Figure 2 shows the pulsed wave simulation with in-house developed software [
12] based on the Point Source Model and programmed in Matlab (Matworks Inc., Natick, MA, USA). The time of flight from each array element to each image pixel is obtained by solving for the fastest path through the interface according to the Fermat principle. Then, synthetic signals generated by each element are delayed accordingly and added for computing the field amplitude at each simulation point, accounting for geometric beam spreading, the transmission coefficient between materials, and elements angular directivity.
The angular sensitivity of an array element can be approximated by [
13]
with
g representing the gap between the array elements and
γ representing the beam propagation angle with regard to the array normal.
The geometric beam spreading and transmission coefficient are computed as [
8]. Let us name
as the incident angle and
as the distances traveled by a ray in materials 1 and 2, respectively. Then, the beam spread factor is:
Let
be the longitudinal wave (L-wave) refraction angle and
be the shear wave (S-wave) refraction angle. The transmission coefficient for an incident L-wave into a refracted L-wave is (also as [
8]):
where
ρ1,
ρ2 are the densities of materials 1 and 2 respectively, and
is the S-wave velocity in material 2.
Figure 2a,b show the maximum amplitude
at each point in the field.
Figure 2c,d show the instantaneous field at
. It can be seen that plane waves are generated. Nevertheless, the lateral extent of the plane wave is limited to the region (PWR, Plane Wave Region) between the entry points of the extreme array elements (black dotted lines in
Figure 2), and only those pixels inside the PWR should be considered for reception beamforming with this plane wave angle. This is the main limitation of plane wave imaging: The effective imaging area for each emission is limited to the projection of the array aperture at the propagation angle.
Another effect is also shown by the simulation: the field amplitude is not constant across the wavefront, which could lead to echo amplitude differences because of a non-controlled apodization. This behavior is produced mainly by the angular sensitivity of the array elements, the geometric spreading of the beam, and the distribution of the entry points along the interface.
A simulation was made for plane waves with angles from −70° to 70° each 10°, to calculate the total insonification map inside the part. This total insonification map represents the average expected acoustic pressure at any point of the part after the emission of a set of plane waves, and it was defined as the summation at each point of the field amplitude (absolute value) generated by each one of the 15 plane waves at that point.
Figure 3 shows the insonification map, which is not homogeneous. Depending on the component geometry, this could lead to the presence of “blind zones” with very low insonification or quite large echo amplitude differences for similar reflectors depending on their position in the image. For parts where the geometry is known and fixed during the inspection, at least approximately, it could be useful to simulate the insonification map to detect possible blind regions. It is the case of parts with complex but constant shape by water immersion or curved parts with customized solid wedges. For example, this might help with choosing an optimal position of the transducer relative to the test part for obtaining an homogeneous insonification map.
3.1. Simplified Estimation of Amplitude Distribution
In order to develop a better understanding of how the amplitude distribution across a plane front wave is not uniform, a simplified approach was considered to estimate the effect of the non-uniform distribution of entry points on the interface.
The plane wavefront is generated by the interference between approximately cylindrical waves with a center at the entry points (xe, ze), which are not evenly distributed for an arbitrarily shaped interface. The closer the entry points are to each other, the higher the amplitude expected for the plane wave, as more of them will contribute in phase at some point of plane wavefront.
Figure 4a shows a schematic representation of the problem. The projections of the entry points over the plane wavefront are
with
Rk representing the distance from the entry point
k to the wavefront following the propagation direction
θ. At point
wk, the wave emitted by the element
j (dotted trace) will travel behind the plane wave at a distance
with
dk,j representing the distance between the entry points of both elements.
Figure 4b shows a schematic representation of the individual wavefields along the propagation line
r, where the wave emitted by the element
j (dotted) is delayed approximately Δ
R with regard to the wave emitted by element
k (solid line), and hence, the interference will not be fully constructive. Furthermore, as
dk,j increases, the contribution of the
jth element to the wavefront at
wk is reduced, because of the wideband nature of the signals. This way, if the entry points are separated, the amplitude of the resultant plane wave is expected to reduce.
Given the impulse response of the array elements
s(r), where
r is the distance from element to field point, the amplitude of the wavefield at point
wk can be obtained by
For estimating the plane wave amplitude distribution, a simple model for a wideband pulse with frequency
f and fractional bandwidth
B is assumed
with
f = 5 MHz and
B = 0.8 are used in this work. Finally, accounting for the three described effects, the amplitude across the wavefield can be approximated by the product of (10), (11), (13), and (17).
Solving (17) for all the array elements requires evaluating (17)
times, where
is the number of emitted plane waves. While this is not expected to be a limitation if the interface geometry is constant (the compensation curves are calculated only once), it could be a problem for water immersion inspections of components with non-constant profiles and auto-focusing algorithms. A first-order approximation is to consider as contributing elements to the point
wk only those whose delays Δ
Rk,j are below λ/2, and hence, their first positive cycles overlap (constructive interference). Modeling the emitted signal as a square pulse of length
λ, the expected amplitude at point
wk is just the number of elements that verify the above condition, and hence, (17) simplifies to
We define two approximations to Equation (20): and . In the second one, we neglect the beam spread and transmission factors.
Figure 5 shows the wavefront amplitude according to the simulation and the two approximations
and
, showing that both follow the trend of the simulation and they are almost equal. This suggests that the most relevant factors affecting the amplitude distribution are angular sensitivity and distribution of entry points along the interface. In
Section 4.2, we will discuss the feasibility of using this approximation as a compensation factor when computing the PWI image.
3.2. Experimental Results
To test the accuracy of the simulation, an experiment was made with an aluminum specimen in the shape of a 90° circular sector and 100 mm radius (
Figure 6b). The specimen was tested in the same conditions as those used in the simulation, with a 5 MHz, 0.65 mm pitch, 128-element array (Imasonic, Voray-sur-l’Ognon, France), and a 128-channel full-parallel phased-array system (Dasel, Madrid, Spain). The interface of the test piece was detected by the pulse-echo method ([
9]), and this interface was the one used in the simulation.
The test piece was positioned so that its backwall was parallel to the array, a 0° plane wave was transmitted (
Figure 6a), and an image around the backwall was generated. The backwall echo in the image should be (ideally) proportional to the amplitude of the field at
Z =
L, where
L is the distance from the array to the specimen backwall. The amplitude of the backwall echo was evaluated in the image to compare it with the simulation as an indirect way of measuring the field amplitude distribution across the wavefront.
Figure 7 shows the comparison between simulation and experiment. We can see that that simulation captures the overall shape of the amplitude distribution across the front-wave, but it is not accurate. This inaccuracy is probably associated with the oversimplifications used in the model. In any case, it could be used as a first-order approximation to reduce the PWR extent to avoid summing low-amplitude signals in the reception beamforming process because of low insonification areas.