PhD student in physical acoustics

E-mail: chiragokani [at] gmail.com

Office: ETC 4.156

CV

- About
- Publications
- Proceedings
- Abstracts
- Practice problems (and solutions)
- Notes
- Code
- Extracurriculars

I am a 3rd-year PhD student in acoustics at the University of Texas at Austin and the Applied Research Laboratories. I am co-advised by Prof. Michael Haberman and Prof. Mark Hamilton. I use principles from continuum mechanics, electrodynamics, and quantum mechanics to describe the diffraction and scattering sound, as well as its coupling to different domains. Applications of my work lie in biomedical, electrical, and mechanical engineering.

As a student member of the Acoustical Society of America (ASA), I enjoy serving as a room monitor and Student Outreach for Networking and Integrating Colleagues mentor at the biannual meetings. Currently, I serve as the Biomedical Acoustics Technical Committee (BATC) Student Council Representative.

I have had a lifelong appreciation for waves—thunder, ocean dynamics, traffic jams, music, etc.—and it is a dream-come-true to get to study what underlies such phenomena at a fundamental level. I share my wonder for this wave-like world through my practice problems, notes, and outreach activities through WiSTEM and IntelliChoice.

- C. A. Gokani, M. R. Haberman, M. F. Hamilton. "Paraxial and ray approximations of acoustic vortex beams," J. Acoust. Soc. Am., in press.

- C. A. Gokani, T. S. Jerome, M. R. Haberman, M. F. Hamilton. "Born approximation of acoustic radiation force used for acoustofluidic separation," Proc. Mtgs. Acoust.,
**48**, 045002 (2022).

- C. A. Gokani, J. M. Cormack, M. F. Hamilton. "Growth rates of harmonics in nonlinear vortex beams," J. Acoust. Soc. Am.
**154**, A328 (2023). - C. A. Gokani, S. P. Wallen, M. R. Haberman. "Reciprocity, passivity, and causality in fully coupled acousto-electrodynamic media," J. Acoust. Soc. Am.
**154**, A118 (2023). - C. A. Gokani, S. P. Wallen, M. F. Hamilton, M. R. Haberman. "Source-driven homogenization theory for electro-momentum coupled scatterers," J. Acoust. Soc. Am.
**153**, A120 (2023). - S. P. Wallen, B. M. Goldsberry, C. A. Gokani, M. R. Haberman. "Computational analysis of sub-wavelength scatterers exhibiting electro-momentum coupling," J. Acoust. Soc. Am.
**153**, A120 (2023). - C. A. Gokani, Y. Meng, M. R. Haberman, M. F. Hamilton. "Analytical solution for a focused vortex beam radiated by a Gaussian source," J. Acoust. Soc. Am.
**152**, A56 (2022). - C. A. Gokani, M. R. Haberman, M. F. Hamilton. "Physical acoustics homework problems written by students: Undisciplined, irreverent, and original," J. Acoust. Soc. Am.
**152**, A168 (2022). - C. A. Gokani, T. S. Jerome, M. R. Haberman, M. F. Hamilton. "Born approximation of acoustic radiation force used for acoustofluidic separation," J. Acoust. Soc. Am.
**151**, A90 (2022).

Acoustics PhD qualifying exam review site: website with everything I need to know for my qualifying exam.

Acoustics I practice problem: an entertaining practice problem I wrote while studying for my Acoustics I midterm exam. It involves a "Gaussian comb" pressure-amplitude profile. [solution].

Acoustics I practice problem: created in preparation for the Acoustics I final. [solution].

Acoustics II practice problem: starring me and Jackson, created in preparation for the Acoustics II midterm. [solution].

Nonlinear acoustics study guide for midterm: problems that review the first half of Dr. Mark Hamilton's nonlinear acoustics course. [solutions].

Nonlinear acoustics study guide for final: problems that review the second half of Dr. Mark Hamilton's nonlinear acoustics course. [solutions].

Angle of Intromission: the angle of intromission is the angle that corresponds to perfect transmission of sound between two media. Its derivation is straightforward but was not discussed in class, so I have shown the steps that lead to Blackstock's Eq. (B-14) in Sec. 5.B.2.a of

*Fundamentals of Physical Acoustics*.Bohr radius from variation principle: based on class notes from Dr. Yuri Gartstein's Quantum I course at UTD.

Concentric Pressure-Release Spheres: based on an Acoustics II homework problem, displayed here are numerically determined combinations of

*ka*and*kb*that correspond to the eigenfrequencies for sound enclosed between two concentric pressure-release spheres. Bright spots on the surface plots correspond to eigenfrequencies.Converging spherical waves: notes to wrap my mind around converging spherical waves.

Coriolis and Centrifugal Forces: slides on fictitious forces.

Double Pressure-Release Parallel Planes: here is the solution for (1) a radially pulsating cylindrical source of sound extending between two pressure-release parallel planes. Dr. Blackstock mentions this kind of waveguide on page 432 in

*Fundamentals of Physical Acoustics*. He solves for (2) a radially pulsating cylindrical source between two rigid, parallel planes on page 430, and assigns the case of (3) a radially pulsating cylindrical source between one rigid boundary and one pressure-release boundary (a 0th-order model of sound in the ocean) as problem 12-13. Interestingly, case (1) and (3) excite many modes, but case (2) excites only the lowest mode.Fourier acoustics: in the

*i*(*kx*-*ωt*) convention.The Fubini solution: The Fubini solution solves the lossless nonlinear approximate evolution equation (accurate to quadratic order) as a Fourier sine series from 0 ≤ σ < 1, where σ is distance nondimensionalized by the shock-formation distance. The expansion coefficients

*B*involve integrating over θ from θ = 0 to θ = π. The phase Φ of the Fubini solution is given by the dimensionless phase of the lossless nonlinear approximate solution, Φ = θ + σ sin(Φ). This page answers the question, "Why does Φ take on the same limits as θ in the integration?"_{n}Green's functions: I show that Green's functions are solutions of an inhomogeneous Helmholtz equation. [How to directly integrate the first integral on the left-hand-side of equation (2)]. Here I re-derive Morse and Ingard's integral equation (7.1.17). This is a derivation of the Sommerfeld radiation condition.

Linear energy density of sound in air. The potential and kinetic energies being equal for progressive waves recovers a special case of the virial theorem.

Intensity integral: evaluation of the integral for intensity in David Blackstock's

*Fundamentals of Physical Acoustics*, section 1E-3 (page 50). Dr. Hamilton's approach, setting*f*= pressure and*g*= particle velocity.Linear Sound Speed Gradient: I use the calculus of variations to show that arcs of circles minimize the travel time between two points in a medium where the sound speed varies linearly (i.e., the upper ocean).

*This recovers the result in "Acoustics: An Introduction to its Physical Principles and Applications" by Allan D. Pierce, section 8-3.*Pappus's theorem: interesting theorem derived in Dr. Mohammad Akbar's undergraduate geometry course.

Planck quantities: based on Prof. Xiaoyan Shi's first Modern Physics lecture at UTD.

Radiation from General Axisymmetric Spherical Source: re-organized for clarity.

Sound in General Axisymmetric Spherical Enclosure: following Dr. Hamilton's approach for radiation due to a general axisymmetric source.

Spherical Symmetry in Acoustics and Quantum Mechanics: physically, why do eigenfrequencies of sound spherical coordinates not depend on the azimuthal index

*m*?Thoughts on the 1D linear wave equation: thoughts from the first week of grad school. It turns out that d'Alembert, Euler, Bernoulli, and others had contentiously debated these very issues 250 years ago!

Three-Medium-Problem Demystified I derive the pressure reflection and transmission coefficients of the three-medium problem in a way that makes sense to me.

Virial theorem for string: a well-known result of Hamilton's formulation of classical dynamics that relates the average kinetic energy of a system to its virial. I have never seen it applied to waves on a string, but in Dr. Mark Hamilton's Acoustics I course, we arrived at a special case of the theorem that showed that the kinetic and potential energies are equal for progressive waves on a string. I derive and apply the virial theorem to show the more general result.

1D wave equation for large deformations on string: a clear step-by-step of Dr. Blackstock's pages 23-24 of

*Fundamentals of Physical Acoustics*.Wave Phenomena notes: my notes from Dr. Hamilton's course, spring 2024..

beamer template: slides using the "Madrid" theme with biblatex implementation, boxed equations, etc. Includes a simple version for quick slide generation, and a long version to demonstrate capabilities.

kpfonts with biblatex and more: my go-to template for everyday document preparation; contains short and long versions and an example bibliography built using biblatex. Contains all the features below and more.

euler-beton template: AMS Euler math and Beton text.

helvetica: format for Haberman group weekly update.

Oscillator with damping: dimensionless code for motion of oscillator with damping.

fig_JASA: figure formatting file to match JASA numbering fonts (Times New Roman).

Fourier transform: using fft in MATLAB.

Laguerre polynomials demo in MATLAB: demo of MATLAB functions for Laguerre polynomials and generalized Laguerre polynomials in MATLAB.

Legendre Polynomials and Associated Legendre Functions in MATLAB: demo of MATLAB functions for Legendre polynomials and associated Legendre functions.

Spherical Bessel, Neumann, and Hankel functions in MATLAB: package that includes user-defined functions for the spherical Bessel, Neumann, and Hankel functions, along with a demo.

Chimes by Chirag: I began handcrafting wind chimes as a sophomore at UTD.

Echo Amphitheater: something I did for fun to better understand the echoic properties of this enchanting natural formation in northern New Mexico.

IntelliChoice SAT Math camp: this course originally served ~100 high school students in the DFW area during summer 2020 (COVID). Daily lessons and homeworks are now publicly available.

Masala cha: traditional Indian tea

WiSTEM: I volunteer over the summer in these diversity initiatives at UT Austin.