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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: an elegant way of calculating the Bohr radius without looking at the Schrödinger equation. Based on class notes from Prof. Yuri Gartstein's Quantum I course at UTD.
Concentric pressure-release spheres: based on an Blackstock's problem 10-10 from Fundamentals of Physical Acoustics. 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. Prof. 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.
Echo Amphitheater: something I did for fun to better understand the echoic properties of this enchanting natural formation in northern New Mexico.
Fourier acoustics: in the \(i(kx-\omega t)\) convention.
Fubini solution: The Fubini solution solves the lossless nonlinear approximate evolution equation (accurate to quadratic order) as a Fourier sine series from \(0 \leq \sigma < 1\), where \(\sigma\) is distance nondimensionalized by the shock-formation distance. The expansion coefficients \(B_n\) involve integrating over \(\theta\) from \(\theta = 0 \) to \(\theta = \pi\). The phase \(\Phi\) of the Fubini solution is given by the dimensionless phase of the lossless nonlinear approximate solution, \(\Phi = \theta + \sigma \sin(\Phi)\). This page answers the question, "Why does \(\Phi\) take on the same limits as \(\theta\) in the integration?"
Green's functions: working out the approach prescribed in Theoretical Acoustics by Morse and Ingard, I show that \(e^{ikR}/4\pi R\) solves the 3D inhomogeneous Helmholtz equation \((\Laplacian + k^2)f = -\delta(\vec{r}-\vec{r}_0)\). [How to directly integrate the first integral on the left-hand-side of Eq. (2)]. Here I re-derive Morse and Ingard's integral equation (7.1.17), which is the Helmholtz-Kirchhoff integral. This is a derivation of the Sommerfeld radiation condition. For much more on Green's functions, see my notes from Prof. Hamilton's Wave Phenomena course.
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 Prof. Blackstock's Fundamentals of Physical Acoustics, Sec. 1E-3 (page 50). Prof. 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.
Masala chai: traditional Indian tea
Nonlinear wave equation for large deformations on string: a clear step-by-step of Prof. Blackstock's pp. 23-24 of Fundamentals of Physical Acoustics.
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 Prof. Hamilton's approach for radiation due to a general axisymmetric spherical source.
Spherical symmetry in acoustics and quantum mechanics: physically, why do eigenfrequencies of sound spherical coordinates not depend on the azimuthal index m? (Confusingly the letters conventionally used for the indices corresponding to spherical harmonics in acoustics are flipped from those used in quantum mechanics!)
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. In Acoustics I, 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.

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