Lecture 16, 18.300, Tue Apr 5, 2022.
Summary
Continue with Gas Dynamics, Acoustics, and Strings.
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#052 Last lecture we computed the wave speed as c = sqrt{gamma*p/rho}
     --- sqrt{g*h} for Shallow water where p = 0.5*g*h^2.
     Do examples: Sound and tidal waves in deep ocean.
Linearize equations. Acoustics. Show it reduces to 1-D wave equations,
where we know solution [pset]. But want a "general method" [in pset we
used a trick].
--- Write equation in vector form, and use the eigenvectors of the matrix
    to diagonalize equations. Get general solution for  Y_t + A*Y_x = 0,
    for any diagonalizable matrix A with REAL eigenvectors.
    Explain why need real, and define HYPERBOLIC.
--- Multiply vector equation by left eigenvector, and get characteristic
    forms.
This method generalizes to the nonlinear case. For example, recall earlier
calculation:
  rho_t + u*rho_x + rho*u_x = 0  times +/- c/rho
  plus
  u_t + u*u_x + (c^2/rho)*rho_x = 0
  --->   (u_t + C*u_x) +/- (rho_t + C*rho_x) = 0  \ Characteristic form.
  where  C = u =/- c  [characteristic speeds].    /

#053 Linear algebra facts. Show that can write L_n*R_m = delta_{n, m}
     for left and right eigenvectors of a real diagonalizable matrix.
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