z)e = 1): shown in Equation (38) that the z-position is𝜎offset thickness of the plate. Just as in the(39) Rayleigh = 0 by at 𝑧the = 0, wave problem, B(−L) , B(+SV ) , and B(−SV ) can be solved by using three of the four BC available (assuming 𝜎 = 0 at 𝑧 = 0, (40) that B(+L) = 1): 𝜎zz = at 𝑧= = 0, 𝐻, (41) σ =00atz (39)

𝜎

(

)

(

)

= 0 at 𝑧 = 𝐻,

σxz = 0atz = 0, (

)

(42)

(40)

= 1, and 𝐵 =𝐵 = −1.0284. As before, Equation (38) can now be It was found that 𝐵 σzz = 0atz = H, used to plot the wave-structure of the A0 mode at this frequency.

(41)

σxz = 0atz = H,

(42)

6.2. S0 Lamb Wave Mode

) It wasThe found that B(−L) =at1,aand B(+SV )of=3.188 B(−SV 1.0284. before, Equation (38) rad/m can now be six eigensolutions frequency × 10= − rad/s and As a wavenumber of 591.57 used toare plot the wave-structure of the A0 mode at this frequency. shown in Table 7, where the shear vertical solutions have real-valued 𝑘 and real-valued

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6.2. S0 Lamb Wave Mode The six eigensolutions at a frequency of 3.188 × 106 rad/s and a wavenumber of 591.57 rad/m are Appl. Sci. 2018, 8, x 13 of 22 shown in Table 7, where the shear vertical solutions have real-valued kz and real-valued eigenvectors. This frequency–wavenumber pair corresponds with the with red-diamond symbol plotted eigenvectors. This frequency–wavenumber pair corresponds the red-diamond symbol plottedin the in the dispersion-curve shown in These solutions correspond withbulk the bulk waves dispersion-curve plot shownplot in Figure 2. Figure These 2.solutions correspond with the waves reflecting the plate waveguide withinreflecting the platewithin waveguide (see Figure 7).(see Figure 7). Table 7. Eigenvalues and normalized eigenvectors from the Christoffel equation for aluminum at

Table 7. Eigenvalues and normalized eigenvectors from the Christoffel equation for aluminum at 𝜔 = 3.188 × 10 rad/s and 𝑘 = 591.57 rad/m. ω = 3.188 × 106 rad/s and k x = 591.57 rad/m. 𝒖𝒚 𝒖𝒙 𝒖𝒛 Partial Waves 𝒌𝒛 , rad/m Partial (+𝑳) Waves kz , rad/m u u uz 312.93𝒊 x−0.884𝒊 y 0 0.468 L) 312.93i 0.468 (+(−𝑳) −312.93𝒊−0.884i 0.884𝒊 0 0 0.468 L) −312.93i 0.468 (−(+𝑺𝑽) 841.13 0.884i −0.818 0 0 0.575 841.13 −0.818 0 0.575 (+SV) −841.13 0.8180.818 0 0 0.575 SV) −841.13 0.575 (−(−𝑺𝑽) SH) 841.13 1 1 (+(+𝑺𝑯) 841.13 0 0 00 (−(−𝑺𝑯) SH) −841.13 0 − 1 −841.13 0 −1 00

Figure 7. Conceptual depiction of the partial waves, which are necessary for the S0 Lamb wave

Figure 7. Conceptual depiction of the partial waves, which are necessary for the S0 Lamb wave problem, problem, overlaid on a plate in vacuum. The angle for the propagation direction of the shear vertical overlaid on a plate in vacuum. The angle for the propagation direction of the shear vertical partial waves partial waves was calculated using tan (𝑘 /𝑘 ). was calculated using tan−1 (k x /kz ). The displacement solution is constructed in a manner analogous to the A0 mode, except now there are bulk waves present:is constructed in a manner analogous to the A0 mode, except now there The displacement solution ( are bulk waves present: 𝑢 (𝑧) = 𝐵

ui (z() = ) 𝐵 )

𝑢

)

𝑒

(

)

(

L) (+ L) −ik(+ B((+ L)) ui e( z) (z− H )

, 𝐵

(assuming that 𝐵 (

) (

and 𝐵

(

)

)

+𝐵

𝐵

+

(

(

) (

)

𝑢

) (

𝑢

)

(

𝑒 (

𝑒

) )

+𝐵

(

.

L) (− L) −ik(− B(− L) ui e z z

+

) (

)

𝑢

𝑒

(

)

(

)

+

(43)

SV ) (−SV ) −ik(− (z− H ) B(−SV ) ui e z +

are determined by using three of the four BCs (Equations (39)–(42))

(+SV ) ( ) (+SV )( −)ik z z = 1), and are found 𝐵 B(+SV to uibe 𝐵 e = −0.998, .

)

(43)

= −0.2139 + 0.4780𝑖 and

= 0.2139 − 0.4779𝑖. As before, Equation () can now be used to plot the wave structure of the 𝐵 SV )this (+ Lmode ) , B(−at frequency. Occasionally, theby plotted will be at (Equations a phase that(39)–(42)) is BS0 and B(+SV ) are determined usingwave threestructure of the four BCs (− L ) (+ L ) (− SV ) inconsistent with the conventional wave structure figures. This can be corrected by multiplying all of (assuming that B = 1), and are found to be B = −0.998, B = −0.2139 + 0.4780i (

)

SVdisplacements ) = 0.2139 − 0.4779i. the by 𝑒 ,As where 𝜙 =Equation tan . This multiplication is equivalent to letting and B(+ before, (43) ( )can now be used to plot the wave structure of the the at wave over some non-zero time. S0 mode thispropagate frequency. Occasionally, thedistance plotted and/or wave structure will be at a phase that is inconsistent with the conventional wave  structure  figures. This can be corrected by multiplying all of the displacements

6.3. Shear Horizontal Guided imag(Wave u ) Mode

by e−iφ , where φ = tan−1 real(u x) . This multiplication is equivalent to letting the wave propagate over x Sincedistance the shearand/or horizontal (SH) guided wave modes are similar to each other, a single discussion some non-zero time. of how each behaves is possible. The dispersion relationship found in Rose [4] is: (𝑀𝜋) =

𝜔ℎ 𝑐

− (𝑘ℎ)

(44)

𝑀 is the guided wave mode number, and ℎ = 𝐻/2. The phase velocity is equal to the shear wave velocity for 𝑀 = 0, and for large 𝜔 values, it approaches the shear wave velocity for all 𝑀 > 0. The

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6.3. Shear Horizontal Guided Wave Mode Since the shear horizontal (SH) guided wave modes are similar to each other, a single discussion of how each behaves is possible. The dispersion relationship found in Rose [4] is:

( Mπ )2 =



ωh cS

2

− (kh)2

(44)

M is the guided wave mode number, and h = H/2. The phase velocity is equal to the shear wave velocity for M = 0, and for large ω values, it approaches the shear wave velocity for all M > 0. The shear wave velocity represents the boundary between regions 1 and 2 (Figure 1a,b). That is, kz = 0 and k x = cω . The SH partial wave exists as a bulk wave in the plate that is propagating parallel to the S traction-free boundaries. When M > 0 and ω is not large, then the SH guided wave modes increase in phase velocity and are highly dispersive. With the increasing phase velocity, the incidence angles of the SH bulk waves decrease as the solution of the Christoffel equation traces the shear slowness curve (the slowness curve representation of this dispersive SH guided wave mode is shown in Auld [2]). 7. The Contribution of Various Partial Waves in Each Lamb Wave 7.1. Symmetry and Opposing Phases Based on the results from the S0 and A0 Lamb wave examples, it would seem as if there is an inherent symmetry to the problem, since the amplitudes of the partial wave pairs often have equal magnitudes or are out of phase by 180◦ . This is due to the problem’s symmetric geometry. This leads to the following simplification to the displacement solution used to calculate the wave structure:   L) L) (± L) −ik(± (∓ L) −ik(∓ ui ( z ) = B ( L) q ui e z (z− H ) + ui e z z   SV) SV) (±SV) −ik(± (∓SV) −ik(∓ (SV ) (z− H ) z z z +B q ui e + ui e

(45)

where q is either +1 or −1. Determining which partial waves should be assigned to the top or bottom of the plate is done in the same way as in the previous examples. As before, it is still implied by the use of the Christoffel equation that ui (x, z, t) = ui (z)ei(−kx x+ωt) . In practice though, a few test calculations reveal that although the amplitudes of the partial-wave pairs are close to equivalent, they are not exactly equivalent. At best, Equation (45) can be used as a general description of the characteristics of many of the Lamb waves, but it is not a mathematically viable solution for our current application. That being said, the general description is invaluable for analyzing the characteristics of Lamb waves. 7.2. Using Symmetry to Summarize the Contribution of Longitudinal and Shear Partial Waves One of the advantages of the partial-wave method approach is that the contribution of each partial wave can be determined for each Lamb wave mode. That is, if the amplitudes are normalized by dividing by:   2 2 2 2 1/2 Bmag = | B(+ L) | + | B(− L) | +| B(+SV ) | + | B(−SV ) | (46) and since |B(+L) | ≈ |B(−L) | due to the symmetry of the problem, a representation of the amount of longitudinal partial wave contribution can be calculated using the average of B(+L) and B(−L) : ( L)

Bavg =

| B(+ L) | + | B(− L) | 2Bmag

(47)

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( L)

This average is used to account for the approximate nature of the solutions. Bavg is representative of the amount of the longitudinal partial waves that make up a given Lamb wave mode, and is overlaid on the dispersion curves as an intensity plot in Figure 8a. Similarly, since |B(+SV ) | ≈ |B(−SV ) | due to the symmetry of the problem, the contribution of the shear vertical partial waves can also be shown by calculating: (SV )

Bavg =

| B(+SV ) | + | B(−SV ) | , 2Bmag

(48)

and plotting it in8,Figure 8b. Appl. Sci. 2018, x

15 of 22

(a)

(b) ( )

Figure 8. Amplitude of the (a) longitudinal (i.e.,(L𝐵)

(

)

) and (b) shear vertical (i.e., 𝐵(SV )) partial waves

Figure 8. Amplitude of the (a) longitudinal (i.e., Bavg ) and (b) shear vertical (i.e., Bavg ) partial waves after after normalizing and averaging the magnitude for various points on the Lamb wave dispersion normalizing and averaging the magnitude for various points on the Lamb wave dispersion curves for curves for a 1-mm thick aluminum plate. a 1-mm thick aluminum plate. In Figure 8, it can be seen that in Region 1 (Figure 1a), the apportionment of longitudinal (i.e., ( ) ( ) ( L) ) and8,shear (i.e., 𝐵 in Region ) partial 1waves are1a), similar. However, in regions 2 and 3 (Figure 𝐵 In Figure it canvertical be seen that (Figure the apportionment of longitudinal (i.e., Bavg ) 1b,c) however, this (isSVno ) longer the case. In Region 2 (Figure 1b), the Lamb waves around 4000 m/s and shear vertical (i.e., Bavg ) partial waves are similar. However, in regions 2 and 3 (Figure 1b,c) however, are almost entirely made up of bulk shear vertical partial waves, and Lamb waves just below 6000 this is m/s no longer theupcase. In Region 2 longitudinal (Figure 1b),partial the Lamb waves around 4000 arehave almost entirely are made of mostly surface waves. Interestingly, Lambm/s modes phase made up of bulk shear vertical partial waves, and Lamb waves just below 6000 m/s are made up of and group velocities of √2𝑐 and are comprised of shear bulk waves propagating at ±45° [14].mostly √ surfaceRegion longitudinal waves. Interestingly, Lamb modes have phase andof group velocities of 2cs 2 (Figurepartial 1b) also has a clear phase velocity dependence on the amount each partial wave, but no frequency In Region 3 (Figure 1c), degree of mode dependence appears, and are comprised of dependence. shear bulk waves propagating ata±certain 45◦ [14]. Region 2 (Figure 1b) also has a clear but it does not appear to differentiate between symmetric and anti-symmetric modes. phase velocity dependence on the amount of each partial wave, but no frequency dependence. In Region 3

(Figure 1c), a certain degree of mode dependence appears, but it does not appear to differentiate between 7.3. Using Opposing Phases to Sort Symmetric and Anti-Symmetric Lamb Wave Modes symmetric and anti-symmetric modes. Another advantage of using the partial-wave method is that the phase difference between the partial waves atPhases the top to and bottom of the plate be different depending on the type of Lamb wave. 7.3. Using Opposing Sort Symmetric and can Anti-Symmetric Lamb Wave Modes

That is, in symmetric modes, the top partial wave will be about 180° out-of-phase with the bottom

Another advantage of using the partial-wave method difference between the partial partial wave. Anti-symmetric modes will typically have is thethat topthe andphase bottom partial waves in-phase. wavesThis at the top and bottom of the plate can be different depending on the type of Lamb wave. That is, can be demonstrated by calculating 𝑞 from Equation (45): ◦ in symmetric modes, the top partial wave will be about ( ) 180 out-of-phase with the bottom partial wave. 𝐵 (49) can be Anti-symmetric modes will typically have the𝑞 = top ( and bottom partial waves in-phase. This ) 𝐵 demonstrated by calculating q from Equation (45): which should be equivalent to: ( L)

BTop 𝐵 (

q𝑞==

((L))

)

B𝐵Bottom

(50)

However, since the frequency–wavenumber pairs are approximations, 𝑞 will vary from −1 to +1 for most Lamb waves. The values of 𝑞 are overlaid on the dispersion curves in Figure 9. It is worth emphasizing that the phase difference between the top and bottom partial waves applies to both the longitudinal and shear vertical partial waves, as shown in Equation (45).

(49)

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which should be equivalent to: (SV )

q=

BTop

(50)

(SV )

BBottom However, since the frequency–wavenumber pairs are approximations, q will vary from −1 to +1 for most Lamb waves. The values of q are overlaid on the dispersion curves in Figure 9. It is worth emphasizing that the phase difference between the top and bottom partial waves applies to16both the Appl. Sci. 2018, 8, x of 22 longitudinal and shear vertical partial waves, as shown in Equation (45).

9. 𝑞 valueiswhich is indicative of thedifference phase difference between and bottom Figure 9.Figure q value which indicative of the phase between the topthe andtop bottom partialpartial waves for waves for various points on the Lamb wave dispersion curves for a 1-mm thick aluminum plate. The various points on the Lamb wave dispersion curves for a 1-mm thick aluminum plate. The color axis is color axis is constrained to +1 and −1. constrained to +1 and −1.

Figure 9 demonstrates the phase difference characteristics of symmetric and anti-symmetric Lamb waves, but it also how those characteristics characteristics break down at high frequencies where Figure 9 demonstrates the shows phase difference of symmetric and anti-symmetric Lamb dispersion At these intersectionbreak points,down a phase difference (either +1 or −1) will waves, but it also curves showsintersect. how those characteristics at high frequencies where dispersion dominate At the these surrounding points of both dispersion curves. The + clearest of this the curves intersect. intersection points, a phase difference (either 1 or −examples 1) will dominate phenomenon are circled in red in Figure 9. surrounding points of both dispersion curves. The clearest examples of this phenomenon are circled in red in Figure8.9. Leaky Characteristics of Lamb Waves Lamb waves travel through a traction-free plate. Leaky Lamb waves are the waves that 8. Leaky Characteristics of Lamb Waves propagate through the same waveguide when the traction-free boundaries are replaced with fluid

Lamb waves travel through literature a traction-free plate. Leaky Lamb waves arecalculation the wavesofthat propagate half-spaces. The theoretical on leaky Lamb waves focuses on the dispersion and determining these curves differ fromboundaries the conventional Lamb wave dispersion curves throughcurves the same waveguidehow when the traction-free are replaced with fluid half-spaces. [15,16]. The focus ofonthis section is not this, focuses but rather characteristics of Lamb curves waves, and The theoretical literature leaky Lamb waves onthe theleaky calculation of dispersion assuming there is no change in the dispersion curves. That is, the dispersion curves of a platefocus determining how these curves differ from the conventional Lamb wave dispersion curves [15,16]. The surrounded by a vacuum will be used as an approximation for the dispersion curves of a plate of this section is not this, but rather the leaky characteristics of Lamb waves, assuming there is no change surrounded by air. This is a universal standard practice. The goal will be to use the partial-wave in the dispersion curves. That is, the dispersion curves of a plate surrounded by a vacuum will be method to determine the characteristics of the acoustic leakage into the fluid from the Lamb wave used as travelling an approximation the dispersion curves of a plate surrounded by air. This is a universal in the platefor waveguide. standard practice. goal will be to use the partial-wave method to determine of the Before The continuing, it should be stated that for fluids, the Eulerian form ofthe the characteristics balance of linear acousticmomentum leakage into fluid from forthe acoustics is: the Lamb wave travelling in the plate waveguide. Before continuing, it should be stated 𝜕 that for𝜕fluids, the 𝜕Eulerian form of the balance of linear 𝜌 𝑣 +𝑣 𝑣 =− 𝑃 (51) momentum for acoustics is: " 𝜕𝑡 𝜕𝑥 !# 𝜕𝑥 ∂ ∂ ∂ ρ v + v j relationship v =of−the acoustic P This complicates the velocity–pressure fluid. To simplify this, it is (51) ∂t i ∂x j i ∂xi assumed that the displacement is time-harmonic, which makes the particle velocity:

This complicates the velocity–pressure relationship 𝑣 = 𝑖𝜔𝑢 . of the acoustic fluid. To simplify this, (52) it is assumed that the displacement is time-harmonic, which makes the particle velocity: It is also assumed that the displacement is spatially harmonic and longitudinal in polarization such that:

vi = iωui .

𝑢 (𝑥, 𝑦, 𝑧, 𝑡) = 𝑈 𝑘 𝑒

,

(53)

where 𝑘 is the unit vector denoting the direction of wave propagation. Using Equations (52) and (53) in Equation (51) results in: 𝑖𝜌|𝑣 |(𝜔 − |𝑘 ||𝑣 |) = 𝑖|𝑘 |𝑃. The Eulerian form can approximate the Lagrangian form with these assumptions if:

(54)

(52)

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It is also assumed that the displacement is spatially harmonic and longitudinal in polarization such that: ui ( x, y, z, t) = Uo kˆ i e−ik j x j +iωt , (53) where kˆ j is the unit vector denoting the direction of wave propagation. Using Equations (52) and (53) in Equation (51) results in: iρ|vi |(ω − |k i ||vi |) = i |k i | P. (54) The Eulerian form can approximate the Lagrangian form with these assumptions if:

Appl. Sci. 2018, 8, x

ω  |k i |ωUo . 𝜔 ≫ |𝑘 |𝜔𝑈 .

17 of 22

(55) (55)

Equation (55) can be rearranged to: Equation (55) can be rearranged to:

cp ω . (56) U 𝜔≪ o. (56) If it is assumed that the phase velocity is around 103 m/s and that displacements are about 10−6 m, 9 rad/s,is If itfrequencies is assumedmuch that the around 10form m/sshould and that displacements are about form. 10 then for lessphase than 10velocity the Eulerian approximate the Lagrangian m, then for frequencies much less than 10 rad/s, the Eulerian form should approximate the As a result, the simpler pressure–velocity relationship: Lagrangian form. As a result, the simpler pressure–velocity relationship: ρc𝜌𝑐 (57) L iωU 𝑖𝜔𝑈o ==P, 𝑃, (57)

which was derived from the Lagrangian of the of balance of linear momentum for fluids which was derived from the Lagrangian form ofform the balance linear momentum for fluids assuming a time-harmonic displacement, canthe befollowing used for the following example. aassuming time-harmonic displacement, can be used for example. Asan anacoustic acousticwave wavein inaa fluid, fluid, the the slowness slownesscurve curvecan canbe be plotted plottedas as aa circle circlewith with aa radius radius equal equal As air to 1/𝑐 . Plotted with respect to aluminum’s slowness curve, the order of magnitude difference in to 1/cL . Plotted with respect to aluminum’s slowness curve, the order of magnitude difference in radii radii makes very dispersive part the A0 mode, very low frequencies, makes all but all thebut verythe dispersive part of the A0of mode, at very lowatfrequencies, emit acousticemit bulkacoustic waves bulkFigure waves 10). At these low frequencies, the A0 mode isand highly andlower has a (see 10).(see At Figure these low frequencies, the A0 mode is highly dispersive, has adispersive, phase velocity phase the wave speed in air. than thevelocity acousticlower wavethan speed in acoustic air.

Figure 10. Depiction of the aluminum slowness curves compared with the slowness curve for air. Figure 10. Depiction of the aluminum slowness curves compared with the slowness curve for air. Slownesscurves curveswith with solid lines denote real-valued 𝑘 solutions, and dotted lines denote imaginarySlowness solid lines denote real-valued kz solutions, and dotted lines denote imaginary-valued valued 𝑘 solutions (figure not to scale). kz solutions (figure not to scale).

The comparison of slowness curves from various media, which are shown in Figure 10, is similar The comparison of slowness curves from various media, which are shown in Figure 10, to an example in Auld [2], but that example discussed the electromagnetic emission from a is similar to an example in Auld [2], but that example discussed the electromagnetic emission from piezoelectric waveguide. The approach relies on Snell’s law, dictating that the wavenumber (i.e., 𝑘 ) a piezoelectric waveguide. The approach relies on Snell’s law, dictating that the wavenumber (i.e., k x ) should be equal at the interface between media. However, if Snell’s law were to be used directly, complex-valued angles (i.e., 𝜃) would need to be used. Based on this slowness curve representation, it can be concluded that the acoustic leakage will be an obliquely travelling bulk wave. The angle of incidence of the acoustic leakage would be between 0° and 20° for most Lamb waves. If these findings are applied to air-coupled transducers, it would suggest that there is an optimum angle of reception for the air-coupled transducers.

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should be equal at the interface between media. However, if Snell’s law were to be used directly, complex-valued angles (i.e., θ) would need to be used. Based on this slowness curve representation, it can be concluded that the acoustic leakage will be an obliquely travelling bulk wave. The angle of incidence of the acoustic leakage would be between 0◦ and 20◦ for most Lamb waves. If these findings are applied to air-coupled transducers, it would suggest that there is an optimum angle of reception for the air-coupled transducers. The partial waves for air can be represented as an obliquely traveling longitudinal wave: uiair ( x, z, t) = A air kˆ i e−ik j x j +iωt ,

(58)

and the partial waves of a Lamb wave in an aluminum plate (see Equations (38) and (43)) can be used with five of the six available interface conditions: Al P air = −σzz atz = 0,

(59)

uzair = uzAl atz = 0,

(60)

Al σxz = 0atz = 0,

(61)

Al P air = −σzz atz = H,

(62)

uzair = uzAl atz = H,

(63)

Al = 0at z = H. σxz

(64)

where an expression for P air can be found by using Equations (57) and (58) to find that: P air ( x, z, t) = iB air ρc L ωe−ik j x j +iωt

(65)

Although stress components are expressly used in the boundary conditions detailed in this paper, it is understood that they describe traction continuity. Equations (59)–(64) represent the traction and displacement continuity conditions indicative of a fluid–solid interface. We assume that B(+L) = 1, B(−L) , B(+SV ) , B(−SV ) , B(+air) , and B(−air) can be calculated as before by using the continuity conditions in Equations (59)−(64). B(+air) and B(−air) in this case would correspond with the amplitudes of the longitudinal waves at the top and bottom of the plate. All of the amplitudes were normalized by dividing each by:   2 2 2 2 2 2 1/2 air Bmag = | B(+ L) | + | B(− L) | + | B(+SV ) | + | B(−SV ) | +| B(+air) | + | B(−air) |

(66)

Equations (38) and (58) can then be used to plot the wave structure, and verify that the interface conditions are satisfied. Since it is now possible to calculate the amplitude of the acoustic leakage, every point of the Lamb wave dispersion curve can be input to determine the ideal guided wave mode for maximum (or minimum) acoustic leakage. Figure 11 shows sample calculations for a 1-mm aluminum plate. The color axis of Figure 11 is the average amplitude of acoustic leakage from Lamb waves: ( air )

Bavg =

| B(+air) | + | B(−air) | air 2Bmag

(67)

(or minimum) acoustic leakage. Figure 11 shows sample calculations for a 1-mm aluminum plate. The color axis of Figure 11 is the average amplitude of acoustic leakage from Lamb waves: 𝐵 Appl. Sci. 2018, 8, 966

(

)

=

𝐵(

)

+ 𝐵( 2𝐵

)

(67) 19 of 22

𝑐

𝑐

Figure Amplitude oflongitudinal the longitudinal waves the air half-space after normalizing and Figure 11.11. Amplitude of the partialpartial waves in the airinhalf-space after normalizing and averaging the for various points on the Lamb wave for a 1-mm thick theaveraging magnitude formagnitude various points on the Lamb wave dispersion curvesdispersion for a 1-mmcurves thick aluminum plate. aluminum plate. The Lamb wave dispersion curves weretraction-free calculated assuming boundary The Lamb wave dispersion curves were calculated assuming boundarytraction-free conditions (BC). conditions (BC).

Assuming there is a direct correlation between the strength of acoustic leakage and the amplitude Assuming there is a direct correlation between the strength of acoustic leakage and the of the acoustic wave coupled to the Lamb wave, Figure 11 suggests several things: Appl. Sci. 2018, 8,ofx the acoustic wave coupled to the Lamb wave, Figure 11 suggests several things: 19 of 22 amplitude • the A0 mode can have a lower amplitude of acoustic leakage than the S0 mode (when normalized by  the A0 mode can have a lower amplitude of acoustic leakage than the S0 mode (when normalized the amplitude of the wave structure); by the amplitude of the wave structure); • the amount of acoustic leakage is not mode-dependent;  the amount of acoustic leakage is not mode-dependent; • comparisons to Figure 8 suggest a potential relationship between the dominant partial waves present  comparisons to Figure 8 suggest a potential relationship between the dominant partial waves and the amplitude of the acoustic leakage; present and the amplitude of the acoustic leakage; the leakage leakage of of the the symmetric symmetric modes modes shown shown in in Figure Figure 11 11 agrees agrees with • when when c𝑐p = = 𝑐cL,, the with the the previous previous analysis by Rose [4](Section 8.7.1), which attributes the lack of leakage to the lack of out-of-plane analysis by Rose [4; Section 8.7.1], which attributes the lack of leakage to the lack of out-of-plane displacement displacementatatthat thatsurface. surface. Perhaps Perhaps most most interesting interesting is is that that the the maximum maximum acoustic acoustic leakage leakage is is concentrated concentrated in in Region Region 22 (Figure 1b) of the dispersion curves, while Region 1 (Figure 1a) has noticeably less acoustic (Figure 1b) of the dispersion curves, while Region 1 (Figure 1a) has noticeably less acoustic leakage. leakage. A possible explanation for why this would be is that the guided wave modes in Region 1 (Figure A possible explanation for why this would be is that the guided wave modes in Region 1 (Figure 1a) 1a) are are composed composed of of surface surface partial partial waves, waves, which whichare areinherently inherentlycoupled coupledto tothe theinterface. interface. In of the the plate. plate. In Region Region 22 (b), (b), bulk bulk shear shear vertical vertical partial partial waves waves propagate propagate through through the the thickness thickness of Incident upon the aluminum–air interface, this can be modeled as an oblique incidence problem Incident upon the aluminum–air interface, this can be modeled as an oblique incidence problem to to determine and transmission characteristics, as shown in Figure 12 where i is the 𝜃incident determinethe thereflection reflection and transmission characteristics, as shown in Figure 12 θwhere is the angle of aangle shearofvertical wave. incident a shearbulk vertical bulk wave.

Figure 12. 12. Depiction Depiction of of the the oblique oblique plane plane wave wave problem problem being being considered. considered. Figure

Displacement terms for each bulk wave are constructed according to particle polarization and wave propagation direction. The same BCs that are detailed in Equations (59)−(64) are applied to this problem. Along with Snell’s law, these BCs are used to calculate the displacement amplitude ratios as a function of incidence angle, 𝜃 , as shown in Figure 13: 

𝑅 is the particle displacement amplitude ratio between the reflected longitudinal wave in the

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Displacement terms for each bulk wave are constructed according to particle polarization and wave propagation direction. The same BCs that are detailed in Equations (59)−(64) are applied to this problem. Along with Snell’s law, these BCs are used to calculate the displacement amplitude ratios as a function of incidence angle, θi , as shown in Figure 13:

• • •

R L is the particle displacement amplitude ratio between the reflected longitudinal wave in the aluminum and the incident shear wave in the aluminum. RS is the particle displacement amplitude ratio between the reflected shear wave in the aluminum and the incident shear wave in the aluminum. TL is the particle displacement amplitude ratio between the transmitted longitudinal wave in the air and the incident shear wave in the aluminum. Appl. Sci. 2018, 8, x

( L)

( L)

20 of 22 ( L)

It is noted that in place of sin θ, the ratio of wavenumbers were used, e.g., sin θ Al = k x /|k i |.

Figure13. 13.(a) (a)The The absolute absolute value, value, (b) (b) real Figure real part, part, and and (c) (c) imaginary imaginarypart partofofthe theamplitude amplituderatios ratiosfor foran incident shear vertical bulk wave at an aluminum–air interface. an incident shear vertical bulk wave at an aluminum–air interface.

By setting up Snell’s law to be an equivalence between a bulk wave and an arbitrary wave in the By setting up Snell’s law to be an equivalence between a bulk wave and an arbitrary wave in the same plate, the relation is: same plate, the relation is:  |  | −1𝑐 sin 𝜃 =| k i | sin 𝜃. (68) cS sin θi = sin θ. (68) ω Considering that 𝜃 is defined with respect to the z-axis, Snell’s law becomes: Considering that θ is defined with respect to the z-axis, Snell’s law becomes: 𝑐 sin 𝜃 = 𝑘 /𝜔 = 𝑐 . (69) 1 c− (69) S sin θi = k x /ω = c p . The black vertical line in Figure 13 corresponds with an incidence angle of about 45.45°, which ◦, translates to a phase velocity 4350 m/s by Snell’s law. For an reference, a phase velocity at the shear The black vertical line inofFigure 13 corresponds with incidence angle of about 45.45 wave speed, 3100 m/s, corresponds with an incidence angle of 90°, and a phase velocity at the which translates to a phase velocity of 4350 m/s by Snell’s law. For reference, a phase velocity longitudinal wave speed, 6350 m/s, corresponds with an incidence angle of 29°. The angles in the vicinity of 45.45° are associated with very little mode conversion to longitudinal modes, but most importantly to a large transmission of acoustic longitudinal waves to the air half-space. The behavior of the waves within the plate is consistent with what is Region 2 of Figure 8b. Region 3 (c) would correspond to smaller angles of incidence as the phase velocity increases. For

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at the shear wave speed, 3100 m/s, corresponds with an incidence angle of 90◦ , and a phase velocity at the longitudinal wave speed, 6350 m/s, corresponds with an incidence angle of 29◦ . The angles in the vicinity of 45.45◦ are associated with very little mode conversion to longitudinal modes, but most importantly to a large transmission of acoustic longitudinal waves to the air half-space. The behavior of the waves within the plate is consistent with what is Region 2 of Figure 8b. Region 3 (c) would correspond to smaller angles of incidence as the phase velocity increases. For angles of incidence that are less than the first critical angle (i.e., k x /ω < 1/c L ), the mode conversion to longitudinal waves is significant with progressively less transmission of acoustic leakage as the angle of incidence decreases. At the larger phase velocities in Region 3 (Figure 1c) (i.e., c p > c L ), the longitudinal bulk waves traveling through the plate would then begin to contribute significantly to any acoustic leakage. In many studies of leaky Lamb waves, the fluid medium in which the wave leaks is water; thus, it is worth mentioning that the dispersion curves do not change significantly when water coupling is included [15,16]. As a result, many of the conclusions made in this example calculation pertaining to the form of the acoustic leakage can be found to agree, to some degree, with the findings in Hayashi and Inoue’s paper [15]. Specifically, the findings that the oblique propagation of the longitudinal acoustic wave in the water and the evanescent propagation of it at a phase velocity less than the longitudinal wave speed in water agree. We reiterate that Hayashi and Inoue [15] and Gravenkamp et al. [16] focus on the study of the dispersion curves of leaky Lamb waves, which is different from what was done in this example calculation. 9. Conclusions This paper exploits the Christoffel equation, slowness curves, and the partial-wave method to establish a foundation on which guided waves can be analyzed, compared, and explained. The approach was used to calculate the wave structures of many types of guided waves, once the frequency–wavenumber pairs were known. It is, of course, possible to calculate the wave structures using the well-known methods for dispersion curve calculation, but those methods can often be calculation-intensive. There are analytical expressions, but those leave little room for conceptual interpretation. Thus, the approach detailed in this paper is seen as a relatively easy, general, and informative method for calculating wave structures of specific modes of interest. Indicative of that are the Lamb wave characteristics observed in the paper, which relate to the phase difference between the top and bottom partial waves, and the amount by which each partial wave contributes to each Lamb wave. It was shown, among other things, that in Region 2 (Figure 1b), there is a phase velocity range in which the Lamb waves consist of partial shear vertical bulk waves traveling at ±45◦ . These are known as Lame waves, and they exhibit the most leakage into the fluid half-spaces surrounding the plate. Likewise, there is a similar band where all of the modes are dominated by longitudinal surface waves. It was also shown that the symmetric and anti-symmetric modes can be characterized by the phase difference between the partial waves associated with the top surface and those associated with the bottom surface. The example calculation of the leaky characteristics of Lamb waves is applicable to air-coupled transducers. The optimum angle calculation is merely Snell’s law, but the calculation of the amplitude of acoustic leakage provides us with a potentially simple methodology to compare the amount of acoustic leakage from various Lamb waves. Also, the potential relationship between the acoustic leakage from Lamb waves and the well-known oblique incidence problems could help to explain some of the acoustic leakage phenomenon using familiar and relatively simple theories. Beyond Lamb waves, a similar approach can be used for Rayleigh waves. For other guided waves such as the Lamb-like waves in a plate on a half-space, the method detailed in this paper should still apply. That is, there will be six partial waves (i.e., four in the plate and two in the half-space) and six BCs (i.e., two traction-free at the top of the plate, and two traction continuity and two displacement continuity at the interface).

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Lastly, the partial-wave method is not at all restricted to isotropic materials; it can also be used with anisotropic materials. If a similar approach is attempted for anisotropic materials, the immediately discernable differences would be that there are four regions instead of three in the dispersion curves, and three slowness curves instead of two. Some symmetry in the eigensolutions of the Christoffel equation may also disappear as a result of the anisotropy. That being said, the general approach is still expected to hold true, although further analysis is needed to confirm this. Author Contributions: C.H. conducted the modeling and drafted the manuscript. C.L. advised C.H. and helped edit the manuscript. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflicts of interest.

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