Archive for the ‘math’ Category

LaTeX now native on WordPress

2007 February 17

Now that WordPress has \LaTeX, let’s play around a little… I can’t call myself a “musical naturalist” without showing you the equations that describe the propagation of sound and music. Behold… the acoustic wave equation and the principles from which it derives:

Conservation of Mass

\dfrac{d\rho}{dt} + \rho (\vec{\nabla} \cdot \vec{v})  = 0

Conservation of Momentum

\rho \dfrac{d\vec{v}}{dt} = - \vec{\nabla} P

Conservation of Energy

\rho \dfrac{dU}{dt} - \dfrac{P}{\rho} \dfrac{d\rho}{dt} = 0

Equation of State:

\dfrac{\partial P}{\partial t} + \vec{v}\cdot\vec{\nabla}P = c^2 \left(\dfrac{\partial \rho}{\partial t}+\vec{v}\cdot\vec{\nabla}\rho\right)

P \sim c^2 \rho

Acoustic Wave Equation:

\nabla^2 P - \dfrac{1}{c^2}\dfrac{\partial^2 P}{\partial t^2} = 0

Simplify by assume time-dependence of form e^{i\omega t}:

P(\vec{x},t) \propto P(\vec{x})e^{i\omega t}

Helmholtz Equation

\nabla^2 P + k^2 P = 0 with k^2 = \dfrac{\omega^2}{c^2}

Solution #1: No boundaries, homogeniuos, isotropic media (spherical coordinates) in the limit |kr|>>1

P(r) \sim P_{0}\dfrac{e^{ikr}}{kr}

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