Structural health monitoring (SHM) based on guided ultrasonic waves has become a powerful tool for detecting incipient damage in plate-like structures across aerospace, civil, and mechanical engineering applications . Conventional linear ultrasonic methods, which rely on wave scattering and mode conversion, are effective for locating and sizing relatively large defects, but they often lack the sensitivity required to detect early-stage micro-damage such as fatigue cracks, diffuse material degradation, and interfacial debonding.

Nonlinear ultrasonic techniques offer a promising route to higher sensitivity by exploiting the nonlinear interaction between guided waves and damage. Among these, frequency-mixing methods – in which two primary waves at the frequencies fa and fb generate intermodulation components at f±=fa±fb – have attracted particular interest because of their enhanced sensitivity to contact-type nonlinearities and microstructural evolution . Nevertheless, the practical deployment of nonlinear mixing techniques is still hindered by a critical bottleneck: the lack of a systematic, physics-informed criterion for selecting excitation frequency pairs that maximise detectability under real hardware constraints .

In many laboratory demonstrations, the two excitation frequencies are tuned manually along a single source–receiver (S → R) path until a clear mixed line appears in the spectrum. While this trial-and-error strategy can work for proof-of-concept studies, it is slow, not readily reproducible, and tightly coupled to one specific layout. In practical structures there are multiple interrogation paths, each with different lengths, boundary proximities, and transducer bandwidths. A tone pair that is perfectly phase-matched on the dispersion diagram may still be undetectable in practice because the piezoelectric transducer or front-end has a notch at f+ or because the two primaries already consume most of the analogue-to-digital converter (ADC) headroom.

What is currently missing is a principled way to move from the large set of all candidate frequency pairs (fa,fb) on a predefined grid to a small subset – typically three to six pairs per path – that are simultaneously (i) physically coherent for nonlinear mixing, (ii) receivable with adequate gain and noise performance, and (iii) safe for the hardware in terms of dynamic range, all before performing spectral phase inversion (SPI-4) or any other leakage-suppressed readout.

Recent literature on nonlinear guided waves shows very active development along four closely related directions: i.

Phase-matching-driven nonlinear mixing. A series of papers over the last 5–6 years has emphasised that cumulative nonlinear guided-wave mixing is achieved only when the phase mismatch Δk=kma(fa)+kmb(fb)-kmt(f±)≈0 so that the effective interaction length is Lint≃π/|Δk|. This requirement has been analysed and demonstrated for both collinear and non-collinear Lamb waves, including Zhu's low-frequency S0–S0 mixing for fatigue damage in plates , Pineda Allen's non-collinear mixing tests , and several recent feasibility studies on phase-matched Lamb wave mixing .

Despite this progress, two important gaps remain. First, most existing selection rules are dispersion-only: they identify which (fa,fb) pairs satisfy Δk≈0, but they do not indicate whether the piezoelectric transducer (PZT)/data acquisition (DAQ) chain can actually pass f+ with adequate signal-to-noise ratio (SNR) or whether the two primaries will overdrive the front-end. Second, almost all demonstrations are path-specific: parameters tuned for one S1 → S3 path on a 0.8m×0.5m plate do not immediately transfer to an orthogonal S2 → S4 path or to a slightly longer path in a different coupon. Bridging these two gaps – by unifying phase-matching physics with measured, channel-aware terms in a single ranking functional – is the specific niche of the present work.

Building on the phase mismatch, receive sensitivity, and dynamic-range analysis developed in the remainder of the paper, we make four main contributions: i.

We introduce a unified, interpretable scoring function J(fa,fb) that combines (a) a dispersion-derived mismatch term |Δk|/kmt, (b) a path-to-path coherence term Lpath/Lint, (c) the measured receive gain Grx(f+), and (d) a dynamic-range penalty that quantifies the cost of two-tone excitation. All required quantities are either computable from the chosen mode triplet or directly measurable on the intended hardware.

Section  formalises the physics-guided selection; derives the expressions for Δk, Lint, Grx, and the dynamic-range penalty; and defines the composite score J(fa,fb). Section  documents the specimen, the transducer layout, the crack growth plan, and the physics- and hardware-guided screening that generate the (fa,fb) maps. Section  presents the experimental results from a deployment viewpoint, including the global correlation between J and detectability, detection reliability versus shortlist size, crack growth sensitivity and repeatability, and path-to-path robustness. Section  concludes the paper and outlines extensions towards temperature-robust and imaging-integrated implementations.

We consider two narrow-band tone bursts at frequencies of fa and fb that predominantly excite the guided-wave modes ma and mb in a plate. Quadratic nonlinearity generates components at f±=fa±fb, received in a target mode mt. The mode-dependent wavenumber is denoted as km(f) and is obtained from the plate dispersion relation (tabulated or numerically extracted and interpolated). For a given source–receiver path, the propagation distance is Lpath. The effective receive chain sensitivity at frequency f (transducer bandwidth, coupling, analogue front end, and digitiser settings) is Grx(f). A dynamic-range penalty DRpen(fa,fb) captures near-saturation risk under two-tone excitation.

During selection, specimen-specific nonlinearity (micro-contact state, local quadratic coefficient, etc.) is treated as an unknown multiplicative factor. Accordingly, the criterion is designed to preserve the ordering induced by the known and/or observable factors rather than to predict an absolute mixed-line amplitude.

For a mode triplet (ma,mb→mt), the phase mismatch is 1Δk(fa,fb)=kma(fa)+kmb(fb)-kmt(f±),f±=fa±fb.

Phase coherence is necessary but not sufficient: the mixed line must also fall into a frequency region that is receivable and measurable under the intended analogue settings.

Equation () implies that, up to the unknown factor Γ, the detectability ordering of candidate pairs is governed by (i) mismatch severity, (ii) whether the available coherence can be exploited by the given path length, (iii) receive sensitivity at f±, and (iv) dynamic-range headroom. We therefore define a composite, dimensionless score as follows: 7J(fa,fb)=w1rΔk(fa,fb)+w2ϕLeffLint(fa,fb)+w3Grx(f±)G0-1+w4DRpen(fa,fb),f±=fa±fb. Here, ϕ(⋅) is a bounded, monotone increasing function that penalises path and/or coherence mismatch without numerical blow-up. In this study we use ϕ(x)=x/(1+x) (equivalently 1/(1+1/x)) so that ϕ→0 when Leff≪Lint (coherence is not yet a limiting factor) and ϕ→1 when Leff≫Lint (cancellation dominates).

The pre-acquisition selection is implemented as follows:

Physics pre-filter. Compute rΔk using Eq. () over the (fa,fb) grid and retain candidates in a low-mismatch corridor while enforcing that f± lies inside the usable receive band. A practical default is rΔk≤τ1 with τ1∈[0.05,0.10] (reported explicitly for traceability).

The score J acts upstream of demodulation and is therefore compatible with SPI-4 phase cycling, coherent averaging, or imaging and/or beam-forming. Operationally, J reduces trial and error by concentrating acquisition on tone pairs that are simultaneously dispersion-coherent and channel-admissible.

Experiments were conducted on a rolled aluminium plate with in-plane dimensions of Lx×Ly=1000mm×900mm and a thickness of h=1.6mm. A right-handed Cartesian frame (x,y) was adopted with the origin at the lower-left corner of the plate, x along the long edge, and y along the short edge. All sensor coordinates, path lengths, and crack tip locations reported in this section are referenced to this global frame. Figure  shows the benchtop experimental setup.

Four surface-bonded PZT wafers (S1–S4) were adhesively mounted on one surface of the plate. Each PZT could act as an actuator or receiver depending on the selected source–receiver configuration. In the global coordinate frame, the four PZTs were positioned at S1: (150,450) mm, S3: (550,450) mm, S2: (350,250) mm, and S4: (350,650) mm. These locations define an interrogation region that contains all source–receiver (S → R) paths considered here and the crack ligament to be monitored.

The primary interrogation path P1 (S1 → S3) crosses the crack ligament and is used for the nonlinear analyses reported in Sect.  and . Additional paths (P2–P4) serve as an orthogonal control path, an off-axis dynamic-range and/or noise-reference path, and a symmetry and/or repeatability path, respectively. The transmitter–receiver pairs, coordinates, nominal path lengths, and qualitative crack crossings are summarised in Table .

A surface-breaking crack was introduced into the ligament between S1 and S3 and then grown incrementally under controlled loading, producing crack length states C0–C6. For each state, the crack length a, crack tip coordinates, and remaining ligament distance along P1 were measured. All acquisitions were conducted under approximately constant laboratory ambient conditions. For field monitoring, the same short calibration sweep used to estimate Grx(f+) and DRpen can be executed periodically to track hardware drift (including temperature-related changes) under identical acquisition settings. The nominal sequence is reported in Table ; these crack states form the crack length axis in the SPI-4 crack growth analysis of Sect. .

All measurements used for the nonlinear analysis were acquired under fixed analogue gain, fixed gate timing, and an identical digitiser full-scale range. Two Hann-tapered narrow-band tones at fa and fb were digitally generated and uploaded to an NI cRIO-9030 (CompactRIO) system. The analogue-output module produced the summed two-tone waveform, which was then linearly amplified by a power amplifier and injected into the transmitting PZT. The received waveform was routed through a low-noise preamplifier and a front-end anti-alias filter and then was digitised by a cRIO analogue-input module at 5–10 MHz for storage and post-processing on the control PC. For each (fa,fb) pair and each crack state, at least N=16 repeats were recorded under the same time gate placed before the first significant back wall reflection.

To document the linear guided-wave behaviour under the same mounting and instrument settings, Fig.  shows representative time domain records on path P1. Figure a displays a gated time response under narrow-band excitation, and Fig. b compares the measured response at 145 kHz with a dispersion-based prediction for the selected Lamb mode. These records confirm that the chosen paths, gate, and sampling settings provide clean and repeatable measurements suitable for the sum frequency analysis reported on later.

We next executed the pre-acquisition screening workflow introduced in Sect.  on path P1 using a dense grid of candidate excitation pairs (fa,fb). The goal was to retain only pairs that are (i) physically favourable for coherent nonlinear mixing (low phase mismatch) and (ii) feasible on the implemented receive chain (sufficient sensitivity and dynamic-range headroom). The screening results underpin Figs.  and .

For screening we focus on the sum frequency component f+=fa+fb, which is received predominantly in a target mode mt. Phase mismatch is quantified by Δk(fa,fb) as defined in Eq. (), evaluated at f+, and the corresponding effective interaction length Lint is computed using Eq. ().

Pairs with sufficiently small rΔk (and, hence, long Lint) and with f+ inside the usable receive band were classed as physics-admissible. Among these, we then imposed hardware admissibility. A short calibration sweep around f+, using analogue settings identical to those of the nonlinear experiment, provided an estimate of the normalised receive chain sensitivity Grx(f+)/G0. In parallel, the dynamic-range penalty DRpen(fa,fb) was evaluated using Eq. (). Only candidates that avoided receive notches and near-saturation operating points were retained.

To prioritise viable candidates for acquisition, we evaluated the composite score J(fa,fb) defined in Eq. () (default weights of wi=1) for each retained pair, with all terms being specialised according to the sum frequency component f+. A low J indicates a combination of favourable phase matching, long effective mixing length relative to the available path length, high receive sensitivity at f+, and sufficient dynamic-range headroom.

All candidates were ranked in ascending J, and the lowest-J subset defines the top-K shortlist carried forward into the SPI-4 crack-monitoring measurements. Figure a plots the J versus candidate index (sorted by J), with the Top-K region highlighted. Figure b shows the corresponding receive response Grx(f+)/G0 and the dynamic-range penalty DRpen(fa,fb), illustrating that top-K pairs combine high receive responses with a low penalty. Figure c connects ranking to performance by plotting the estimated SNR(f+) versus J.

All nonlinear measurements used for crack growth analysis on P1 were acquired using an SPI-4 phase-cycling protocol combined with a single-shot same-spectrum readout. Four phase states were recorded by toggling the relative phase between the two drive tones at fa and fb and then were linearly combined to suppress deterministic leakage at fa and fb while coherently reinforcing the quadratic sum frequency component at f+=fa+fb. All four states were acquired under identical analogue gain, gating, and trigger conditions so that dynamic-range constraints were consistent across candidates.

For each SPI-4 combined record, a single fast Fourier transform (FFT) was computed, and the magnitudes at fa, fb, and f+ were extracted. A local noise floor was estimated from a nearby band free of deterministic lines. The detectability metric was defined as 8SNR(f+)=20log⁡10A(f+)Anoise,rms, where A(f+) is the amplitude at the sum frequency, and Anoise,rms is the root-mean-square noise level in that band.

Figure a illustrates leakage suppression for a shortlisted top-K pair, (fa,fb)=(145,171) kHz, measured on path P1. After SPI-4 combination, leakage at fa and fb is strongly suppressed, while the quadratic component at f+≈316 kHz emerges as a distinct line above the local noise floor. Figure b tracks the mixed-frequency response versus crack growth from C0 to C6 (Table ), showing that low-J shortlisted pairs cross the detection threshold earlier and exhibit stronger growth with crack extension than non-shortlisted, higher-J pairs.

The screening workflow of Sect.  was executed, without manual retuning, on each S → R path listed in Table . Candidate tone pairs were first filtered by physics (phase mismatch and receive band constraints) and then were screened by hardware using the measured receive response and dynamic-range headroom and were finally ranked by the composite score J defined in Eq. () to produce a top-K shortlist.

Repeatability and traceability were enforced as follows. For every shortlisted pair and for every crack length state C0–C6 (Table ), we acquired N=16 SPI-4 phase-cycled repeats under an identical time gate placed before the first back wall reflection. Along with the raw waveforms, we logged the S → R geometry (Table ), crack metadata (Table ), analogue gain and digitiser full-scale settings, gate timing, FFT window and zero-padding, and the noise band used for estimating Anoise,rms in Eq. (). These metadata provide an audit trail that enables independent reconstruction of the shortlisted pairs and reproduction of the reported mixed-frequency SNR(f+) trends.

This subsection evaluates whether the composite score J can be used as a global predictor of nonlinear crack detectability rather than as a path-specific tuning number. For every admissible tone pair (fa,fb) across all source–receiver paths and all crack states, we record (i) its score J, which penalises phase mismatch, short interaction length, weak receive gain at f+=fa+fb, and limited dynamic-range headroom, and (ii) the gated SPI-4 same-spectrum response level at f+, expressed as SNR(f+) in dB.

Figure a plots SNR(f+) versus J for all paths, with a single global linear fit and a horizontal detection threshold (20 dB). A clear inverse trend is observed: low-J tone pairs consistently lie above the threshold, whereas high-J pairs cluster close to the noise floor. Importantly, each acquisition path follows the same trend despite differences in propagation distance and coupling. This shows that J generalises across hardware configurations and crack lengths.

To quantify robustness, the full dataset is binned by score decile (Fig. b). The lowest-J bins exhibit high SNR(f+) with tight spread, indicating not only a stronger average response but also repeatability. As J increases, both the mean SNR(f+) and its stability deteriorate, and many tone pairs fall on or below the detection threshold. This separation defines a practically useful “low-J regime”: if a pair falls into that regime, it is very likely to yield a clear nonlinear peak.

Finally, we treat J as a binary selector. Tone pairs whose SNR(f+) exceeds the threshold are labelled as “detectable”, and we sweep a cutoff Jcut to predict detectability. The resulting receiver operating characteristic (ROC) curve in Fig. c shows a high area under the curve, confirming that a single scalar threshold on J can discriminate between usable and unusable frequency pairs with a favourable true-detect/false-alarm balance.

We further examine the sensitivity of this global trend to the weight vector in J. Starting from the default uniform setting ωi=1, we apply relative random perturbations ωi←ωi(1+ϵi) with ϵi∼N(0,σ2) and recompute the ranking. Figure a shows that the detectability–prediction area under the curve (AUC) degrades gracefully with increasing σ, while Fig. b and Table  summarise shortlist and ranking stability. These results indicate that the conclusions in Fig. a are not driven by a finely tuned weight choice.

In summary, J is not only physically interpretable; it is operationally predictive and remains stable under moderate weight perturbations. Ranking candidate tone pairs by J and keeping the lowest few provide a reliable shortlist of f+ without exhaustive brute-force scanning. This motivates the deployment-oriented analysis in Sect. .

The analysis in Sect.  shows that low-J tone pairs are, on average, measurably better. A practical question remains: how many of these low-J pairs must actually be transmitted and acquired in order to achieve a reliable detection on a given path and crack state? This is the key question for field deployment because each additional tone pair costs acquisition time and, in multiplexed systems, downlink bandwidth.

To answer this, all admissible tone pairs on all paths were first ranked in ascending J. We then constructed, for N=1, 2, etc., the following detection metric: 9Phit(N)=Pr⁡i≤Ns.t.SNR(f+)i≥SNRth, i.e. the probability that at least one of the first N tone pairs in the J-sorted list produces a nonlinear mixed-frequency component whose gated SPI-4 SNR exceeds the detection threshold SNRth (20 dB in this study). Figure  plots Phit(N) against N using all paths and crack lengths as independent trials.

Three features are noteworthy. First, the curve in Fig.  rises steeply for small N: using only the best tone pair (N=1) already yields a non-trivial hit probability because the score places genuinely coherent, instrument-compatible pairs at the top. Second, the curve exhibits an early saturation: for this dataset, N=3–5 is sufficient to reach a plateau where adding more, progressively worse (higher-J) candidates brings only marginal benefit. Third, the saturation level remains high when trials from different paths are pooled, indicating that the ranking imposed by J is not overfitted to a single geometry.

Table  reports the same analysis per path. Paths with longer propagation or slightly poorer coupling naturally start from a lower Phit(1), but all of them converge to high reliability once three to five of the lowest-J pairs are exercised. This path-by-path view is important for operators: it shows that a fixed shortlist size can be prescribed a priori (e.g. “always fire top 4”) without re-tuning the selection strategy for each sensor line.

Overall, this experiment confirms the deployment value of the ranking score: it not only tells us which tone pairs are promising but also tells us how many we need to test to make detection robust.

To assess whether the proposed frequency selection strategy is useful for tracking rather than merely detecting cracks, we compared a low-J (shortlisted) tone pair with a medium-J (non-shortlisted) pair on the same source–receiver path and over the same sequence of crack lengths (Table ). For each crack state we acquired 16 gated SPI-4 records under identical windowing, and we converted the mixed component at f+=fa+fb to a normalised response SNR(f+) using the noise band defined in Sect. .

Figure a shows SNR(f+) versus crack length. For the shortlisted pair, the response rises monotonically and crosses the 20 dB detection line at an early crack length (typically at or one step above the second crack state). A piecewise-linear fit to the pre-saturation part of the curve gives a pooled slope of about 2–3 dB mm⁻¹ across the inspected paths. By contrast, the non-shortlisted pair exhibits a shallower and sometimes non-monotonic trend (< 1 dB mm⁻¹) and often hovers around the detection threshold even at the largest crack state. The practical consequence is that the ranked pair does not only “see” the nonlinearity earlier but also amplifies the contrast per millimetre of growth, which is the quantity needed for sizing or trend-based alarms.

Figure b summarises the 16-repeat acquisitions at two representative crack sizes (near-threshold and mid-range). For the shortlisted pair, the within-state standard deviation is ≈ 1–1.5 dB, and the coefficient of variation stays below 7 %, indicating that the amplitude scatter is much smaller than the growth-induced increment (2–3 dB mm⁻¹). The non-shortlisted pair shows a broader spread (2–3 dB, CV 10 %–15 %), which is consistent with a response that sits closer to the noise floor and is more affected by small coupling changes. In other words, for the ranked pair, the signal change due to crack growth is larger than the measurement noise, while, for an arbitrary pair, the two are of comparable magnitude.

These results extend the conclusion of Sects.  and . The same low-J pairs that rank highest in detectability are also the pairs that (i) cross the detection threshold earliest, (ii) provide the steepest growth-to-response conversion, and (iii) do so with repeatability better than 2 dB under re-acquisition. This combination of early crossing, steep slope, and low scatter is what makes the selection rule usable for in-service monitoring, where operators will not be able to re-optimise tone pairs for every small crack increment.

Table  summarises the performance of the proposed selection scheme over all inspected S → R paths. Despite noticeable differences in path length and coupling, two features are consistent. First, the ranking generalises: for well-coupled, short paths, exciting only the top-3 lowest-J tone pairs already yields a hit probability close to 1.0; for the less favourable paths, extending the shortlist to three to five pairs restores the hit probability to above 0.8–0.9 while keeping SNR(f+) clearly above the 20 dB detection line. Second, the repeatability penalty on difficult paths is modest (only ∼1 dB larger within-state scatter than the best path), and so the growth-related increments reported in Sect.  remain observable.

This cross-path consistency stems from the definition of J: the physics-driven part (phase matching, effective interaction length) keeps the preferred (fa,fb) region stable across paths, while the channel-aware part (receive band gain and dynamic-range margin) prevents obviously unusable pairs from entering the shortlist. As a result, a single system-wide policy – “fire the first three to five ranked tone pairs” – can be issued without path-specific retuning.

A practical caveat is that very long or strongly reflective paths may require a slightly larger shortlist (top 6) or a refreshed receive band estimate, but these appear as outliers in Table  rather than as the dominant case.

Temperature influences the ranking primarily by shifting (a) the dispersion-based terms through k(f,T) and (b) the receive chain terms through Grx(f) and available headroom. A practical deployment policy is therefore to refresh the channel terms via the short calibration sweep whenever the operating temperature changes beyond a prescribed threshold and to re-evaluate Δk(fa,fb,T) and Lint(T) using a temperature-indexed dispersion table when the expected dispersion shift is non-negligible. This preserves the interpretability of J while maintaining shortlist consistency under slowly varying environments.

The present experiments focus on surface-breaking fatigue cracks, where contact acoustic nonlinearity produces a strong mixed component. For other defect classes (internal cracks, interfacial debonding, corrosion pits), the dominant nonlinearity and the effective source strength can differ, and so the absolute SNR(f+) level and the earliest detectable size may change. However, the role of J is to pre-select tone pairs that are physically phase-coherent and practically measurable on a given path and hardware chain; these constraints remain relevant irrespective of the defect mechanism. In applications where a defect class is known a priori, mode-triplet selection and modest weight re-tuning can be used to bias the shortlist towards frequencies with higher expected nonlinear radiation efficiency for that class while retaining the same screening workflow.