Self focusing or natural focusing is an effect observed in the emitted field by an ultrasonic transducer. The effect is known and atrributed to the wavelets generated at the edge of the crystal, interfereing in a coherent manner and that, maximizes at the boundary of nearfield / farfield.
It actually reminds me of the Poisson or Arago spot, where a bright spot shows up in the middle of the shadow of opaque object (on-axis) blocking the wave (light) .
Both are taking place in the transition zone, being explainable to some degree by Fresnel diffraction. Yet, I couldnt find anyone analogy in between , and wondering if I am not doing my bibliography right!
There is a nice text on acoustics by Eugen Skudrzyk ("The Foundations of Acoustics"), long out of print I believe, that has a nice treatment of this effect, using rather different terminology, roughly along the following lines:
In other words, the transducer looks approximately like piston in a baffle, plus an edge diffraction component around the edge of the transducer, which accounts for the fact that there is no baffle.
As I recall Skudrzyk does this for sound through an aperture in an infinite baffle, but much the same applies for a transducer face in free space by Babinet's Principle, which is also treated nicely by Skudrzyk.
The original work by Rabinowitz is very old. I had to dig it up on microfiche some time ago.
For non-circular transducers (square or rectangular faces), the edge-diffracted field is dominated by points of stationary phase in the Rabinowitz integral. These lead to the edge-diffracted rays.
I used this method to estimate the edge diffraction effect by (audio) transducers in finite baffles with good success when compared with experiment.
But use the Skudrzyk text with some care because it was translated (from Russian perhaps) and has typos in its equations which are easily corrected if you work things out yourself.
I hope that the Skudrzyk and Rabinowitz names help in literature search.
Ronald
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