Using the Partial Wave Method for Wave Structure Calculation and the Conceptual Interpretation of Elastodynamic Guided Waves

applied sciences

Article 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

Appl. Sci. 2018, 8, 966; doi:10.3390/app8060966 www.mdpi.com/journal/applsci Appl. Sci. 2018, 8, 966 2 of 22 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. Sections2 and3 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. Section4 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. Sections5 and6 apply the new method of interpretation to a variety of guided waves, which include Rayleigh waves, Stoneley waves, Lamb waves, and shear horizontal waves. Section7 focuses on Lamb waves, and a method developed in Section6 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. Section8 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 = Cijklekl (2) 1 e = u + u (3) ij 2 i,j j,i Appl. Sci. 2018, 8, 966 3 of 22

i(−kjrj+ωt) ui(x, y, z, t) = Uie (4)

Here σij, eij, and ui are the stress, strain, and displacement; ρ and Cijkl are the mass density and elastic stiffness; kj 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:

2 CijklkkUlkj − ρω 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 = λekkδij + 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:

L i(−kx x+ωt) ui (x, t) = Uoxeˆ (7) or: S i(−kx x+ωt) ui (x, t) = Uozeˆ (8) 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:

2 λ + 2µ = ρcL (9)

2 µ = ρcS (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 −1 −1  (λ + 2µ)cx + µcz − ρ 0 cx cz (λ + µ)  −2 −2  A =  0 µ cx + cz − ρ 0  (12) −1 −1 −2 −2 cx cz (λ + µ) 0 µcx + (λ + 2µ)cz − ρ

−1 Equations (11) and (12) represent a system of linear homogeneous equations, where cx = kx/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) cx + cz ρ − µ cz + cx = 0 (14)

The solutions to the characteristic equation (Equation (14)) are: ρ c−2 + c−2 = (15) x z (λ + 2µ) Appl. Sci. 2018, 8, 966 4 of 22

ρ c−2 + c−2 = (16) x z µ If Equations (9) and (10) are considered, the right-hand side of Equations (15) and (16) become −2 −2 cL and cS , 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.

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 kx/ω axis instead of the cz 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-kx) 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 kx, in which the partial waves have specific characteristics. Also, in Auld [2], it is recognized that at an interface, kx 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 kx 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 < cp < cS All six partial waves are surface waves (imaginary-valued kz) (see Figure1a). • Region 2: cS < cp < cL: Two partial waves are surface waves (imaginary-valued kz), and four are bulk waves (real-valued kz) (see Figure1b). • Region 3: cL < cp: All six partial waves are bulk waves (real-valued kz) (see Figure1c).

−1 −1 The inverses of the shear and longitudinal group velocities (cS and cL ) 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 kZ 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. Appl. Sci. 2018, 8, 966 5 of 22

Appl. Sci. 2018, 8, x 5 of 22

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 an isotropic solid. Slowness curves with solid lines denote real-valued k solutions, solutions, and dotted lines denote imaginary-valued 𝑘 solutions. This figure is a compilation ofz and dottedelements lines denote from the imaginary-valued figures in Auld [2] andkz solutions.Crandall [6].This figure is a compilation of elements from the figures in Auld [2] and Crandall [6]. 4. Reinterpreting the Dispersion Curves 4. ReinterpretingAs mentioned the Dispersion previously, Curves the Christoffel equation is only dependent upon the material properties of the elastic solid in which the plane waves propagate. The BCs are not used until the As mentionedpartial-wave previously, method is applied. the Christoffel Not only that, equation but the is boundary only dependent conditions uponare not the limited material to the properties of the elasticconventional solid in traction-free which the BCs plane (e.g., waves Lamb waves). propagate. As a result, The the BCs partial-wave are not method used until is not thelimited 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 only some, that, such but as the the boundaryshear horizontal conditions guided arewaves, not this limited approach to theis more conventional traction-freecumbersome BCs (e.g., than Lamb the standard waves). derivation. As a result, However, the partial-wave its value lies in method how it provides is not a limited fundamental to any specific type of waveguide.foundation Infor fact,all of mostthe guided types waves, of guided and a common waves method should for be interpreting representable and comparing using slowness each curves. For some,one. such When as theconsidering shear horizontalreflection/transmission, guided waves, mode conversion, this approach or multi-mode is more propagation, cumbersome this than the perspective is quite beneficial. standard derivation.In modern However, literature, itsperhaps value the lies most in commonly how it provides used dispersion a fundamental curves are of foundation Lamb waves forin all of the guided waves,an aluminum and a commonplate in terms method of phase for velocity interpreting versus the and frequency–thickness comparing each product. one. WhenUsing the considering reflection/transmission,three regions depicted mode in conversion, Figure 1, we or can multi-mode divide the phase propagation, velocity dispersion this perspective curves into is quitethree beneficial. In modernregions, literature,as shown in perhapsFigure 2. By the doing most this, commonlyit is possible to used view dispersionhow the dispersion curves curve are solutions of Lamb waves match up with the partial waves’ characteristics. For example, this is a good way to dispel the in an aluminumpotential plate misconception in terms that of guided phase waves velocity are composed versus theof multiple frequency–thickness reflecting bulk waves. product. That is, Using the three regionsthe depictedoriginal assumption in Figure of1, plane we can wave divide propagat theion phase (Equation velocity (4)), dispersion which is fundamental curves into to threethe regions, as shown inChristoffel Figure2 .equation, By doing does this, not it exclude is possible the surface to view wave how solution the dispersion (i.e., imaginary-valued curve solutions 𝑘) as match 𝑘 up with the partialincreases. waves’ characteristics.It would be more accurate For example, to say that this guided is a good waves way are a tosuperposition dispel the of potential reflecting misconceptionbulk waves and surface waves with elliptical particle motion (this will be shown in the later calculations that guidedand waves examples). are composed of multiple reflecting bulk waves. That is, the original assumption of plane wave propagationIn fact, guided (Equation waves composed (4)), which solely is of fundamental reflecting bulk to waves the Christoffelexist only in Region equation, 3 (c), does where not exclude the surfacethey wave are highly solution dispersive. (i.e., imaginary-valued However, there is onek zex)ception as kx increases. to this rule: there It would exists bea range more of phase accurate to say that guidedvelocities waves in are Region a superposition 2 (Figure 1b), ofwhere reflecting only bulk bulk waves waves propagate, and surface which will waves be discussed with elliptical in a particle later section. motion (this willThe be A0 shown mode and in Rayleigh the later waves calculations exist in Region and examples). 1 (Figure 1a), where each guided wave can be In fact,represented guided waves in terms composed of six surface solely waves. of reflecting The low-frequency bulk waves S0 mode exist is only in Region in Region 2 (b), 3and (c), is where they are highlycomposed dispersive. of two However, surface waves there and is one two exceptionbulk waves to(asthis will rule:be shown there later exists in the a paper), range excluding of phase velocities in Region 2the (Figure two shear1b), horizontal where only partial bulk waves. 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 A0many mode Lamb and wave Rayleigh modes appear waves near existly non-dispersive in Region near 1 (Figure it. That1 is,a), the where higher-order each guidedLamb waves wave can be representedapproach in terms the of shear six surfacewave speed, waves. and Thethe A0 low-frequency and S0 modes S0approach mode the isin Rayleigh Region wave 2 (b), speed and for is composed of two surfacelarge waves𝜔 values. and The two shear bulk horizontal waves modes (as will also be converge shown to later this inline the for paper), large 𝜔 excludingvalues. The theline two shear horizontal partial waves. The line separating Region 1 (Figure1a) and Region 2 (Figure1b) 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 (Figure1b) and Region 3 (Figure1c) has many higher-order Lamb modes, which appear nearly non-dispersive close to it as well. Appl. Sci. 2018, 8, x 6 of 22

Appl. Sci.separating2018, 8, 966 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.

FigureFigure 2. Lamb 2. Lamb wave wave dispersion dispersioncurves curves for an an aluminum aluminum plate plate divided divided according according to the tothree the regions threeregions shownshown in Figure in Figure1. 1.

The A0 and S0 Lamb wave modes’ tendency to approach the Rayleigh wave phase velocity can The A0 and S0 Lamb wave modes’ tendency to approach the Rayleigh wave phase velocity can also be explained using this reinterpretation, since increasing 𝑘 will increase the magnitude of the also beimaginary-valued explained using 𝑘 (seethis Figure reinterpretation, 1a). Once 𝑘 since increases increasing sufficiently,kx will the increasesurface waves the magnitudedecay with of the imaginary-valuedthe depth fromk zthe(see surface Figure before1a). Onceinteractingkz increases with the sufficiently, opposite traction-free the surface boundary, waves decay thereby with the depthimitating from the a surfaceRayleighbefore wave. interactingThis also explains with the why opposite quasi-Rayleigh traction-free waves boundary,[9] have similar thereby wave imitating a Rayleighstructures wave. to Rayleigh This also waves. explains That whyis, the quasi-Rayleigh quasi-Rayleigh waveswave and [9] the have Rayleigh similar wave wave are structures both to Rayleighcomposed waves. of Thatpartial is, waves the quasi-Rayleigh (with slight differences wave and in amplitude the Rayleigh and exponential wave are both decay). 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 Many1b), this higher-order could be explained Lamb modesby surface have waves wave affecting structures the thatwave are structure sinusoidal near throughthe traction-free the thickness, but haveboundaries, slightly while larger the or reflecting smaller amplitudesbulk waves create near the sinusoidal traction-free appearance. boundaries. In Region 2 (Figure1b), this couldThis be explainedreinterpretation by surface of guided waves waves affecting provides the wavefor the structure relatively nearsimple the calculation traction-free of wave boundaries, whilestructures the reflecting based bulk on the waves superposition create the of sinusoidal the partial-wave appearance. eigensolutions. Based on the preceding Thisconcepts reinterpretation and with wave of propagation guided waves in the provides 𝑥-direction, for the 𝑐 relatively=𝜔/𝑘 =𝑐 simple. Solving calculation Equations of wave (15) and structures based(16) on for the 𝑐 superposition results in: of the partial-wave eigensolutions. Based on the preceding concepts and with −1 wave propagation in the x-direction, cp = ω/kx = cx. Solving Equations (15) and (16) for cz results in: 𝑐 =±𝑐 −𝑐 (17) q and: −1 −2 −2 cz = ± cL − cx (17)

𝑐 =±𝑐 −𝑐 (18) and: q −1 −2 −2 respectively. Equations (9), (10) and (17)cz can= then± becS substituted− cx into Equation (11) to get matrix 𝐴 (18) as a function of 𝑐 and 𝑐: 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 cx and c : 0 ±𝑐 −𝑐𝑐(𝜆 + 𝜇) L 0 𝐴() ≔ −(𝜆+𝜇)𝑐 (19) 0  ±𝑐 −𝑐−2 𝑐−2(𝜆+𝜇) 0 q  (λ + µ) c − cx (𝜆+𝜇)(−−𝑐2 )−2 −1 L 0 ± cL − cx cx (λ + µ) (L)  −2  AndA Equations:=  (9), (10) and0 (18) can be substituted−(λ + intoµ)c Equation () to get matrix0 𝐴 as a function (19)  q L  − − − −2 of and 𝑐 and 𝑐 : 2 2 1 0 ( + ) − ± cL − cx cx (λ + µ) λ µ cx

And Equations (9), (10) and (18) can be substituted into Equation (11) to get matrix A as a function of −1 −1 and cx and cS :  q  (λ + 2µ)c−2 + µ c−2 − c−2 − µc−2 −2 −2 −1 x S x S 0 ± cS − cx cx (λ + µ) ( )   A S :=  0 0 0  (20)  q  −2 −2 −1 0 −2 + ( + ) −2 − −2 − −2 ± cS − cx cx (λ + µ) µcx λ 2µ cS cx µcS Appl. Sci. 2018, 8, 966 7 of 22

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:

−1  cx  q m c−2−c−2 (+L)  L x  u =   (21) i  0  m

While for the negative kz eigenvalue of the longitudinal wave, the eigenvector is:

−1  cx  − q m c−2−c−2 (−L)  L x  u =   (22) i  0  m

For the positive kz eigenvalue of the shear wave, the eigenvector is:

q  −2 −2  − cS −cx −1 m (+S)  cx  u =   (23) i  p  m

While for the shear wave’s negative kz eigenvalue, the eigenvector is:

q  −2 −2  cS −cx −1 m (−S)  cx  u =   (24) i  p  m

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 −1 Examining Equations (21) and (22), it is clear that for cx > cL , the square root becomes −1 −1 imaginary-valued, and for cx < cL , 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 cx cL or cx cS and positive kz, the corresponding eigenvector is:   −im (+) =   ui  0  (25) m and for negative kz it is:   im (−) =   ui  0  (26) m

−1 Hence, for very large cx 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 kx 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): q (±L) −2 −2 kz = ±ω cL − cx (27) Appl. Sci. 2018, 8, 966 8 of 22

q (±S) −2 −2 kz = ±ω cS − 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 Table1 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] 3 ρB, kg/m 2700 19,250 1.2 λB, GPa 56.98 199.4 - µB, GPa 25.95 158.6 - cL, m/s 6350 5180 343 cS, m/s 3100 2870 -

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 Table2. 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 6 ω = 3.188 × 10 rad/s and kx = 1103.4 rad/m.

Partial Waves kz, rad/m ux uy uz (+L) 982.6i −0.747i 0 0.665 (−L) −982.6i 0.747i 0 0.665 (+SV) 400.0i −0.341i 0 0.940 (−SV) −400.0i 0.341i 0 0.940 (+SH) 400.0i 0 1 0 (−SH) −400.0i 0 −1 0

There are several notable results in Table2:

• All of the kz values are imaginary, since Rayleigh waves exist in Region 1 (Figure1a); (±L) (±SV) (±SH) • kz is larger in magnitude than kz and kz because of the hyperbolic shape of the imaginary-valued slowness curves; (+SV) (−SV) (+SH) (−SH) • kz , kz , kz , and kz have a multiplicity due to the isotropic material properties; • The eigenvectors are complex, which suggests elliptical particle motion [9]; (+SH) (−SH) • ui and ui are not complex-valued. As shown in Figure3, only the exponentially decaying solutions are necessary for the Rayleigh wave solution, because the domain is a half-space. Appl. Sci. 2018, 8, 966 9 of 22 Appl. Sci. 2018, 8, x 9 of 22

FigureFigure 3. Conceptual 3. Conceptual depiction depiction of of the thepartial partial waves, which which are are necessary necessary for forthe theRayleigh-wave 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 Althoughthey match the the derivation values of 𝑞 methodand 𝑠 given is different, by Viktorov closer [9]. inspectionIn the conventional of the k derivation,z solutions the shows Rayleigh that they matchwave the values solution of isq aand superpositions given by of Viktorov two exponentially [9]. In the decaying conventional terms that derivation, are dependent the Rayleigh upon the wave solutionphase is avelocity superposition and material of two properties, exponentially so this cl decayingose agreement terms is thatexpected. are dependent A superposition upon of the the phase ( ) ( ) () velocitytwo and partial material waves, properties, when 𝑢 𝑥, so 𝑧, this 𝑡 =𝑢 close 𝑧 𝑒 agreement, can is expected.be represented A superposition as: of the two partial i(−kx x+ωt)() () () () u (x z t) = u (z)e () () waves, when i , , 𝑢i(𝑧) =𝐵 𝑢 , can𝑒 be represented+𝐵 𝑢 as: 𝑒 (29)

() () The amplitudes of each partial wave, 𝐵(−L) and 𝐵 , are solved for (by−SV substituting) into one (−L) (−L) −ikz z (−SV) (−SV) −ikz z of the BCs: ui(z) = B ui e + B ui e (29)

𝜎 =0 at 𝑧=0, (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 approach is taken in lieu of satisfying both BCs (by setting the σzz = 0 at z = 0, (30) determinant of the coefficients matrix equal to zero), since 𝑘 is an approximate value, and the determinant is never exactly zero, so mostσxz attemp= 0 attsz at= getting0, 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 and assuming B(−L) = 1. This approach is taken in lieu of satisfying both BCs (by setting the normalized as necessary. 𝜎 =0 could be used in place of the other BC, but only one is necessary. determinantSolving for of the𝐵() coefﬁcients, it is found that matrix 𝐵() equal= −1.154 to zero),. Plotting since Equationkx is (29) an approximatewith the calculated value, values and the determinantof 𝐵() isand never 𝐵() exactly gives the zero, same so wave most structure attempts for ata Rayleigh getting wave a reduced as that found row echelon in Viktorov form [9]. of the matrix result in the trivial solution of the eigenvector. To avoid this dilemma, we set B(−L) = 1, 5.2. Stoneley Waves and normalized as necessary. σxz = 0 could be used in place of the other BC, but only one is necessary. Solving forTheB( −sameSV), approach it is found can that be appliedB(−SV) to= generalize−1.154. Plottingd Rayleigh Equation waves. Generalized (29) with theRayleigh calculated waves values of B(−referL) and to Bguided(−SV) giveswaves thethat same demonstrate wave structure Rayleigh forwave-like a Rayleigh characteristics, wave asthat such found as forin example, Viktorov [9]. Stoneley waves, Scholte waves, and Love waves [11–13]. For this example, Stoneley waves will be the 5.2. Stoneleyfocus, but Waves a similar approach can be applied to the others as well. Stoneley waves, which propagate along the interface between two half-spaces, will be analyzed Theas an same example. approach Stoneley can waves be applied typically to have generalized a non-dispersive Rayleigh phase waves. velocity Generalized that is slightly Rayleigh less than waves refer tothe guided denser material’s waves that bulk demonstrate shear wave velocity, Rayleigh which wave-like means that characteristics, it is in Region such 1 (Figure as for 1a). example, A StoneleyRayleigh waves, wave-like Scholte wave waves, structure and Love should waves exist [ 11in –the13 ].denser For this half-space example, [2]. StoneleyA tungsten waves half-space will be the focus,rigidly but a similarconnected approach to an aluminum can be applied half-space to thewill othersbe used as for well. the example. Using Auld’s example Stoneley(i.e., on pages waves, 104–105 which in propagate [2]), the six along eigensolutions the interface are calculated between for two a half-spaces,frequency of will3.188 be × analyzed10 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 thefrequency denser and material’s wavenumber bulk are shear shown wave in tables velocity, 3 and which 4 for tungsten means and that aluminum, it is in Region respectively. 1 (Figure 1a). 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 Tables3 and4 for tungsten and aluminum, respectively. Appl. Sci. 2018, 8, 966 10 of 22 Appl. Sci. 2018, 8, x 10 of 22

TableTable 3. Eigenvalues 3. Eigenvalues and and normalized normalized ei eigenvectorsgenvectors from from the Christoffel the Christoffel equation equation for tungsten for at tungsten 𝜔= at 6 ω = 3.1883.188× ×10 10 rad/srad/s and and 𝑘kx== 1172.11172.1 rad/m. rad/m.

Partial Waves 𝒌𝒛, rad/m 𝒖𝒙 𝒖𝒚 𝒖𝒛 Partial Waves kz, rad/m ux uy uz (+𝑳) 997.6𝒊 −0.762𝒊 0 0.648 (+L) 997.6i −0.762i 0 0.648 (−(−𝑳)L) −997.6−997.6i 𝒊 0.762 0.762i 𝒊 0 0.6480.648 (+(+𝑺𝑽)SV) 374.5374.5i 𝒊 −0.304−0.304i 𝒊 0 0.9530.953 (− ) − (−𝑺𝑽)SV 374.5−374.5i 𝒊 0.304 0.304i 𝒊 0 0.9530.953 (+SH) 374.5i 0 1 0 (−(+𝑺𝑯)SH) −374.5374.5i 𝒊 0 0 − 11 0 0 (−𝑺𝑯) −374.5𝒊 0 −1 0 Table 4. Eigenvalues and normalized eigenvectors from the Christoffel equation for aluminum at Table 4. Eigenvalues6 and normalized eigenvectors from the Christoffel equation for aluminum at ω = 3.188 × 10 rad/s and kx = 1172.1 rad/m. 𝜔 = 3.188 × 10 rad/s and 𝑘 = 1172.1 rad/m.

PartialPartial Waves Waves kz𝒌,𝒛 rad/m, rad/m ux𝒖𝒙 𝒖uy𝒚 𝒖𝒛u z (+(+𝑳)L) 1058.11058.1i 𝒊 −0.742−0.742i 𝒊 0 0.6710.671 (−(−𝑳)L) −1059.1−1059.1i 𝒊 0.7420.742i 𝒊 0 0.6710.671 (+SV) 562.5i −0.433i 0 0.902 (−(+𝑺𝑽)SV) −562.5562.5i 𝒊 0.433−0.433i 𝒊 0 0.9020.902 (+(−𝑺𝑽)SH) 562.5−562.5i 𝒊 0.4330𝒊 10 0.902 0 (− − − (+𝑺𝑯)SH) 562.5562.5i 𝒊 0 0 11 0 0 (−𝑺𝑯) −562.5𝒊 0 −1 0 For both tungsten and aluminum, the Stoneley wave is in Region 1 (Figure1a). To construct the Stoneley waveFor both displacement tungsten and solution, aluminum, the the following Stoneley wave partial is in wavesRegion are1 (Figure needed: 1a). To solutions construct( −theL ) and (−𝐿) (−SV)Stoneleyfrom aluminum’s wave displacement Christoffel solution, equation, the following and solutions partial waves(+L are) and needed:(+SV solutions) from the tungsten’s and (−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 decayingexponentially solutions decaying for +z values, solutions while for + solutions𝑧 values, while(+L )solutionsand (+ SV(+𝐿)) fromand (+𝑆𝑉) the tungsten from the are tungsten considered to be exponentiallyare considered to decaying be exponent solutionsially decaying for −z solutionsvalues. for These −𝑧 values. are shown These in are Figure shown4 within Figure respect 4 to 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(−kxx+ωt) Theshown superposition in Equations of (32) partial and (33). waves For for tungsten each material, (using values when fromui( xTables, z, t) =3): ui(z)e , is shown in () () () () Equations (32) and (33). For tungsten() (using values from Table() 3): 𝑢 (𝑧) =𝐵 𝑢 𝑒 +𝐵 𝑢 𝑒 (32)

(+L) (+SV) while for aluminumW (using values(+L) from(+L )Table−ikz 4): z (+SV) (+SV) −ikz z ui (z) = B ui e + B ui e (32) while for aluminum (using values from Table4):

(−L) (−SV) Al (−L) (−L) −ikz z (−SV) (−SV) −ikz z ui (z) = B ui e + B ui e (33) Appl. Sci. 2018, 8, x 11 of 22

Appl. Sci. 2018, 8, 966 11 of 22 () () () () () () 𝑢 (𝑧) =𝐵 𝑢 𝑒 +𝐵 𝑢 𝑒 (33)

Since the two half-spaces are assumed to be welded at the interface, interface, the three of the four available available interface conditions:conditions: Al W σ = σ atz = 0, (34) 𝜎zz =𝜎zz at 𝑧=0, (34) Al W σ = σ atz = 0, (35) 𝜎xz =𝜎xz at 𝑧=0, (35) Al = W = 𝑢uz =𝑢uz atz 𝑧=00,, (36) (36) Al W 𝑢ux =𝑢= ux atz 𝑧=0= 0,, (37) (37) can becan used be used to solve to solve for the for amplitudes the amplitudesB(−L )𝐵, B()(+SV, 𝐵)(), and, Band(−SV 𝐵)()(i.e., (i.e., assuming assumingB(+L )𝐵=()1).=1). Using thesethese interfaceinterface conditions,conditions, itit waswas found found that thatB (−𝐵L()) ==− −0.34650.3465, , B𝐵(+()SV) = −1.2181−1.2181,, 𝐵B()(−SV)== 0.31080.3108.. EquationsEquations (32)(32) and and (33) (33) can can now now be plotted be plotted to get to the get Stoneley the Stoneley wave’s wavewave’s structure, wave structure,as shown inas Figureshown5 .in Figure 5. Coordinate Along the Z-axis (m) Z-axis the Along Coordinate

Figure 5. 5. WaveWave structure structure of ofa Stoneley a Stoneley wave wave calculated calculated using using the thepartial-wave partial-wave method method at an at aluminum/tungstenan aluminum/tungsten interface. interface. Blue Blue lines lines are the are coordinate the coordinate axis. axis. −𝑧 is− inz theis in tungsten the tungsten half-space, half-space, and +and𝑧 is + inz is the in thealuminum aluminum half-space. half-space.

6. Lamb Waves and Shear Horizontal Waves Consider a 1-mm thick aluminum plate with traction-free BCs, and the same frequency as was analyzed inin the the previous previous section. section. The The wavenumbers wavenumbers are calculated are calculated based onbased the Lamb on the wave Lamb dispersion wave dispersioncurve solutions curve at solutions that frequency. at that frequency.

6.1. A0 Lamb Wave Mode The six eigensolutions at at a a frequency frequency of of 3.1883.188 × 106 rad/s rad/s and and a a wavenumber wavenumber of of 1688.7 rad/mrad/m are shown in Table5 5.. ThisThis frequency–wavenumberfrequency–wavenumber pairpair correspondscorresponds withwith thethe red-trianglered-triangle symbolsymbol plotted in the dispersion-curve plot shown in Figure2 2.. SimilarSimilar toto thethe ChristoffelChristoffel equationequation solutionssolutions used for the Rayleigh -wave case, the solutions in Ta Tableble5 5 are are typicaltypical ofof RegionRegion 11 (Figure(Figure1 1a),a), in in which which the A0A0 mode mode exists. exists. Unlike Unlike the Rayleighthe Rayleigh wave case,wave there case, is nowthere a is bottom now surface,a bottom so thesurface, exponentially so the exponentiallyincreasing eigensolutions increasing areeigensolutions now necessary, are although now necessary, they will be although written asthey exponentially will be written decreasing as exponentiallyfrom the bottom decreasi surfaceng from upward the bottom as e−ik surfacez(z−H). upward Using the as 𝑒 coordinate (). Using system the shown coordinate in Figure system6, shownassignment in Figure to the 6, top assignment or bottom to of the the top plate or meansbottom that of the the plate partial means waves that will the be partial deﬁned waves using wille−ik zbe(z) −ik (z−H) definedor e z using, respectively. 𝑒 () or For𝑒 simplicity, () , respectively. the rule used For in simplicity, this paper the to categorize rule used thein partialthis paper waves to categorizebetween the the top partial or bottom waves of between the plate the is showntop or bottom in Table of6. 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 Appl. Sci. 2018, 8, 966 12 of 22

Table 5. Eigenvalues and normalized eigenvectors from the Christoffel equation for aluminum at 6 ω = 3.188 × 10 rad/s and kx = 1688.7 rad/m. Appl. Sci. 2018, 8, x 12 of 22

Partial Waves kz, rad/m ux uy uz (+L)(+𝑳) 1612.41612.4i −𝒊 0.723 −i0.723𝒊 00 0.691 0.691 (−L)(−𝑳) −1612.4−1612.4i 𝒊0.723 i 0.723𝒊 00 0.691 0.691 (+SV(+𝑺𝑽)) 1339.51339.5i −𝒊 0.621 −i0.621𝒊 00 0.784 0.784 (−SV) −1339.5i 0.621i 0 0.784 (+SH(−𝑺𝑽)) 1339.5−1339.5i 𝒊 0 0.621𝒊 10 0.784 0 (−SH(+𝑺𝑯)) −1339.51339.5i 𝒊 0 0 −1 1 0 0 (−𝑺𝑯) −1339.5𝒊 0 −1 0 Table 6. Method for assigning partial waves to the top or bottom of the plate using the coordinate systemTable shown 6. Method in Figure for6 assigning and assuming partial wavese−ikz z .to the top or bottom of the plate using the coordinate system shown in Figure 6 and assuming 𝑒.

Top Top of Plate of Plate Bottom Bottom of of Plate Plate (+L) (+SV) (−L) (−SV) Real-valued (bulk waves) k ()and k () k ()and k () Real-valued (bulk waves) z 𝑘 andz 𝑘 𝑘z andz 𝑘 (−L) (−SV) (+L) (+SV) Imaginary-valued (surface waves) kz ()and kz () kz ()and kz () Imaginary-valued (surface waves) 𝑘 and 𝑘 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 Thesymmetry superposed of the displacementproblem: 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 + () () () (38) (+L) 𝐵 𝑢 𝑒 (−L), (+SV) (+L) (+L) −ikz (z−H) (−L) (−L) −ikz z (+SV) (+SV) −ikz (z−H) ui(z) = B ui e + B ui e + B ui e + 𝑢 (𝑥, 𝑧, 𝑡) =𝑢(𝑧)𝑒() where . For partial waves existing(−SV) on the bottom surface of the plate, it is (38) (−SV) (−SV) −ikz z shown in Equation (38) that the 𝑧-positionB is offsetui bye the thickness, of the plate. Just as in the Rayleigh () () () wave problem, 𝐵 , 𝐵 , and 𝐵 can be solved by using three of the four BC available ()i(−kxx+ωt) where(assumingui(x, z, t) that= u 𝐵i(z)e =1): . For partial waves existing on the bottom surface of the plate, it is shown in Equation (38) that the z-position is offset by the thickness of the plate. Just as in the Rayleigh 𝜎 =0 at 𝑧=0, (39) 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): σ𝜎zz =0= 0at at z𝑧=𝐻= 0,, (41) (39)

𝜎 =0 at 𝑧=𝐻, (42) σxz = 0atz = 0, (40) () () () It was found that 𝐵 =1, and 𝐵 =𝐵 = −1.0284. As before, Equation (38) can now be σ = 0atz = H, (41) used to plot the wave-structure of the A0 modezz at this frequency.

σxz = 0atz = H, (42) 6.2. S0 Lamb Wave Mode (−L) = (+SV) = (−SV) = − It wasThe found six eigensolutions that B at1, a and frequencyB of 3.188B × 10 rad/s1.0284. and Asa wavenumber before, Equation of 591.57 (38) rad/m can now be used toare plot shown the wave-structurein Table 7, where of the the A0 shear mode vertical at this solutions frequency. have real-valued 𝑘 and real-valued Appl. Sci. 2018, 8, 966 13 of 22

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 Table7, where the shear vertical solutions have real-valued kz and real-valued eigenvectors. This frequency–wavenumbereigenvectors. This frequency–wavenumber pair corresponds pair co withrresponds the with red-diamond the red-diamond symbol symbol plotted plotted in the dispersion-curvein the dispersion-curve plot shown plot in Figureshown 2in. Figure These solutions2. These solutions correspond correspond with the with bulk the wavesbulk waves reflecting withinreflecting the plate within waveguide the plate (see waveguide Figure7). (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 = 3.1886 × 10 rad/s and 𝑘 = 591.57 rad/m. ω = 3.188 × 10 rad/s and kx = 591.57 rad/m. Partial Waves 𝒌𝒛, rad/m 𝒖𝒙 𝒖𝒚 𝒖𝒛 Partial(+𝑳) Waves kz, rad/m312.93𝒊 ux−0.884𝒊 uy 0 0.468uz (+(−𝑳)L) 312.93−312.93i 𝒊− 0.8840.884i 𝒊 00 0.468 0.468 (−(+𝑺𝑽)L) −312.93841.13i 0.884−i0.818 00 0.575 0.468 (+SV) 841.13 −0.818 0 0.575 (−(−𝑺𝑽)SV) −841.13−841.13 0.8180.818 00 0.575 0.575 (+(+𝑺𝑯)SH) 841.13841.13 00 11 0 0 (−(−𝑺𝑯SH) ) −841.13−841.13 0 0 −1−1 0 0

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). −1 was calculated using tan (kx/kz). The displacement solution is constructed in a manner analogous to the A0 mode, except now Thethere displacement are bulk waves solution present: is constructed in a manner analogous to the A0 mode, except now there () () () () () () () () () are bulk waves present: () () 𝑢(𝑧) =𝐵 𝑢 𝑒 +𝐵 𝑢 𝑒 +𝐵 𝑢 𝑒 + () () () (43) (+L) 𝐵 𝑢 𝑒 (−L). (−SV) (+L) (+L) −ikz (z−H) (−L) (−L) −ikz z (−SV) (−SV) −ikz (z−H) ui(z()) = B() u e() + B u e + B u e + 𝐵 , 𝐵 andi 𝐵 are determined byi using three of the four BCsi (Equations (39)–(42)) (+SV) (43) () (+SV) (+SV)()−ikz z () (assuming that 𝐵 =1), and are foundB tou ibe 𝐵 e = −0.998. , 𝐵 = −0.2139 + 0.4780𝑖 and 𝐵() = 0.2139 − 0.4779𝑖. As before, Equation () can now be used to plot the wave structure of the BS0(+ Lmode), B(− atSV )thisand frequency.B(+SV) areOccasionally, determined the byplotted using wave three structure of the fourwill be BCs at (Equationsa phase that (39)–(42)) is (assuminginconsistent that B with(−L) the= conventional1), and are wave found structure to be figures.B(+L) This= can−0.998 be corrected, B(−SV by) multiplying= −0.2139 all+ of0.4780 i ( ) (+theSV displacements) = − by 𝑒, where 𝜙=tan . This multiplication is equivalent to letting and B 0.2139 0.4779i. As before, Equation( (43)) can now be used to plot the wave structure of the S0 modethe atwave this propagate frequency. over Occasionally, some non-zero the distance plotted and/or wave structuretime. 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 ( Wave) Mode by e−iφ, where φ = tan−1 imag ux . This multiplication is equivalent to letting the wave propagate over real(ux) some non-zeroSince distancethe shear and/orhorizontal time. (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: 𝜔ℎ (𝑀𝜋) = − (𝑘ℎ) (44) 𝑐 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 Appl. Sci. 2018, 8, 966 14 of 22

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:

ωh 2 (Mπ)2 = − (kh)2 (44) cS

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 (Figure1a,b). That is, kz = 0 ω and kx = . The SH partial wave exists as a bulk wave in the plate that is propagating parallel to the cS 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) (±L) −ikz (z−H) (∓L) −ikz z ui(z) = B q ui e +ui e (45) (±SV) (∓SV) (SV) (±SV) −ikz (z−H) (∓SV) −ikz z +B q ui e +ui e

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 i(−kxx+ωt) Christoffel equation that ui(x, z, t) = ui(z)e . 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: 1/2 (+L) 2 (−L) 2 (+SV) 2 (−SV) 2 Bmag = |B | + |B | +|B | + |B | (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) (−L) (L) |B | + |B | Bavg = (47) 2Bmag Appl. Sci. 2018, 8, 966 15 of 22

(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 Figure8a. 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) (−SV) (SV) |B | + |B | Bavg = , (48) 2Bmag and plottingAppl. Sci. 2018 it in, 8 Figure, x 8b. 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., B ) and (b) shear vertical (i.e., B ) partial waves after after normalizing and averaging the magnitudeavg for various points on the Lambavg wave dispersion normalizingcurves for and a 1-mm averaging thick aluminum the magnitude plate. for various points on the Lamb wave dispersion curves for 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., () () 𝐵 ) and shear vertical (i.e., 𝐵 ) partial waves are similar. However, in regions 2 and 3 (Figure (L) In Figure8, it can be seen that in Region 1 (Figure1a), the apportionment of longitudinal (i.e., Bavg) 1b,c) however, this (isSV no) longer the case. In Region 2 (Figure 1b), the Lamb waves around 4000 m/s and shear vertical (i.e., B ) partial waves are similar. However, in regions 2 and 3 (Figure1b,c) however, are almost entirely avgmade up of bulk shear vertical partial waves, and Lamb waves just below 6000 this ism/s no longerare made the up case. of mostly In Region surface 2 longitudinal (Figure1b), pa thertial Lamb waves. waves Interestingly, around 4000Lamb m/s modes are have almost phase entirely made up of bulk shear vertical√2𝑐 partial waves, and Lamb waves just below 6000 m/s are made±45° up of mostly and group velocities of and are comprised of shear bulk waves propagating at [14]. √ surfaceRegion longitudinal 2 (Figure partial 1b) also waves. has a clear Interestingly, phase velocity Lamb dependence modes have on the phase amount and of group each partial velocities wave, of 2cs and arebut comprised no frequency of dependence. shear bulk wavesIn Region propagating 3 (Figure 1c), at a± certain45◦ [14 degree]. Region of mode 2 (Figure dependence1b) also appears, has a clear phase velocitybut it does dependence not appear to on differentiate the amount be oftween each symmetric partial wave, and but anti-symmetric no frequency modes. dependence. In Region 3 (Figure1c), 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 7.3. Usingpartial Opposing waves at Phasesthe top toand Sort bottom Symmetric of the plate and can Anti-Symmetric be different depending Lamb Wave on the Modes type of Lamb wave. That is, in symmetric modes, the top partial wave will be about 180° out-of-phase with the bottom Anotherpartial wave. advantage Anti-symmetric of using themodes partial-wave will typicall methody have isthe that top theand phase bottom difference partial waves between in-phase. the partial wavesThis at the can top be anddemonstrated bottom of by the calculating plate can 𝑞 be from different Equation depending (45): on the type of Lamb wave. That is, ◦ in symmetric modes, the top partial wave will be about() 180 out-of-phase with the bottom partial wave. 𝐵 Anti-symmetric modes will typically have the𝑞= top and bottom partial waves in-phase. This(49) can be 𝐵() demonstrated by calculating q from Equation (45): which should be equivalent to: (L) () 𝐵BTop q = (49) 𝑞= (()) (50) 𝐵 L 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 to both the longitudinal and shear vertical partial waves, as shown in Equation (45). Appl. Sci. 2018, 8, 966 16 of 22 which should be equivalent to: (SV) BTop q = (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 Figure9. It is worth emphasizing that the phase difference between the top and bottom partial waves applies to both the Appl. Sci. 2018, 8, x 16 of 22 longitudinal and shear vertical partial waves, as shown in Equation (45).

Figure 9.Figureq value 9. 𝑞 which value is which indicative is indicative of the phase of the difference phase difference between between the top the and top bottom and bottom partial partial waves for various pointswaves for on various the Lamb points wave on the dispersion Lamb wave curves disper forsion a 1-mmcurves thickfor a 1-mm aluminum thick aluminum plate. The plate. color The 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 FigureLamb9 demonstrateswaves, but it also the phaseshows differencehow those characteristicscharacteristics break of symmetric down at andhigh anti-symmetricfrequencies where Lamb waves, butdispersion it also curves shows intersect. how those At these characteristics intersection break points, down a phase at highdifference frequencies (either +1 where or −1 dispersion) will curves intersect.dominate Atthe thesesurrounding intersection points points, of both a phase disper differencesion curves. (either The +clearest1 or − examples1) will dominate of this the 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

8. Leaky CharacteristicsLamb waves travel of Lamb through Waves a traction-free plate. Leaky Lamb waves are the waves that propagate through the same waveguide when the traction-free boundaries are replaced with fluid Lambhalf-spaces. waves travelThe theoretical through literature a traction-free on leaky plate. Lamb Leaky waves Lamb focuses waves on the are calculation the waves ofthat dispersion propagate throughcurves the same and determining waveguide how when these the curves traction-free differ from boundaries the conventional arereplaced Lamb wave with dispersion fluid half-spaces. curves The theoretical[15,16]. The literature focus of on this leaky section Lamb is not waves this, focusesbut rather on the the leaky calculation characteristics of dispersion of Lamb curveswaves, and determiningassuming how thesethere curvesis no change differ fromin the the dispersion conventional curves. Lamb That wave is, the dispersion dispersion curves curves [15 ,of16 ].a Theplate focus of this sectionsurrounded is not by this, a vacuum but rather will the be leaky used characteristicsas an approximation of Lamb for waves,the dispersion assuming curves there of is a no plate 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 astravelling an approximation in the plate forwaveguide. the dispersion curves of a plate surrounded by air. This is a universal standard practice.Before Thecontinuing, goal will it should be to use be stated the partial-wave that for fluids, method the Eulerian to determine form of thethe characteristicsbalance of linear of the acousticmomentum leakage into for the acoustics fluid from 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 momentum for acoustics is: 𝜌 𝑣 +𝑣 𝑣 = − 𝑃 (51) " 𝜕𝑡 𝜕𝑥!# 𝜕𝑥 ∂ ∂ ∂ This complicates the velocity–pressureρ v + v relationshipv = of− the acousticP fluid. To simplify this, it is (51) ∂t i j ∂x i ∂x assumed that the displacement is time-harmonic,j which makes thei particle velocity:

This complicates the velocity–pressure relationship𝑣 =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. (52) 𝑢(𝑥,𝑦,𝑧,𝑡)=𝑈𝑘𝑒 , (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. (54) The Eulerian form can approximate the Lagrangian form with these assumptions if: Appl. Sci. 2018, 8, 966 17 of 22

It is also assumed that the displacement is spatially harmonic and longitudinal in polarization such that: −ikj xj+iωt ui(x, y, z, t) = Uokˆie , (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 − |ki||vi|) = i|ki|P. (54) The Eulerian form can approximate the Lagrangian form with these assumptions if: Appl. Sci. 2018, 8, x 17 of 22

ω |ki|ωUo. (55) 𝜔≫|𝑘|𝜔𝑈. (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, then forIf it frequencies is assumed much that the less phase than 10 velocity9 rad/s, is the around Eulerian 10 form m/s should and that approximate displacements the Lagrangian are about form.10 Asm, athen result, for the frequencies simpler pressure–velocity much less than relationship: 10 rad/s, the Eulerian form should approximate the Lagrangian form. As a result, the simpler pressure–velocity relationship:

ρcLiωUo = P, (57) 𝜌𝑐𝑖𝜔𝑈 =𝑃, (57) whichwhich was derivedwas derived from thefrom Lagrangian the Lagrangian form of form the balance of the ofbalance linear momentumof linear momentum for fluids assumingfor fluids aassuming time-harmonic a time-harmonic displacement, displacement, can be used forcan the be followingused for the example. following example. AsAs anan acousticacoustic wavewave inin aa fluid,fluid, thethe slownessslowness curvecurve cancan bebe plottedplotted asas aa circlecircle withwith aa radiusradius equalequal air toto1 /1/𝑐cL. Plotted. Plotted with with respect respect to aluminum’sto aluminum’s slowness slowness curve, curve, the orderthe order of magnitude of magnitude difference difference in radii in makesradii makes all but all the but very the dispersive very dispersive part of the part A0 of mode, the A0 at verymode, low at frequencies, very low frequencies, emit acoustic emit bulk acoustic waves (seebulk Figure waves 10 ).(see At Figure these low 10). frequencies, At these low the A0frequencie mode iss, highly the A0 dispersive, mode is andhighly has adispersive, phase velocity and lowerhas a thanphase the velocity acoustic lower wave than speed the in acoustic air. wave speed in 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. Slowness curves with solid lines denote real-valued 𝑘 solutions, and dotted lines denote imaginary- Slowness curves with 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 ) should be equal at the interface between media. However, if Snell’s law were to be used directly,x 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:

𝑢 (𝑥, 𝑧, 𝑡) = 𝐴𝑘𝑒 , (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:

𝑃 =−𝜎 at 𝑧=0, (59) Appl. Sci. 2018, 8, 966 18 of 22 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:

air ˆ −ikj xj+iωt ui (x, z, t) = Aairkie , (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:

air Al P = −σzz atz = 0, (59)

air Al uz = uz atz = 0, (60) Al σxz = 0atz = 0, (61) air Al P = −σzz atz = H, (62) air Al uz = uz atz = H, (63) Al σxz = 0at z = H. (64) where an expression for Pair can be found by using Equations (57) and (58) to ﬁnd that:

air air −ikj xj+iωt P (x, z, t) = iB ρcLωe (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:

1/2 air (+L) 2 (−L) 2 (+SV) 2 (−SV) 2 (+air) 2 (−air) 2 Bmag = |B | + |B | + |B | + |B | +|B | + |B | (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:

|B(+air)| + |B(−air)| (air) = Bavg air (67) 2Bmag Appl. Sci. 2018, 8, x 18 of 22

𝑢 =𝑢 at 𝑧=0, (60) 𝜎 =0 at 𝑧=0, (61) 𝑃 =−𝜎 at 𝑧=𝐻, (62) 𝑢 =𝑢 at 𝑧=𝐻, (63) 𝜎 =0 at 𝑧 = 𝐻. (64) where an expression for 𝑃 can be found by using Equations (57) and (58) to find that:

𝑃 (𝑥, 𝑧, 𝑡) =𝑖𝐵 𝜌𝑐𝜔𝑒 (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() =1, 𝐵() , 𝐵() , 𝐵() , 𝐵() , and 𝐵() can be calculated as before by using the continuity conditions in Equations (59)−(64). 𝐵() and 𝐵() 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: / () () () () () () 𝐵 =𝐵 +𝐵 +𝐵 +𝐵 +𝐵 +𝐵 (66)

() () () 𝐵 +𝐵 𝐵 = (67) 2𝐵 Appl. Sci. 2018, 8, 966 19 of 22

𝑐

FigureFigure 11. 11.Amplitude Amplitude of the oflongitudinal the longitudinal partial partial waves inwaves the air in half-space the air half-space after normalizing after normalizing and averaging and theaveraging magnitude the for magnitude various points for various on the Lamb points wave on the dispersion Lamb wave curves dispersion for a 1-mm curves thick aluminumfor a 1-mm plate. thick Thealuminum Lamb wave plate. dispersion The Lamb curves wave were dispersion calculated curves assuming weretraction-free calculated assuming boundary traction-free conditions (BC). boundary 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.amplitude Sci. 2018, 8of, x the acoustic wave coupled to the Lamb wave, Figure 11 suggests several things:19 of 22 • 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 Figure8 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; • when cp = cL, the leakage of the symmetric modes shown in Figure 11 agrees with the previous  when 𝑐 =𝑐, the leakage of the symmetric modes shown in Figure 11 agrees with the previous analysisanalysis byby RoseRose [[4;4](Section Section 8.7.1),8.7.1], which attributes thethe lacklack ofof leakage to the lack of out-of-plane displacementdisplacement at at that that surface. surface. PerhapsPerhaps mostmost interestinginteresting isis thatthat thethe maximummaximum acousticacoustic leakageleakage isis concentratedconcentrated inin RegionRegion 22 (Figure(Figure1 1b)b) of of the the dispersion dispersion curves, curves,while whileRegion Region 11(Figure (Figure1 a)1a) has has noticeably noticeably less less acoustic acoustic leakage. leakage. AA possiblepossible explanationexplanation forfor whywhy thisthis wouldwould bebe isis thatthat thethe guidedguided wave modes in Region 1 (Figure1 1a)a) areare composedcomposed ofof surfacesurface partialpartial waves,waves, whichwhich areare inherentlyinherently coupledcoupled toto thethe interface.interface. InIn RegionRegion 22 (b),(b), bulkbulk shearshear verticalvertical partialpartial waveswaves propagate through the thickness of the plate. IncidentIncident uponupon thethe aluminum–airaluminum–air interface,interface, thisthis cancan bebe modeledmodeled asas anan obliqueoblique incidenceincidence problemproblem toto determine the reﬂection and transmission characteristics, as shown in Figure 12 where θ is the incident determine the reflection and transmission characteristics, as shown in Figure 12 wherei 𝜃 is the angleincident of aangle shear of vertical a shear bulk vertical wave. bulk wave.

Figure 12.12. Depiction of the oblique plane wavewave problemproblem beingbeing considered.considered.

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 aluminum and the incident shear wave in the aluminum.

 𝑅 is the particle displacement amplitude ratio between the reflected shear wave in the aluminum and the incident shear wave in the aluminum.

 𝑇 is the particle displacement amplitude ratio between the transmitted longitudinal wave in the air and the incident shear wave in the aluminum.

() () () It is noted that in place of sin 𝜃, the ratio of wavenumbers were used, e.g., sin 𝜃 =𝑘 /|𝑘 |. Appl. Sci. 2018, 8, 966 20 of 22

• RL 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 20 of 22 (L) (L) (L) It is noted that in place of sin θ, the ratio of wavenumbers were used, e.g., sin θAl = kx /|ki |.

FigureFigure 13. 13.( a(a)) TheThe absoluteabsolute value,value, (b) real part, and and ( (cc)) imaginary imaginary part part of of the the amplitude amplitude ratios ratios for for an anincident incident shear shear vertical vertical bulk bulk wave wave at at an an aluminum–air aluminum–air interface. 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 =|ki| sin𝜃. (68) c sin θ = sin θ. (68) S i 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 sin θi = kx/ω = cp. (69) The black vertical line in Figure 13S corresponds with an incidence angle of about 45.45°, which translatesThe black to a phase vertical velocity line in of Figure4350 m/s13 correspondsby Snell’s law. with For anreference, incidence a phase angle velocity of about at the 45.45 shear◦, whichwave translatesspeed, 3100 to am/s, phase corresponds velocity of with 4350 an m/s incidence by Snell’s angle law. of 90° For, reference,and a phase a phase velocity 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/𝜔 < 1/𝑐), 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 >𝑐), 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 Appl. Sci. 2018, 8, 966 21 of 22 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 Figure8b. 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., kx/ω < 1/cL), 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 (Figure1c) (i.e., cp > cL), 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 (Figure1b), 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). Appl. Sci. 2018, 8, 966 22 of 22

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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