Wave Properties — From Sound to Ultrasonic NDT and Acoustic FEM

1. Waves in Engineering: More Than Just Sound

Wave physics is essential in multiple CAE domains: acoustic analysis (noise in vehicles, HVAC systems, concert halls), ultrasonic non-destructive testing (NDT), seismic engineering (seismic waves propagating through soil and structures), and electromagnetic simulation (antenna design, radar cross-section). Understanding wave behavior — speed, reflection, refraction, and superposition — translates directly into setting up these simulations correctly.

2. The Wave Equation and Wave Speed

The 1D wave equation:

Where $u(x,t)$ is the wave displacement and $c$ is wave speed. General solution: any function of $(x - ct)$ or $(x + ct)$ — waves traveling right or left at speed $c$. A sinusoidal wave:

Wave parameters and their relationships:

3. Transverse vs Longitudinal Waves

Longitudinal waves (also called P-waves or compression waves): particle motion is parallel to wave propagation. Sound in air, blast pressure waves, P-waves in earthquakes, and most ultrasonic NDT probes use longitudinal waves. They can propagate in solids, liquids, and gases.

Transverse waves (S-waves or shear waves): particle motion is perpendicular to wave propagation. They can only propagate in solids (liquids have no shear stiffness). Shear wave transducers in ultrasonic NDT are particularly effective for detecting vertically oriented cracks because the shear wave scatters strongly off crack faces oriented perpendicular to the beam.

In a solid, the longitudinal wave speed:

For steel ($E=210$ GPa, $\nu=0.3$, $\rho=7800$ kg/m³): $c_L \approx 5900$ m/s, $c_S \approx 3200$ m/s. Note: $c_L/c_S = \sqrt{2(1-\nu)/(1-2\nu)} \approx 1.84$ for steel.

4. Superposition and Standing Waves

When two waves exist simultaneously in the same medium, their displacements add algebraically (principle of superposition). Two sinusoidal waves of equal amplitude and frequency traveling in opposite directions create a standing wave:

The spatial pattern $\sin(kx)$ is fixed — nodes (zero displacement) and antinodes (maximum displacement) don't move. Standing waves in a structural cavity or duct create resonant acoustic modes — exactly the noise problem in car cabins and HVAC ducts that acoustic FEM addresses.

Room Acoustics and Acoustic Modes

In a rectangular room of dimensions $L_x \times L_y \times L_z$, room modes (standing wave resonances) occur at:

At these frequencies, sound is severely colored (amplified). Car cabin acoustic FEM uses exactly this analysis to predict booming frequencies and guide acoustic treatment placement.

5. Acoustic Impedance and Wave Reflection

Acoustic impedance $Z = \rho c$ (kg/m²·s, or Rayl) determines how much of a wave is reflected vs. transmitted at a boundary between two media:

Where $R$ is the reflected power fraction and $T$ is the transmitted power fraction. For steel ($Z = 45 \times 10^6$ Rayl) vs. air ($Z = 413$ Rayl):

A steel-air interface reflects essentially all sound — this is why air cracks and delaminations are perfect reflectors in ultrasonic NDT, and why noise isolation through a steel wall is not about the steel's mass but about maintaining airtight seals (any gap allows sound through).

6. Snell's Law and Wave Refraction

When a wave crosses a boundary at an angle, it refracts — changes direction. Snell's Law governs the angle change:

Since longitudinal speed $c_L >$ shear speed $c_S$ in steel, an incident longitudinal wave at a surface can generate refracted shear waves at a different angle. This is mode conversion — important in ultrasonic NDT because angled probes deliberately exploit mode conversion to steer shear waves at specific angles for crack detection.

At critical angles, $\theta_2 = 90°$ and the refracted wave travels along the surface. The first critical angle (for longitudinal to longitudinal) and second critical angle (for longitudinal to shear) are key design parameters for angle-beam ultrasonic probes. Beyond the second critical angle, total internal reflection occurs — the basis for Lamb wave generation in plate inspection.

7. Ultrasonic NDT: Finding Cracks with Waves

Ultrasonic NDT uses sound waves (typically 1–10 MHz) to inspect structural components non-destructively. Common techniques:

The minimum detectable crack size in ultrasonic NDT is limited by wavelength: reliable detection requires defect size ≥ λ/2. At 5 MHz in steel ($c_L = 5900$ m/s): $\lambda = c/f = 5900/5\times10^6 = 1.18$ mm. Minimum detectable crack ≈ 0.6 mm. For smaller defects, you need higher frequency — but higher frequency also means shorter penetration depth due to attenuation. This frequency-penetration trade-off governs probe selection.

8. Acoustic FEM and BEM

Acoustic FEM solves the Helmholtz equation (the frequency-domain wave equation) in fluid domains:

Where $p$ is the acoustic pressure field. This requires meshing the fluid domain (air inside a car cabin, water around a submarine) with elements small enough to resolve the shortest wavelength of interest — typically 6 elements per wavelength minimum. For car interior acoustic analysis up to 1000 Hz, element size must be ≤ $c/(6f) = 343/(6000) \approx 57$ mm.

BEM vs. FEM for Acoustics

For exterior acoustic problems (noise radiating into free space), the Boundary Element Method (BEM) is often preferred because it only meshes the structure surface, not the infinite fluid domain. Interior problems (car cabin, muffler) are better handled by FEM. Coupled FEM-BEM (or FEM-FEM with infinite elements) handles both simultaneously for problems like vehicle pass-by noise.

9. Cross-Topics