Elastic Waves & Ultrasonic NDT

In this chapter

28The Elastic Wave Equation
29Mode Conversion & Snell's Law
30The Angle Beam NDT Setup
31Cracks as Scatterers
32The NDT Operator Learning Problem

28The Elastic Wave Equation

In Chapters 1–5 every PDE we studied was scalar: Darcy flow produces a single pressure value at each point, Burgers' equation tracks a single velocity component, and even Navier–Stokes in vorticity form reduces to a scalar field. Now we enter the world of vector wave propagation in solids, where the unknown is a displacement or velocity vector at every point in space and every instant in time.

This chapter and the next draw on two sources: the COMSOL "Angle Beam Nondestructive Testing" tutorial, which provides a complete 2D simulation of ultrasonic crack inspection; and the review by Mehtaj & Banerjee (2025), which surveys the application of FNO, DeepONet, and PINNs to elastic and acoustic wave propagation problems. Together they motivate a concrete FNO pipeline for crack detection.

The governing equations come from Newton's second law applied to a continuous medium. In the velocity–strain formulation (used by COMSOL's "Elastic Waves, Time Explicit" physics interface), the system reads:

Elastic Wave Equation (velocity–strain form)

$$ \rho \frac{\partial \vv}{\partial t} = \nabla \cdot \vsig + \mathbf{F} \tag{11} $$ $$ \frac{\partial \veps}{\partial t} = \frac{1}{2}\!\left(\nabla \vv + (\nabla \vv)^T\right) $$ $$ \vsig = \mathbf{C} : \veps $$

where $\rho$ is density, $\vv$ is the velocity vector, $\vsig$ is the Cauchy stress tensor, $\veps$ is the strain tensor, $\mathbf{C}$ is the fourth-order elasticity tensor, and $\mathbf{F}$ is a body force.

The first equation is momentum conservation (force = mass × acceleration per unit volume). The second is the kinematic relation between velocity and strain rate. The third is the constitutive law — it tells us how stress relates to strain for a particular material.

For an isotropic material, the full elasticity tensor $\mathbf{C}$ reduces to just two independent parameters: the Lamé constants $\lambda$ and $\mu$. The stress–strain relation becomes:

$$ \vsig = \lambda\, (\text{tr}\,\veps)\, \mathbf{I} + 2\mu\, \veps $$

where $\text{tr}\,\veps = \varepsilon_{xx} + \varepsilon_{yy}$ in 2D. These two parameters control the two fundamental wave types in solids:

P-wave and S-wave speeds

$$ \cp = \sqrt{\frac{\lambda + 2\mu}{\rho}} \qquad\text{(compressional / P-wave)} \tag{12} $$ $$ \cs = \sqrt{\frac{\mu}{\rho}} \qquad\text{(shear / S-wave)} $$

Since $\lambda + 2\mu > \mu$ for all real materials, we always have $\cp > \cs$.

P-waves (primary, pressure, compressional) have particle motion parallel to the propagation direction — like a slinky being pushed and pulled. S-waves (secondary, shear) have particle motion perpendicular to propagation — like shaking a rope side-to-side. In metals, P-waves travel roughly twice as fast as S-waves.

Figure 6.1. P-waves compress and rarefy material along the propagation direction; S-waves displace particles transversely. In aluminum, $\cp \approx 6200$ m/s and $\cs \approx 3120$ m/s.

For the angle beam NDT problem we'll study, the relevant material properties are:

Property	Aluminum	Acrylic (wedge)
Density $\rho$ (kg/m³)	2700	1190
P-wave speed $\cp$ (m/s)	6200	2080
S-wave speed $\cs$ (m/s)	3120	1000
Acoustic impedance $Z = \rho \cp$ (MRayl)	16.7	2.5

Physics note: Unlike Darcy flow (scalar, steady-state), elastic waves are vector-valued and time-dependent. The output isn't one number per grid point — it's a velocity vector $(v_x, v_y)$ at every point at every time step. This fundamentally changes the FNO architecture requirements, as we'll see in Chapter 7.

29Mode Conversion & Snell's Law

When a wave travelling through one material hits an interface with a different material at an oblique angle, something remarkable happens: a single incident wave can generate four outgoing waves — reflected P, reflected S, refracted P, and refracted S. The relationship between all these angles is governed by a generalized form of Snell's law.

Snell's Law for Elastic Waves

$$ \frac{\sin\alpha}{c_{p1}} = \frac{\sin\alpha'}{c_{p1}} = \frac{\sin\beta'}{c_{s1}} = \frac{\sin\beta}{c_{p2}} = \frac{\sin\gamma}{c_{s2}} \tag{13} $$

where $\alpha$ = incident P-wave angle, $\alpha'$ = reflected P-wave angle, $\beta'$ = reflected S-wave angle, $\beta$ = refracted P-wave angle, $\gamma$ = refracted S-wave angle, and all speeds are in the respective materials.

Let's use this for the angle beam NDT geometry. The transducer generates a P-wave in the acrylic wedge that hits the acrylic–aluminum interface. We want the refracted S-wave in the aluminum to travel at 45° to the interface normal — this is the standard NDT inspection angle.

Computing the wedge angle. We need $\gamma = 45°$:

$$ \frac{\sin\alpha}{c_{p1}} = \frac{\sin\gamma}{c_{s2}} \quad\Longrightarrow\quad \sin\alpha = \frac{c_{p1}\sin(45°)}{c_{s2}} = \frac{2080 \times 0.707}{3120} = 0.471 $$ $$ \alpha \approx 28.1° $$

This sets the angle of the wedge face relative to the aluminum surface.

Now, what happens to the refracted P-wave in the aluminum? Its angle $\beta$ satisfies:

$$ \frac{\sin\beta}{c_{p2}} = \frac{\sin\alpha}{c_{p1}} \quad\Longrightarrow\quad \sin\beta = \frac{c_{p2}\sin\alpha}{c_{p1}} = \frac{6200 \times 0.471}{2080} = 1.40 $$

Since $\sin\beta > 1$, there is no real solution — the P-wave undergoes total internal reflection. We can verify this with the critical angle:

$$ \sin\alpha_c = \frac{c_{p1}}{c_{p2}} = \frac{2080}{6200} = 0.335 \quad\Longrightarrow\quad \alpha_c \approx 19.6° $$

Our wedge angle $\alpha = 28.1°$ exceeds $\alpha_c = 19.6°$, so the P-wave is evanescent in the aluminum. Only the S-wave propagates.

Figure 6.2. Snell's law at the acrylic–aluminum interface. The incident P-wave at $\alpha = 28°$ exceeds the P-wave critical angle ($19.6°$), so only the refracted S-wave at $\gamma = 45°$ propagates into the aluminum. This is by design.

Key idea: The wedge angle is not arbitrary — it's engineered so that only S-waves propagate into the aluminum. The P-wave is totally internally reflected. This gives a clean, single-mode beam for inspection.

30The Angle Beam NDT Setup

Now let's describe the complete physical setup from the COMSOL "Angle Beam Nondestructive Testing" tutorial. This is a standard 2D simulation of ultrasonic crack inspection that we'll use as our target for FNO surrogate modeling.

Figure 6.3. Cross-section of the angle beam NDT setup. The PZT-5H transducer drives a P-wave into the acrylic wedge. After mode conversion at the interface, a 45° S-wave beam propagates through the aluminum toward the crack. Scattered waves return to the transducer for detection.

The transducer excitation is a Gaussian-modulated sinusoidal pulse:

Transducer voltage source

$$ V_0(t) = 100 \exp\!\left(-\left(\frac{t - 2T_0}{T_0/2}\right)^{\!2}\right) \sin(2\pi f_0\, t) \tag{14} $$

with center frequency $f_0 = 1.5$ MHz and period $T_0 = 1/f_0 \approx 0.667\;\mu$s. The PZT-5H converts this voltage into mechanical displacement via piezoelectric coupling.

Figure 6.4. The transducer pulse $V_0(t)$: a Gaussian envelope (dashed) modulating a 1.5 MHz sinusoid. The pulse is compact in both time (~2 μs duration) and frequency (~0.5 MHz bandwidth).

Timeline of wave propagation — what happens during a single NDT measurement:

$t \approx 0$–$4\;\mu$s: P-wave propagates through the acrylic wedge from the transducer face toward the aluminum interface.
$t \approx 4$–$6\;\mu$s: Wave hits the acrylic–aluminum interface. Mode conversion occurs — the P-wave generates a refracted S-wave at 45° (and an evanescent P-wave that decays rapidly).
$t \approx 6$–$10\;\mu$s: The S-wave beam travels at 45° through the aluminum toward the crack location.
$t \approx 8$–$12\;\mu$s: S-wave hits the crack. Scattered waves are generated — specular reflections, tip diffractions, and mode-converted signals.
$t \approx 12$–$18\;\mu$s: Scattered waves propagate back through the aluminum, through the wedge, and arrive at the transducer. The transducer converts mechanical motion back into a measurable voltage.

Physics note: The entire round trip — pulse emission, mode conversion, propagation to crack, scattering, return — takes about 15–18 μs. At COMSOL's typical explicit time step of $\Delta t \approx 0.3$ ns (set by the CFL condition), that's roughly 50,000 time steps. This is why a surrogate model is attractive: a single FEM simulation takes minutes, and inverse problems require thousands of forward evaluations.

31Cracks as Scatterers

In the COMSOL model, the crack is represented as a Fracture boundary condition: a zero-thickness interior surface where the traction (force per unit area) vanishes on both faces.

Fracture boundary condition

$$ \vsig \cdot \vn = \mathbf{0} \qquad\text{on both faces of the crack} \tag{15} $$

The stress is zero at the crack faces (stress-free surfaces), but the displacement can be discontinuous across the crack.

When the incident S-wave beam encounters this free surface, it produces several types of scattered signals:

Specular reflection: A mirror-like bounce off the crack face. This is the strongest signal and depends on the crack orientation relative to the beam. A crack oriented perpendicular to the beam produces the strongest specular echo.
Tip diffraction: Cylindrical waves emanating from each crack tip. These are weaker than specular reflections but are diagnostic of crack size — the two tip signals arrive at slightly different times, and the time difference encodes the crack length.
Mode-converted scattering: The incident S-wave can generate P-waves at the crack faces (and vice versa). These arrive at different times due to the different wave speeds and provide additional information about the crack geometry.

Figure 6.5. Scattering of an incident S-wave from a crack. The specular reflection (blue) is the strongest signal. Tip diffractions (purple circles) emanate from each crack tip. Mode-converted S→P waves (red, dashed) travel at a different speed and arrive at different times.

How the transducer signal depends on crack parameters:

Position $(x_c, y_c)$: Changes the time of flight — the arrival time of the scattered signal at the transducer. A deeper crack means a later echo.
Size (length $L$): Larger cracks produce stronger reflections and more separated tip diffraction signals. The time difference between the two tip echoes is proportional to $L$.
Orientation (angle $\theta$): Changes the scattering pattern. A crack normal to the beam gives the strongest specular signal. A tilted crack redirects more energy away from the transducer.

Key idea: The crack's signature is encoded in the scattered wavefield — specifically in when the echoes arrive, how strong they are, and what pattern they form. This is exactly the mapping an FNO needs to learn: crack parameters in, signal characteristics out.

32The NDT Operator Learning Problem

We now have all the physics. Let's cast the angle beam NDT problem in the operator learning language from Chapters 1–5.

Domain. $D = \Omega \times [0, T]$ where $\Omega \subset \R^2$ is the 2D aluminum cross-section (~35 mm × 32 mm) and $T \approx 20\;\mu$s is the total simulation time.

Input function space $\mathcal{A}$. We parameterize each straight-line crack by four numbers:

$$ a = (x_c,\, y_c,\, L,\, \theta) \in \R^4 $$

where $(x_c, y_c)$ is the crack center, $L$ is the crack length, and $\theta$ is the orientation angle measured from vertical. This gives $\da = 4$ — compact, interpretable, and directly the quantities an NDT practitioner wants to infer. The sampling distribution and parameter ranges are defined precisely in §33.

Output function space $\mathcal{U}$. Two options, depending on what we want to predict:

Full wavefield: $u(\mathbf{x}, t) = (v_x, v_y)$ at every spatial point and time step — a tensor $\in \R^{N_x \times N_y \times N_t \times 2}$.
Transducer signal: $V(t) \in \R$ — the voltage measured at the transducer, a 1D time series. This is what an NDT practitioner actually measures.

The NDT operator

Full-field version:

$$ G^\dagger: a \mapsto \vv(\mathbf{x}, t), \qquad a \in \mathcal{A},\;\; \vv: \Omega \times [0, T] \to \R^2 $$

Signal-level version:

$$ G^\dagger: a \mapsto V(t), \qquad a \in \mathcal{A},\;\; V: [0, T] \to \R $$

In both cases, $G^\dagger$ maps a crack configuration to the resulting wavefield or signal — the same abstract framework from Chapter 1, now with richer physics.

Why operator learning? Each COMSOL simulation takes 2–5 minutes. An FNO forward pass takes milliseconds. This speedup enables:

Inversion: Given a measured signal $V_{\text{obs}}(t)$, infer the crack parameters $(x_c, y_c, L, \theta)$ via Bayesian inference (same idea as §26) — requires thousands of forward evaluations.
Design: Optimize inspection parameters (transducer frequency, wedge angle, position) for a given class of defects.
Real-time inspection: Deploy a trained FNO on an edge device for in-field crack characterization.

Let's compare this to the Darcy flow problem from Parts I–III:

Property	Darcy Flow (Ch 1–5)	Ultrasonic NDT (Ch 6–7)
PDE type	Scalar, elliptic, steady-state	Vector, hyperbolic, time-dependent
Domain $D$	$(0,1)^2$ (2D spatial)	$\Omega \times [0, T]$ (2D spatial + time)
Input $a$	Permeability field $a(x)$ ($\da = 1$, smooth)	Crack parameters $(x_c, y_c, L, \theta)$, $\da = 4$
Output $u$	Pressure (scalar, smooth)	Velocity vector or signal (sharp wavefronts)
Boundaries	Dirichlet ($u = 0$)	Absorbing + free surface
Key challenge	Piecewise-constant $a(x)$	High-frequency scattering, non-periodic
Simulation cost	~seconds	~minutes

Physics note: This is the same abstract framework from Chapter 1 — just with richer physics. The domain $D$ is now spatiotemporal, the output is vector-valued, and the "coefficient field" is a crack geometry instead of a permeability field. The FNO architecture needs only modest modifications to handle these differences, as we'll see in Chapter 7.

References

Li et al. (2020) — "Fourier Neural Operator for Parametric Partial Differential Equations." arXiv:2010.08895. arxiv.org/abs/2010.08895 — The primary FNO paper this course is based on (Chapters 1–5).
Mehtaj & Banerjee (2025) — "Scientific Machine Learning for Elastic and Acoustic Wave Propagation: Neural Operator and Physics-Guided Neural Network." Sensors, 25(11), 3588. doi.org/10.3390/s25113588 — Review of FNO, DeepONet, and PINNs applied to wave problems; documents challenges and architecture variants for elastic wave scattering.
COMSOL (2024) — "Angle Beam Nondestructive Testing" tutorial (model ID 19585). comsol.com/model/angle-beam-nondestructive-testing-19585 — Complete 2D simulation of P-wave–to–S-wave mode conversion through an acrylic wedge into aluminum with a zero-thickness fracture defect.