PSI - Issue 80

Guangxiao Zou et al. / Procedia Structural Integrity 80 (2026) 93–104

95

Author name / Structural Integrity Procedia 00 (2023) 000–000

3

where d ref is a reference distance used for calibration, and X ( ω ) is the frequency-domain representation of the excitation signal x ( t ). Then, in time domain, it can be expressed as

d d ref

i ω d cp ( ω ) }

) − 1 / 2 F − 1 { X ( ω )exp −

u ( t ) = (

(2)

Here u ( t ) is the inverse Fourier transform of U ( ω )and F − 1 is the inverse Fourier transform operation. Now, consider a single-mode Lamb wave excited at location e = [ e 1 , e 2 ] and a small scatterer present at location s = [ s 1 , s 2 ]. The wave arriving at the scatterer after propagating from the excitation source can be expressed, in the frequency domain, as:

e − s ∥ d ref

U [ ω ; e , s ] = ( ∥

i ω ∥ e − s ∥ cp ( ω )

) − 1 / 2 X ( ω )exp −

(3)

where ∥ e − s ∥ is the distance between the emitter and the scatterer. The presence of a small scatterer at a given location can be modeled as a secondary source that, in the far field, scatters the incident wave omni-directionally according to some scattering pattern ψ [ ω ; θ in ,θ out ]. This pattern dictates the amplitude and phase of the scattered wave based on the angles of incidence and reflection, as well as the frequency. The frequency spectrum of the scattered wave, measured at location r = [ r 1 , r 2 ], which accounts for propagation from the source to the scatterer and then to the receiver, is givenby:

s − r ∥ d ref

= ( ∥

i ω ∥ s − r ∥ cp ( ω )

) − 1 / 2 ( ψ [ ω ; θ in ,θ out ] × U [ ω ; e , s ])exp −

U scattered [ ω ; e , r ]

e − s ∥ d ref

s − r ∥ d ref

) − 1 / 2 ( ∥

in ,θ out ] × ( ∥

i ω ∥ e − s ∥ cp ( ω ) exp −

i ω ∥ s − r ∥ cp ( ω )

) − 1 / 2 X ( ω )exp −

= ψ [ ω ; θ

(4)

In the time domain , the scattered wave can be expressed as:

e − s ∥ d ref

s − r ∥ d ref

) − 1 / 2 ( ∥

= ( ∥

i ω ∥ e − s ∥ cp ( ω ) exp −

i ω ∥ s − r ∥ cp ( ω ) }

) − 1 / 2 F − 1 { ψ [ ω ; θ in ,θ out ] × X ( ω )exp −

u scattered [ t ; e , r ]

(5)

This equation provides a forward model that predicts the received scattered signal based on the excitation signal, x(t). The model accounts for the phase shifts and amplitude changes that occur during propagation and scattering, a relationship that is fundamental to the imaging algorithms discussed subsequently. In practice, the signal measured on a damaged plate is a superposition of several wave components. In addi tion to the desired damage-scattered signal u scattered [ t ], the measurement also includes the direct wave from the transmitter to the receiver u direct [ t ], reflection waves from structural boundaries u boundary − reflection [ t ], and white noise w measured − damage [ t ]. Therefore, the complete measured signal can be expressed as:

u measured − damage [ t ]

direct [ t ]

scattered [ t ]

boundary − reflection [ t ]

measured − damage [ t ]

= u

+ u

+ u

+ w

(6)

To isolate the signals scattered by damage, a technique known as baseline subtraction is commonly used. This involves taking a set of measurements from the structure in its damaged state and subtracting a corresponding set of baseline measurements taken when the structure was known to be undamaged. Because the corresponding baseline signal only contains direct wave u direct [ t ], reflection waves from structural boundaries u boundary − reflection [ t ], and white