13.5 Acoustic Wave Mechanics, Refraction & Angle-Beam Sound-Path Geometry

13.3 Ultrasonic Testing (UT): Wave Propagation, Angle-Beam Shear, PAUT & TOFD

Quick Answer: Ultrasonic Testing (UT) uses high-frequency acoustic waves ($0.5\text{ to }15\text{ MHz}$) to detect internal planar and volumetric weld discontinuities. Longitudinal waves ($c_L \approx 5900\text{ m/s}$) are deployed for straight-beam thickness and lamination testing, while refracted transverse shear waves ($c_S \approx 3240\text{ m/s}$) are generated via Lucite wedges cut between the first ($27.5^\circ$) and second ($57.4^\circ$) critical angles for angle-beam weld inspection ($45^\circ, 60^\circ, 70^\circ$). AWS D1.1 Clause 8 evaluates welds using the decibel rating formula $d = a - b - c$, where lower or negative $d$ ratings represent more severe flaws. Advanced methods include Phased Array UT (PAUT) for electronic beam steering / S-scans and Time-of-Flight Diffraction (TOFD) for sub-millimeter flaw height sizing via crack-tip diffraction.

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1. Acoustic Wave Mechanics: Wave Types, Velocities & Acoustic Impedance

Ultrasonic testing introduces mechanical vibrational waves into elastic media. In industrial welding metallurgy, piezoelectric transducers convert electrical pulses into mechanical sound packets at center frequencies typically between $2.0\text{ and }5.0\text{ MHz}$.

                    ACOUSTIC WAVE PROPAGATION MODES IN STEEL

    1. LONGITUDINAL (COMPRESSION) WAVE: c_L ≈ 5900 m/s
       Particle Displacement Parallel to Wave Vector
       <===> <===> <===> <===> <===>
       -----------------------------> Wave Propagation Direction

    2. TRANSVERSE (SHEAR) WAVE: c_S ≈ 3240 m/s
       Particle Displacement Perpendicular to Wave Vector
         ^     |     ^     |
         |     V     |     V
       -----------------------------> Wave Propagation Direction

    3. SURFACE (RAYLEIGH) WAVE: c_R ≈ 2900 m/s
       Retrograde Elliptical Particle Motion Along Boundary

Primary Wave Modes in Ferritic Steel

• Longitudinal (Compression) Waves ($c_L \approx 5900\text{ m/s} = 0.232\text{ in/}\mu \text{s}$): Particle motion is parallel to wave propagation direction. Propagates through solids, liquids, and gases. Deployed in $0^\circ$ straight-beam testing for plate thickness measurement, cladding debonding, and base metal lamination screening.
• Transverse (Shear) Waves ($c_S \approx 3240\text{ m/s} = 0.128\text{ in/}\mu \text{s}$): Particle motion is strictly perpendicular to wave propagation direction. Shear waves require shear modulus ($G > 0$) and therefore cannot propagate in liquids or gases. Transverse wave velocity is roughly $55\%$ of longitudinal velocity ($c_S / c_L \approx 0.55$). Because shear waves possess shorter wavelengths ($\lambda = c / f$) at identical frequencies, they exhibit far higher sensitivity to microscopic planar flaws.
• Rayleigh (Surface) Waves ($c_R \approx 0.90 c_S \approx 2900\text{ m/s}$): Oscillate in a retrograde elliptical orbit along a free surface; energy attenuates exponentially with depth, penetrating roughly one wavelength ($\lambda$).

Acoustic Impedance & Interface Boundaries

Acoustic impedance ($Z$) defines a material's resistance to acoustic particle motion:

$$Z = \rho \cdot c$$

where $\rho$ is material density ($\text{kg/m}^3$) and $c$ is acoustic sound velocity ($\text{m/s}$).

• Carbon Steel: $Z_1 \approx 7850 \times 5900 \approx 46.3 \times 10^6\text{ Rayls}$ ($\text{kg}/(\text{m}^2\cdot\text{s})$)
• Water: $Z_2 \approx 1000 \times 1480 \approx 1.48 \times 10^6\text{ Rayls}$
• Air: $Z_3 \approx 1.2 \times 340 \approx 408\text{ Rayls}$
• Lucite (Polymethyl methacrylate): $Z_4 \approx 1180 \times 2730 \approx 3.22 \times 10^6\text{ Rayls}$

Reflection and Transmission Coefficients

At normal incidence across a flat boundary between medium 1 and medium 2, the pressure reflection coefficient ($R$) and transmission coefficient ($T$) are:

$$R = \frac{Z_2 - Z_1}{Z_2 + Z_1}, \qquad T = \frac{2 Z_2}{Z_2 + Z_1}$$

• Steel-to-Air Boundary: $$R = \frac{408 - 46,300,000}{408 + 46,300,000} \approx -0.99998 \approx -100\%$$ Nearly $100\%$ of the acoustic energy is reflected at any steel-air boundary. This physical fact explains two fundamental operational principles:
  1. Ultrasonic waves cannot cross an air gap between transducer and steel; a liquid couplant (cellulose gel, light oil, or water) is mandatory to displace air.
  2. Ultrasonic testing is exceptionally sensitive to cracks and delaminations, because even a sub-micron air gap acts as an absolute $100\%$ acoustic mirror.

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2. Refraction, Mode Conversion & Angle-Beam Lucite Wedge Design

When a longitudinal wave propagating through a plastic wedge strikes an inclined interface with steel at an oblique incident angle ($\theta_1$), both refraction and mode conversion occur. The incident wave splits into two distinct refracted waves: a refracted longitudinal wave and a refracted transverse (shear) wave.

                  MODE CONVERSION & SNELL'S REFRACTION AT WEDGE

                           Piezoelectric Element
                                +---------+
                                | Transd. |
                                +----+----+
                                     | Incident Longitudinal Wave (c1 = 2730 m/s)
                    Lucite Wedge     | Angle: θ1
                                     V
     --------------------------------+-------------------------------- Steel Surface
                                    / \
                                   /   \
                                  /     \ Refracted Shear Wave (cS = 3240 m/s)
                                 /       \ Angle: θS (e.g., 45°, 60°, 70°)
                                /         V
       Refracted Longitudinal  /
       Wave (cL = 5900 m/s)   V
       Angle: θL

Snell's Law of Acoustic Refraction

The angular relationship is governed by Snell's Law:

$$\frac{\sin \theta_1}{c_{\text{wedge}, L}} = \frac{\sin \theta_{L2}}{c_{\text{steel}, L}} = \frac{\sin \theta_{S2}}{c_{\text{steel}, S}}$$

where:

• $\theta_1$ = Incident angle in the Lucite wedge
• $c_{\text{wedge}, L} = 2730\text{ m/s}$ (longitudinal velocity in Lucite)
• $\theta_{L2}$ = Refracted longitudinal angle in steel ($c_{\text{steel}, L} = 5900\text{ m/s}$)
• $\theta_{S2}$ = Refracted shear angle in steel ($c_{\text{steel}, S} = 3240\text{ m/s}$)

The Two Critical Angles

Because sound travels faster in steel than in Lucite, increasing the wedge angle $\theta_1$ refracts the waves closer to the surface plane ($90^\circ$):

     0° < θ1 < 27.5°              27.5° < θ1 < 57.4°               θ1 > 57.4°
  ------------------------    ----------------------------    ----------------------
   Dual Mode Region:           OPERATIONAL ANGLE-BEAM WINDOW:  Total Reflection:
   Both Longitudinal &         Longitudinal wave vanishes;     Both modes vanish;
   Shear waves coexist         PURE SHEAR WAVE IN STEEL        Rayleigh surface waves
   in steel. (Confusing)       (Standard: 45°, 60°, 70°)       only.

1. First Critical Angle ($\theta_{c1}$): The incident angle at which the refracted longitudinal wave in steel refracts to exactly $90^\circ$ (parallel to the plate surface): $$\sin \theta_{c1} = \frac{c_{\text{wedge}, L}}{c_{\text{steel}, L}} = \frac{2730\text{ m/s}}{5900\text{ m/s}} = 0.4627 \implies \theta_{c1} = \arcsin(0.4627) \approx 27.56^\circ$$ At angles beyond $27.56^\circ$, the longitudinal wave in steel is completely eliminated via total internal reflection.
2. Second Critical Angle ($\theta_{c2}$): The incident angle at which the refracted shear wave in steel reaches $90^\circ$: $$\sin \theta_{c2} = \frac{c_{\text{wedge}, L}}{c_{\text{steel}, S}} = \frac{2730\text{ m/s}}{3240\text{ m/s}} = 0.8426 \implies \theta_{c2} = \arcsin(0.8426) \approx 57.42^\circ$$ At angles beyond $57.42^\circ$, shear waves cease to propagate into the body of the steel, converting entirely into surface Rayleigh waves.

The Operational Angle-Beam Window

To conduct clean angle-beam shear wave inspection without ghost signals or confusing dual-mode reflections, wedge angles must fall between $27.6^\circ$ and $57.4^\circ$. Standard industrial refracted shear angles in steel are:

• $45^\circ$ Shear Wedge: Incident wedge angle $\theta_1 = \arcsin\left( \frac{2730}{3240} \sin 45^\circ \right) = \arcsin(0.8426 \times 0.7071) = \arcsin(0.5958) \approx 36.6^\circ$.
• $60^\circ$ Shear Wedge: Incident wedge angle $\theta_1 = \arcsin(0.8426 \times 0.8660) = \arcsin(0.7297) \approx 46.9^\circ$.
• $70^\circ$ Shear Wedge: Incident wedge angle $\theta_1 = \arcsin(0.8426 \times 0.9397) = \arcsin(0.7918) \approx 52.4^\circ$.

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3. Straight-Beam Lamination Screening & Sound Path Geometry

Base Metal Lamination Screening (AWS D1.1 Clause 8, Part F)

Prior to performing angle-beam examination on any CJP groove weld, AWS D1.1 Clause 8 strictly mandates that the base metal through which the angle beam must travel be scanned using a $0^\circ$ straight-beam longitudinal transducer.

• Engineering Rationale: Rolling mills can produce laminar inclusions (elongated manganese sulfides) aligned parallel to the plate surface. If an angle beam enters a plate containing a lamination, the acoustic shear wave strikes the lamination boundary and reflects away prematurely. The ultrasound never reaches the weld fusion zone, producing a fatal false-negative result.

                 BASE METAL LAMINATION SHIELDING WELD DEFECT

                         Angle-Beam Transducer
                           +---------+
                            \  Wedge  \
                             +---------+
                               \  Acoustic Beam
                                \ 
     ============================\========================[===WELD===]==== Base Plate
                                  \  Base Metal Lamination
                                   \====[LAMINATION]====  (Unfused Root Defect
                                   /                       is NEVER reached
                                  / Reflected Away!        by Sound Beam!)
     ============================/========================================

Skip Distance & Scanning Legs

Angle-beam testing steers sound through the plate via boundary reflections off the top and bottom surfaces:

                        ANGLE-BEAM SKIP DISTANCE GEOMETRY

                Transducer
                +--------+
                 \       \
                  +-------+
                     \                                       Top Surface
       ===============+=======================================+==========
                       \                                     /
                 Leg 1  \       Thickness (t)               / Leg 2
                (Direct) \                                 / (Skipped)
                          \                               /
       ====================+=============================+================
                        Bottom Surface / Backwall
                       |<--------- Skip Distance (S) --------->|

• Leg 1 (Half Skip / Direct Path): The sound path traveling from the transducer index point directly to the bottom plate surface. $$\text{Depth} = \text{Sound Path} \times \cos(\theta)$$ $$\text{Surface Distance} = \text{Sound Path} \times \sin(\theta)$$ $$\text{Leg 1 Surface Reach} = t \times \tan(\theta)$$
• Leg 2 (Full Skip): The sound path reflecting off the bottom surface and traveling upward toward the top surface. $$\text{Total Skip Distance (Full Skip)} = 2 \times t \times \tan(\theta)$$

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