CAPITULO 2: DESCRIPCIÓN DE LA SOLUCIÓN PROPUESTA
2.8 De lo abstracto a lo concreto
Circular Elliptical
Non-Circular Non-Circular
(a) Isotropic material θ
θ θ
θ
(b) Unknown material
(Perfect ellipse, specified aspect ratio)
(c) Possibly centrifugally cast
stainless steel (d) Possibly unidirectional composite material
cp cp
cp
cp
Figure 3.2. Sample longitudinal or shear wave velocity surfaces in anisotropic media. (Polar representation of cp and θ)
3.4 On Velocity, Wave, and Slowness Surfaces
We will now contemplate several possible phase velocity profiles for anisotropic media, all nonspherical in nature. Several possible velocity surfaces are illustrated in Figure 3.2.
If we are given specific phase velocity values as a function of wavevector direction or a specific angle θ then we can visualize a slowness profile, which is simply a plot of 1/cp versus k or θ. Consider the sketch in Figure 3.3, which was generated by solving the Christoffel equation discussed previously. Useful information can then be extracted from the resulting curve.
If we are given the wave surface and the k or θ direction, then we can calculate cp and φ. For example, the energy velocity and group velocity vectors are the same for lossless media. This can be seen by examining a normal to the slowness profile – as illustrated in Figure 3.3, which also gives us the skew angle φ. If we were now able to plot (on a point-by-point basis) a locus of points as a function of θ, we would be able to produce group velocity and skew angle as a function of θ. If we were instead to plot cE versus the sum of θ plus φ (which we call ψ), then we would obtain the wave surface term illustrated in Figure 3.4. This gives us the group or energy velocity variation with angle (with no mention of launch angle if we didn’t know θ and φ).
For some physical insight, imagine the actual wave surface propagating from an acoustic source, sending waves in one particular direction that could be sensed with some sensor for a field distribution. As a result of a superposition of all of the phase velocity contributions from different directions, if the source had some finite size then the energy velocity vector would be normal to the slowness profile for the material. See Love (1926), Musgrave (1959), or Pollard (1977) for more details.
We can now explore briefly the fact that the group velocity is normal to the slowness profile. For many different kinds of problems, we’ll be extracting information similar to that found to produce a frequency spectrum of ω(k). For example, ω(kx) is obtained for the one-dimensional string problem; ω(kx, ky) is the result for the anisotropic media problem in one plane. In general, ω(kx, ky, kz) could be obtained for a three-dimensional problem.
Let’s consider ω(kx, ky). This result could be plotted in three dimensions: for a particular ω value, a plane parallel to the (kx, ky)-plane could be used to intersect the conical-like ω(kx, ky)-surface. For anisotropic media, an intersection plane parallel to the (kx, ky)-plane does not produce a circle; see Figure 3.5.
Any ω could be selected, since only a scale factor would be changed and the general shape of the curve would be the same. Note that the line drawn from the origin to a point in question would give cp in that particular direction. From three-dimensional calculus, the normal to the tangent plane to the ω(kx, ky) function at the point of interest would have a slope equal to (∂ω/∂kx, ∂ω/∂ky, 1). The intersection of the plane could achieve the slowness curve, (1/cp)(kx, ky); this projection onto the (kx, ky)-plane is what we normally extract from the Christoffel equation solution
Derived from 1
ψ = θ + φ θ
φ cE
Cp k
versus θ
Figure 3.4. Sample wave surface showing cE versus ψ.
φ is skew angle cE cosφ = cp
θ
φ cE
k
Figure 3.3. Sample slowness profile of 1/cp versus ψ.
49 3.4 On Velocity, Wave, and Slowness Surfaces
results. The projection of n onto the (kx, ky)-plane or dot product is simply (∂ω/∂kx,
∂ω/∂ky, 0), which we recognize as the group velocity.
It is easy to show that this group velocity vector direction is normal to the slowness profile. Imagine an intersection of a plane parallel to the (kx, ky)-plane with the tangent plane to the ω(kx, ky) function at the point in question. This projects as a line onto (kx, ky)-space that is tangent to the curve (1/cp)(kx, ky). Consider t as the tangent line with slope (tx, ty, 0). Then t n• = 0 and
t k t
x k
x y
y
∂
∂
+ ∂
∂
+ =
ω ω 0 0.
An alternative argument is presented by Auld (1990) whereby k + Δk is considered in a limit process to show that cg is normal to the slowness profile.
In the wave propagation diagrams of Figure 3.6, wave packets are sent out in all directions but the wavevector is in the phase velocity direction k only. If we view (via Huygens’s principle) the acoustic source as a series of point sources, then potential constructive interference path directions are as illustrated in Figure 3.6. Note that an infinite number of point sources is contemplated for a plane wave. Waves across the face of the transducer emanate in all directions. If the velocity surface, or phase velocity profile, is symmetric with respect to the direction D (sometimes called the director), giving us a sense of the orientation of an anisotropic material, then there would be no skew angle in that particular direction and the phase velocity would be equal to the energy velocity. On the other hand, if the wavevector k is not in line with the director of the material then the wave interference pattern becomes much more difficult to evaluate. The waves propagating from each point source are nonspherical in nature, but in this case are inclined at an angle Γ. The selected wave path for energy concentration will be (a), (b), (c), or some other path, depending on the actual phase velocity profile, the inclined angle Γ, and the resulting interference patterns.
Consequently, from a piston or point source (for example), three wave surfaces are produced – each with its own phase velocity variation with direction. The faster
ω
ωc
n
ky
kx
Figure 3.5. Typical intersection of the ω(kx, ky)-surface with a plane parallel to the (kx, ky)-plane, showing slowness profile and projection onto the (kx, ky)-plane.
wave surface is for longitudinal waves, followed by two shear wave surfaces, possibly with different phase velocity profiles. All of this is possible owing to interference phenomena and to the changes taking place as a result of wave velocity variations with direction, which modify the interference patterns.
For a particular direction k, consider the sketch in Figure 3.7. The distances illustrated to times ts1, ts2, and tL are proportional to the phase velocities for the directions k. The three skew angles are also shown. The distances tqs1, tqs2, and tqL
are proportional to the group velocities for the three wave types.
A number of interesting presentations can now be made. For example, examining in Figure 3.8 a wave surface profile of cE versus θ + φ shows us how to extract θ and the phase velocity value if needed. Because k is normal to the surface at the point in question for a particular cE, all information is available. If we wanted to achieve a certain energy velocity value – or, more importantly, a specific energy velocity direction – this technique shows us how to place a transducer at a specified θ or phase velocity direction, as illustrated in the figure.
That is, if we are given cE or the angle of cE, we can find out what transducer angle could be used to achieve this. We need only carry out the following steps, which are illustrated in Figure 3.8.
tqs2 tqs1 ts1
ts2
tqL tL
Quasi-longitudinal Radiation source
k Quasi-shear1
φ1
φ2
φ3 Quasi-shear2
Figure 3.7. Wave propagation into an anisotropic solid in a particular wavevcctor direction.
Acoustic source
(c)
k (a) (b)
γ D
Figure 3.6. Possible directions of wave propagation from an acoustic source in anisotropic media.
51 3.5 Exercises
(1) Plot CD tangent to wave surface at cE tip.
(2) Plot AB parallel to CD.
(3) We now know θ and cp (where cp = | cE | cos ϕ) and hence the k direction from known θ.
(4) The angle ϕ between k and the cE direction is also known and plotted.
3.5 Exercises
1. Derive the following expressions for the velocities of longitudinal and transverse waves traveling in the [1, 1, 1] direction in a cubic crystal:
c C C C
c c C C C
1 11 12 44
2 3 11 12 44
2 4 3
3
= + +
= = − +
( ) / ,
( ) / .
ρ ρ
2. For Problem 1, define the wave types c1, c2, c3 as pure or quasi-longitudinal and as pure or quasi-shear.
3. Verify the formula
c1= C11/ ,ρ c2= C44/ ,ρ c3= (C11−C12)/2ρ
for longitudinal and shear waves traveling in the [1, 0, 0] direction in a hexagonal crystal.
4. Carefully list all steps necessary to produce a computer program that calculates points on a curve of phase velocity versus angle in the (x, y)-plane for a transversely isotropic material. What assumptions are made in the analysis?
How can you define the modes and the pure or quasi nature of the modes? Is the acoustic tensor symmetric?
5. For a cubic tungsten crystal, plot approximately the three slowness profile curves.
k
A
B C
D cp
cE
Ultrasonic transducer
θ φ
φ
Figure 3.8. Transducer location angle calculation for a specific energy velocity magnitude or direction.
6. From the cubic tungsten slowness profile, graphically estimate the wave surface of cE versus ψ or θ + ϕ for the quasi-longitudinal case.
7. Graphically estimate skew angle ϕ versus θ for the quasi-longitudinal case in cubic tungsten.
8. What angle θ is required to produce a resulting energy velocity at 45° for the quasilongitudinal case in cubic tungsten?
9. Calculate the resulting phase velocity in the [1, 1, 0] direction for transversely isotropic unidirectional graphite epoxy.
10. For a plane wave in a transversely isotropic crystal, derive an expression for the acoustic tensor and the wave velocities in the [1, 0, 0] direction.
11. For cubic silicon, develop a velocity surface of c1, c2, and c3 as a function of wavevector direction in the (x1, x2)-plane. Use at least three points followed by interpolating estimates.
12. Calculate the phase velocity in a particular direction for transversely isotropic titanium or a barium titanate crystal.
13. Calculate the phase velocity in a particular direction for orthotropic barium sodium niobate. [1, 1, 0]
14. Solve for the elastic constants of a cubic diamond material if the wave velocities were measured for plane waves in the [1, 1, 0] direction as follows:
c1 = 17,890 m/s; c2 = 11,930 m/s; c3 = 10,330 m/s.
15. For Plexiglas over tungsten, determine all critical angles. Do likewise for tungsten over Plexiglas.
16. For the slowness profiles in isotropic media 1 and 2 (shown in Figure 3.9), measure critical angles for L and S input and also for total reflection (1 to 2 and 2 to 1).
17. (a) For the slowness profile shown in anisotropic media (Figure 3.10), calculate the skew angle and energy velocity at the wavcvector direction shown.
1
2 1
2
Figure 3.9. Exercise 16.
53 3.5 Exercises
(b) Estimate the energy velocity value in the wavevector direction shown.
(c) How could this result be used to produce a wave surface?
18. Given an ultrasonic transducer emitting plane wave segments on the surface of an anisotropic medium with the slowness curves shown in Figure 3.11, determine the ideal location (of a receiving transducer in through-transmission) to receive maximum energy from quasi-longitudinal wave QL and from quasi-transverse waves QT1 and QT2.
19. How could you produce an entire slowness profile for an anisotropic material?
20. Plot skew angle versus k for a given 1/cp curve (versus ϕ in a plane).
21. Plot energy velocity versus k for a given 1/cp curve (versus ϕ in a plane).
22. Select a slowness profile for a realistic material; use any reference or research paper to obtain constants. Evaluate energy velocity and skew angle profiles.
23. Estimate or use a realistic wave surface for this problem: From a given wave surface, calcuate the phase velocity and skew angle for a particular k or wave-traveling direction.
1 inch = .25 µsec/mm
20°
Figure 3.10. Exercise 17.
x
d x
y
0.25 µsec/mm
Slowness surface QT1
QL QT2
Figure 3.11. Exercise 18.
54
4.1 Introduction
Wave reflection and refraction considerations are fundamental to the study of stress wave propagation in solids. This chapter presents basic concepts with an emphasis on physical phenomena. In this chapter we examine normal beam incidence reflection factors as well as computation of refraction angles. Reflection factor concepts are outlined first, followed by angle beam analysis and mode conversion as an ultrasonic wave encounters an interface between two materials. For more details, see Auld (1990), Brekhovskikh (1960), Graff (1991), or Kolsky (1963).