Mathematical Derivation for Wave Propagation and Scattering in Random Media by Akira Ishimaru: A step-by-step electromagnetic wave mathematical derivation to accompany the text written by Ishimaru

Ever become frustrated when the classic textbook skip and jump steps in formulas derivation in the teaching of important assumption and approximation? If you are like countless students being stuck at, a particular mathematical calculus-working step or steps while reading a simple undergraduate physics text, then this manuscript may help in your journey. This manuscript discusses the core chapter 2 on electric field and magnetic field propagation and scattering in spherical coordinate system of Ishimaru’s Wave Propagation and Scattering in Random Media, a classic IEEE reissue. It includes calculus of partial differential equations, spherical Bessel’s functions of the first kind, spherical Hankel’s functions of the first kind, associated Legendre polynomial, and integration by parts.This text provides a detail step-by-step symbolical derivation workouts that are omitted or incomplete in between the presented physics formulas or definitions. All the mathematical derivations and expansions involve the trigonometry functions and identities, first and second order time independent partial differential equations, calculus as well as simultaneous equations solving via matrix or direct substitution method. Readers with/without fundamental handle on electromagnetism and/or undergraduate level mathematics proficiency but wish to study the physics and/or the applied mathematics, can now use this text as a step-by-step solution reference and derivation counter-checking resources. Readers can follow and learn the essence of scalar electric field and magnetic field Hertz’s vectors application. There are 12 subchapters of electromagnetics theory derivation. Students will learn about cross sections and scattering amplitudes theory in subchapter 2-1. Next, are subchapter 2-5 on Rayleigh scattering, subchapter 2-6 Rayleigh-Debye scattering, which is also known as Born approximation, and subchapter 2-7 WKB interior wave number approximation. These subchapters are essential electromagnetic wave approximation and forward scattering theorem knowledge for serious physics students. The Mie scattering derivation and its approximation subchapter 2-8 carries the most “essence” for any physics students to learn fundamental of electric and magnetic field vector and scalar Laplacians in spherical coordinates and scalar Hertz’s vectors in the undergraduate level as well as a fresher course for graduate students. Followed by Subchapter 2-9 on Elliptic polarization and the Stokes parameters explore the actual representation found in real settings. The remains subchapters consist of subchapter 2-12 Scattering amplitude functions and the Stokes matrix solution, subchapter 2-13 Transformation of the Stokes parameters for rotation about the axis, subchapter 2-14 Particle size distribution. These chapters are the applied physics knowledge in electromagnetic wave propagation physics. Last of all are subchapter 2-15 Acoustic waves and subchapter 2-16 Acoustic scattering simple derivation.Many critics commented the mathematical derivation and working found on the text is very redundant. The author is truly glad that these critics are well equipped with the mathematical and tools background to comprehend the wonderful electromagnetic wave scattering physics described in the original manuscripts. To the author this rudimentary text is meant for those minority few that are not that fortunate on mathematical upbringing training or skill to be well versed to learn, follow, brush up and play catch up in order to sincerely appreciate what is delivered on the original manuscripts, without getting lost in the mathematics. Read more