Ultrasonic system models and measurements

Abstract

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A Study on the Model of Detecting the Liquid Level of Sealed Containers Based on Kirchhoff Approximation Theory

“…At any point B on the actual transducer, the sound pressure ${\mathrm{p}}_{\mathrm{e}}$ radiated by the circular section can be calculated by Equation (1); furthermore, by integrating the sound pressure ${\mathrm{p}}_{\mathrm{e}}$ on the entire surface of the transducer [14,15,16], the average echo sound pressure $\overline{{\mathrm{P}}_{\mathrm{s}}}$ can be obtained approximately by Equation (3):${\overline{\mathrm{p}}}_{\mathrm{normals}}\left(\mathrm{normalR},\mathrm{sans-serif\theta }\right)=\left(\frac{\mathrm{sans-serif\pi }{\left(\mathrm{normald}/2\right)}^2}{\mathrm{sans-serif\lambda }\mathrm{normalR}}\right)\left[\frac{2{\mathrm{J}}_1\left(\mathrm{normalk}\left(\frac{\mathrm{d}}{2}\right)\sin \mathrm{sans-serif\theta }\right)}{\mathrm{normalk}\left(\frac{\mathrm{d}}{2}\right)\sin \mathrm{sans-serif\theta }}\right]·\overline{p_r}·{\mathrm{sans-serif\pi }\mathrm{normala}}^2$ where $\mathsf{\lambda }$ is the wave length of ultrasonic waves in a medium, $\mathrm{a}$ is the transducer radius, $\mathrm{k}$ is the wave number, ${\mathrm{J}}_1$ is the first order Bessel function, $\mathsf{\theta }$ is the angle between R and the Z-axis, ${\mathrm{R}}_{\mathrm{w}}$ is the reflection coefficient at the inner surface of the wall, and $\overline{{\mathrm{p}}_{\mathrm{r}}}={\mathrm{p}}_0{\mathrm{e}}^{-\mathrm{sans-serif\alpha }\mathrm{normalL}}{\mathrm{R}}_{\mathrm{w}}·4{\mathrm{a}}^2/{\mathrm{d}}^2$.…”

“…In Figure 5 , the circular section at the inner surface can be approximately regarded as a round transmitting transducer whose average initial pressure is . At any point B on the actual transducer, the sound pressure radiated by the circular section can be calculated by Equation (1); furthermore, by integrating the sound pressure on the entire surface of the transducer [ 14 , 15 , 16 ], the average echo sound pressure can be obtained approximately by Equation (3): where is the wave length of ultrasonic waves in a medium, is the transducer radius, is the wave number, is the first order Bessel function, is the angle between R and the Z-axis, is the reflection coefficient at the inner surface of the wall, and .…”

Ultrasonic modeling for complex geometries and materials

“…The computational inefficiency of numerical methods has led many authors to consider approximate wave scattering models such as elastodynamic ray theory [66,67], low frequency expansions [68,69], and the Kirchhoff and Born approximations [70][71][72][73][74][75][76][77][78][79][80][81][82][83][84][85]. The Kirchhoff approximation in particular has been found to be very useful and widely studied in the literature [70][71][72][73][74].…”

“…However, experiments of Gray [62] and recent experimental benchmark studies [75] have shown that the Kirchhoff approximation accurately predicts the pulse-echo response of a circular crack at relatively high angles from normal incidence. Also, recent model-based benchmark studies [77][78] have suggested that the Kirchhoff approximation can work well at much lower frequencies/sizes where kb »1 is violated. In Chapter 5 it will be shown that this apparent disagreement of earlier and more recent studies on when the Kirchhoff approximation is valid can be explained by the fact that the accuracy of the Kirchhoff approximation depends on both the non-dimensional wave number kb and the bandwidth of the system.…”