Amplitude, isobar and grey-scale imaging of ultrasonic shadows behind rigid, elastic and gaseous spheres

The theory of wave reflection from spherical obstacles was applied for determination of the cause of the shadow created by plane wave pulses incident on rigid, steel, gaseous spheres and on spheres made of kidney stones. The spheres were immersed in water which was assumed to be a tissue-like medium. Acoustic pressure distributions behind the spheres with the radii of 1 mm, 2.5 mm and 3.5 mm were determined at the frequency of 5 MHz. The use of the exact wave theory enabled us to take into account the diffraction effects. The computed pressure distributions were verified experimentally at the frequency of 5 MHz for a steel sphere with a 2.5-mm radius. The experimental and theoretical pulses were composed of about three ultrasonic frequency periods. Acoustic pressure distributions in the shadow zone of all spheres were shown in the amplitude axonometric projection, in the grey scale and also as acoustic isobar patterns. Our analysis confirmed existing simpler descriptions of the shadow from the point of view of reflection and refraction effects; however, our approach is more general, also including diffraction effects and assuming the pulse mode. The analysis has shown that gaseous spherical inclusions caused shadows with very high dynamics of acoustic pressures that were about 15 dB higher in relation to all the other spheres. The shadow length, determined as the length at which one observes a 6-dB drop of the acoustic pressure, followed the relation r-6dB = 3.7a2/lambda with the accuracy of about 20% independent of the sphere type. lambda denotes the wavelength and a the sphere radius. Thus, a theoretical possibility of differentiating between gaseous and other inclusions and of estimation of the inclusion size in the millimeter range from the shadow was shown. The influence of the frequency-dependent attenuation on the shadow will be considered in the next study.