Wave Physics for Technical Decisions

Learning outcomes

By the end of this chapter, the learner should be able to relate frequency, wavelength and velocity; explain propagation modes and particle motion; apply travel-time, impedance and reflection relationships; use Snell's law and critical-angle concepts correctly; and recognise the limits of simple isotropic models.

2.1 Frequency, wavelength and velocity

For a given mode in a uniform elastic material:

lambda = v / f

Velocity is principally a property of the material, wave mode and condition. Increasing frequency in a non-dispersive material shortens wavelength; it does not normally increase the bulk-wave velocity. A shorter wavelength can improve response to small reflectors, while higher frequency often increases attenuation and reduces penetration.

Example: for a longitudinal velocity of 5.90 mm/us and frequency of 5 MHz, wavelength is 5.90 / 5 = 1.18 mm.

2.2 Wave modes

Longitudinal waves have particle motion mainly parallel to propagation and can travel through solids, liquids and gases. Shear waves have particle motion perpendicular to propagation and require shear rigidity, so they do not propagate as bulk shear waves through ordinary liquids or gases. Rayleigh waves travel near a free surface, with energy concentrated within roughly one wavelength of that surface. Plate and guided modes arise when boundaries constrain propagation and can be dispersive.

Mode choice changes velocity, wavelength, reflection behaviour, sensitivity to orientation and coupling requirements. A procedure must therefore identify the mode rather than referring only to "sound velocity."

2.3 Transit time and location

In pulse-echo testing, the measured time normally represents a round trip. For a straight beam:

one-way distance = velocity x transit time / 2

For an angle beam, the one-way sound path may be resolved into depth and surface distance using trigonometry. The reference point must be defined: probe index, beam exit point, component origin or encoded coordinate. Location values without a coordinate convention are not traceable.

2.4 Acoustic impedance and normal-incidence reflection

Acoustic impedance is:

Z = rho x v

At a flat boundary and normal incidence, the ideal intensity reflection coefficient is:

R = ((Z2 - Z1) / (Z2 + Z1))^2

For a lossless boundary, transmitted intensity fraction is approximately T = 1 - R. Real interfaces add roughness, imperfect contact, curvature, mode conversion and attenuation. The equation is a model, not a universal calibration method.

Air between a contact probe and metal creates a severe impedance mismatch, which is why a couplant is needed. Couplant reduces the intervening air layer; it does not make the interfaces disappear.

2.5 Refraction and critical angles

For ideal isotropic materials, Snell's law can be written:

sin(alpha) / v1 = sin(beta) / v2

The incident and refracted angles are measured from the normal. When a longitudinal wave in a slower wedge enters a faster solid, both refracted longitudinal and shear modes may be generated. The first critical angle occurs when the refracted longitudinal angle reaches 90 degrees. Beyond it, no propagating refracted longitudinal bulk wave exists in the second medium. The second critical angle occurs when the refracted shear angle reaches 90 degrees.

Between the two critical angles, the intended bulk mode is commonly shear. Close to a critical angle, small changes in wedge velocity, temperature or actual angle can cause large changes in the refracted condition. A Level 3 should avoid treating nominal wedge labels as measurements.

2.6 Anisotropy and dispersion

In anisotropic material, phase direction, energy direction and polarisation may not align. Velocity varies with direction, and a simple isotropic Snell calculation can give an incorrect beam path. Austenitic weld metal, textured forgings and composites may require measured velocities, modelling, representative trials or alternative techniques.

Guided waves can be dispersive: velocity depends on frequency-thickness product and mode. A single material velocity is then insufficient. The selected mode and operating point must be controlled.

Failure modes and decision control

Weak practice commonly includes using longitudinal velocity for a shear-wave path; omitting the round-trip factor in pulse-echo calculations; measuring angles from the surface instead of the normal; treating couplant as lossless; applying isotropic calculations to strongly anisotropic material without verification; and quoting a nominal critical angle without temperature or material basis.