Wave motion is the transfer of energy through a medium by a periodic disturbance that travels without permanent net displacement of the medium itself. The marine sonar and ultrasonic NDT industries depend on it to inspect hulls and welds without cutting them open. The disturbance can be transverse — particles moving perpendicular to travel — or longitudinal, with particles oscillating along the line of propagation. The result: energy moves from source to target at the medium's wave speed, typically 343 m/s in air, 1480 m/s in water, and 5900 m/s in steel.
Wave Motions Interactive Calculator
Vary wave speed, frequency, and feature size to see wavelength, period, and how many wavelengths span a defect or target feature.
Equation Used
The calculator uses the wave relation lambda = v / f. Wave speed v is the medium velocity, f is excitation frequency, and lambda is the wavelength. Comparing feature size d to wavelength estimates how many wavelengths span the flaw or target feature.
- Uniform, non-dispersive medium.
- Phase velocity equals the entered wave speed.
- Feature comparison uses straight-line wave travel.
- Frequency is entered in MHz and feature size in mm.
The Wave Motions in Action
A wave starts when a source perturbs a medium that has both inertia and a restoring force. The inertia stores kinetic energy, the restoring force stores potential energy, and the two trade back and forth as the disturbance hands itself off from one element of the medium to the next. That handoff is what makes a wave a wave — energy moves, but the medium itself only oscillates around a fixed point. In a steel rail, a tap from a hammer launches a longitudinal stress wave at roughly 5900 m/s while the steel atoms themselves displace by less than a micron.
The shape of the wave depends on how particles move relative to the direction of travel. In a transverse wave, like a vibrating guitar string or a Rayleigh surface wave on a railhead, particles move at right angles to propagation. In a longitudinal wave, like sound in air or the P-wave from an earthquake, particles compress and rarefy along the travel axis. Phase velocity tells you how fast a single wavefront moves; group velocity tells you how fast the energy envelope moves, and these two are not the same in dispersive media — that mismatch is why you can hear a thunderclap as a long rumble even though it started as a sharp crack.
Things go wrong when the medium is wrong for the frequency. Drive a 40 kHz ultrasonic horn into a part with poor acoustic coupling and the wave reflects at the interface instead of penetrating, so you get heating at the contact face and no useful energy delivery. Get the wavelength comparable to a feature size — say a 6 mm flaw in a part inspected at 5 MHz in steel where wavelength is around 1.2 mm — and you start seeing diffraction artefacts that look like real defects. Standing waves form when reflections from a boundary interfere with the incoming wave, which is useful in a resonator and a problem in a duct that whistles.
Key Components
- Source / Driver: Inputs energy into the medium at a controlled frequency and amplitude. In an ultrasonic transducer this is typically a piezoelectric stack driven at 20-40 kHz with displacement amplitudes of 5-50 µm. The source impedance must match the medium impedance within roughly 20% or most energy reflects rather than launching.
- Propagating Medium: Carries the disturbance via its inertia and restoring stiffness. Speed depends on the ratio: c = √(stiffness / density). For steel that gives ~5900 m/s longitudinal, for water 1480 m/s, for air 343 m/s at 20 °C. The medium must be continuous — voids or delaminations larger than λ/4 cause scattering.
- Wavefront: The surface of constant phase moving through the medium. Plane wavefronts dominate far from a source; spherical wavefronts dominate near a point source within roughly 1 wavelength. Phase coherence across the wavefront sets how tightly the wave can be focused.
- Boundary or Interface: Where the wave reflects, transmits, or mode-converts. Reflection coefficient depends on the acoustic impedance mismatch Z2/Z1. A steel-to-air boundary reflects roughly 99.99% of incident energy, which is why ultrasonic flaw detectors need couplant gel to push the mismatch down.
- Receiver / Sensor: Converts the returning wave back into a measurable signal. Bandwidth must cover the wave's frequency content — a 5 MHz transducer needs at least 2 MHz analog bandwidth in the receive chain or you lose pulse-edge resolution. Time-of-flight accuracy of ±10 ns is achievable with modern flaw detectors.
Who Uses the Wave Motions
Wave motion shows up anywhere you need to send energy or information through a medium without physical contact end-to-end. The application set spans medical imaging, structural inspection, communications, geophysics, and process measurement. The right frequency and the right medium are everything — push too high and attenuation kills your range, push too low and you cannot resolve the feature you are trying to find.
- Medical Imaging: GE Vivid E95 cardiac ultrasound systems use 1.5-4 MHz phased-array probes to image heart valves through tissue at depths of up to 200 mm.
- Non-Destructive Testing: Olympus EPOCH 650 ultrasonic flaw detectors run 2-10 MHz transducers to find weld defects as small as 0.5 mm in pressure-vessel steel.
- Marine Survey: Kongsberg EM 2040 multibeam echosounders map seabed bathymetry at 200-400 kHz with depth accuracy better than 0.05% of water depth.
- Industrial Welding: Branson 2000X ultrasonic plastic welders fuse polypropylene parts at 20 kHz with horn amplitudes of 30-80 µm peak-to-peak.
- Seismic Exploration: Schlumberger WesternGeco vibroseis trucks sweep 5-80 Hz P-waves into the ground for oil and gas reservoir mapping at depths beyond 5 km.
- Telecommunications: Corning SMF-28 single-mode optical fibre carries electromagnetic waves at 1310 and 1550 nm with attenuation below 0.2 dB/km.
The Formula Behind the Wave Motions
The basic wave relation links speed, frequency, and wavelength. It tells you which combination of frequency and medium will give you the resolution and penetration you need. At the low end of typical industrial frequencies — say 20 kHz in air — wavelength is around 17 mm, fine for power delivery in ultrasonic cleaning but useless for resolving small features. At the high end — 10 MHz in steel — wavelength is about 0.6 mm, giving sharp NDT resolution but with heavy attenuation past 50 mm of material. The sweet spot for general-purpose ultrasonic flaw detection sits at 2-5 MHz in steel, balancing penetration against detectable flaw size.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| c | Wave propagation speed in the medium | m/s | ft/s |
| f | Frequency of the wave | Hz | Hz |
| λ | Wavelength in the medium | m | in |
Worked Example: Wave Motions in an ultrasonic thickness gauge on a pipeline survey
You are calibrating a Cygnus 6+ ultrasonic thickness gauge on an offshore gas pipeline inspection in the Bass Strait, measuring wall thickness on API 5L X65 carbon steel pipe with a nominal wall of 12.7 mm. The gauge runs a 5 MHz dual-element transducer. You need to confirm the wavelength in steel and decide whether the gauge can reliably resolve a 0.8 mm corrosion pit in the pipe wall.
Given
- f = 5 MHz
- c = 5900 m/s (longitudinal in carbon steel)
- Pit depth target = 0.8 mm
Solution
Step 1 — at the nominal 5 MHz operating frequency, compute wavelength in carbon steel:
Resolution rule of thumb for pulse-echo ultrasonics is roughly λ/2, so at 5 MHz you can resolve features down to about 0.6 mm in steel. A 0.8 mm pit sits comfortably above that threshold — you will see it cleanly on the A-scan.
Step 2 — check the low end of the typical NDT band, 2 MHz, where you might drop to gain penetration on thicker pipe:
At 2 MHz, λ/2 resolution is around 1.5 mm — your 0.8 mm pit disappears into the noise floor. You would gain penetration to perhaps 200 mm of steel but lose the ability to find the corrosion you came to find.
Step 3 — push to the high end of the practical band, 10 MHz:
Now λ/2 is about 0.3 mm and you would resolve sub-millimetre pits with no problem. The catch — attenuation in carbon steel scales roughly with frequency squared, so 10 MHz gives you usable echoes only out to about 25 mm of pipe wall, not the 50-100 mm range you would get at 5 MHz. For a 12.7 mm wall this is fine, but for a thicker manifold it would not be.
Result
The nominal wavelength at 5 MHz in carbon steel is 1. 18 mm, giving a theoretical minimum detectable feature size around 0.6 mm — so the 0.8 mm corrosion pit is detectable with margin. Across the operating range, 2 MHz blurs the pit out entirely while 10 MHz resolves it sharply but cuts useful penetration depth by half; 5 MHz is the sweet spot for 12.7 mm pipe wall. If your measured echo amplitude comes back 30% below the calibration block reading, the most likely causes are: (1) couplant gel film too thin or air-bubbled at the transducer-to-pipe interface, dropping transmitted energy by 6-10 dB, (2) probe wedge worn flat with surface Ra above 3 µm causing scattering, or (3) the steel grain size has coarsened in a heat-affected zone near a girth weld and is scattering the 5 MHz signal — drop to 2.25 MHz to get past the grain noise.
Wave Motions vs Alternatives
Wave-based sensing and energy transfer is one option among several for moving information or force across a gap. The right choice depends on the medium you have, the resolution you need, and the cost you can absorb. Compared against direct contact measurement and electromagnetic alternatives, mechanical wave methods sit in a specific zone of the design space.
| Property | Mechanical Wave (Ultrasonic) | Electromagnetic Wave (Radar/Optical) | Direct Contact Measurement |
|---|---|---|---|
| Typical operating frequency | 20 kHz - 50 MHz | 1 GHz - 500 THz | DC - 10 kHz |
| Resolution in steel | 0.3 - 5 mm | Cannot penetrate metal | Limited by probe tip geometry |
| Penetration depth in metal | 10 mm - 500 mm | Surface only (skin depth <1 mm) | Surface only |
| Equipment cost | $2k - $50k for industrial NDT | $5k - $200k | $100 - $5k |
| Couplant or contact required | Yes — gel, water, or dry contact | No — works through air | Yes — physical touch |
| Speed of measurement | Microseconds per shot | Nanoseconds per shot | Seconds per point |
| Best application fit | Subsurface flaw detection, thickness gauging | Distance ranging, imaging through air | High-accuracy dimensional metrology |
Frequently Asked Questions About Wave Motions
The transducer face is flat and the pipe surface is curved, so contact area shrinks and couplant film thickness becomes uneven across the footprint. Energy that should launch as a coherent plane wavefront instead spreads at varying angles, and the first-arriving echo comes from whichever sliver of the footprint has the best coupling — usually not the deepest path.
Fix it by using a curved-face transducer matched to the pipe OD, or by using a delay-line probe with a shaped wedge. On pipes below 50 mm OD with a flat probe, expect readings biased low by 0.1-0.3 mm.
It comes down to the trade between resolution and grain noise. Weld metal and the heat-affected zone often have coarser grain than the parent plate, and grain scattering scales steeply with frequency — roughly the fourth power of frequency once grain size approaches λ/10.
If you see a hashy, grassy A-scan baseline at 5 MHz, the grain is winning. Drop to 2.25 MHz and the noise floor usually falls 10-15 dB. You lose some resolution on small flaws but you gain back the ability to see anything at all in austenitic or coarse-grained welds.
Acoustic impedance mismatch and horn resonance both clamp the useful output. The horn is designed as a half-wavelength resonator at a specific frequency — drive it harder and you increase amplitude, but only up to the point where the horn's mechanical Q and the load impedance allow. Beyond that, extra electrical input dissipates as heat in the piezo stack and the booster.
If your weld energy plateaus, check the horn's no-load resonant frequency on the welder's diagnostic — a shift of more than 50 Hz from spec means the horn is detuning under load, often from a worn tip or a cracked stud. Replace the horn rather than chasing power.
Reflection at an interface is governed by the acoustic impedance ratio Z = ρ × c. Steel sits around 46 × 106 kg/m²s, water around 1.5 × 106, and air around 415. The reflection coefficient is ((Z2 − Z1) / (Z2 + Z1))2.
Steel-to-air gives roughly 99.99% reflection — nearly perfect mirror. Steel-to-water gives about 88% reflection, which still sounds high but leaves 12% transmitted, plenty for imaging. This is the entire reason ultrasonic NDT needs couplant gel: removing the air film between probe and part is the difference between zero signal and a usable one.
Two things commonly cause this. First, the gauge's zero offset — the time the pulse spends in the transducer wear face and any delay line — may not be calibrated. Most gauges need a two-point calibration on a known step block to subtract this offset, typically 0.3-0.8 µs.
Second, the assumed wave speed may be wrong for your alloy. Generic aluminium is quoted at 6320 m/s, but 7075-T6 runs closer to 6250 m/s and cast aluminium can be down at 6100 m/s. A 3% speed error gives a 3% thickness error directly. Always calibrate on a sample of the same alloy and temper as the part.
Rayleigh waves travel along the surface and decay exponentially with depth — by one wavelength down, amplitude is roughly 10% of surface amplitude. They are the right choice when the defect you are hunting is at or near the surface, like fatigue cracks at fastener holes on aircraft skins, and you need to inspect under a curved or obstructed geometry where a normal-incidence bulk wave probe cannot sit.
They are the wrong choice for subsurface flaws below about λ deep, for through-thickness measurements, and on rough or painted surfaces where surface texture scatters the wave directly. Aerospace MRO shops use Rayleigh-wave probes routinely on landing gear; pressure-vessel shops almost never do.
References & Further Reading
- Wikipedia contributors. Wave. Wikipedia
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