By Yuri M. Galperin
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Extra info for An Introduction to Modern Solid State Physics
E-r Oz' If - 7'[ - 1 / ? f~cei[t'~(x-x')+k 27r ~ _ ~ . 65) Comparison with Eq. 66) Finally we insert this result into Eq. 62) to yield R a y l e i g h ' s s e c o n d i n t e g r a l formula: p(x,y,z) - 1 / ? 67) with z > z'. This formula relates the spatial pressure in one plane to the spatial pressure in another plane. It gives a forward propagation formula by convolving the pressure in the plane z' =constant with a propagator gp projecting the field to a more distant plane (z > z'). Later we will look at the inverse of this equation so that we can backpropagate the spatial acoustic pressure field.
6: Evanescent wave traveling parallel to x axis, decaying exponentially in the vertical direction. In k-space this wave falls outside the radiation circle, as shown. All plane waves outside the radiation circle are subsonic. From Eq. 38) so t h a t the intensity is f IAl2e -21'': ~ - 2wpo (k~i + kyj). 39) F r o m this it is clear t h a t the power flows parallel to the (x, y) plane in the direction kx~ + ku), decaying exponentially in the z direction. T h e direction of the evanescent 26 CHAPTER 2.
The sinusoidal variation of pressure along the two axes is shown gray scale C H A P T E R 2. P L A N E I I'AVES 22 Figure 2 . 3 : Plane wave illustration showing trace matching. Snapshot at t - 0. The insert on the right shouts the equivalent k-space representation. The trace wave is supersonic, and lies within the radiation circle of radius k. coded, with white-gray representing positive pressure and black-gray negative pressure. Since pressure must be continuous throughout space (no sources), adjacent regions along the two axes must have the same pressure.
An Introduction to Modern Solid State Physics by Yuri M. Galperin