latex/problems/Serway_and_Jewett_4_wking/problem13.12.T.tex

   1 \begin{problem}
   2 Show that the all functions of the form $y(x,t) = f(x \pm vt)$ for any
   3 function $f(z)$ satisfy the linear wave equation (Equation 13.19)
   4 \begin{equation}
   5   \npderiv{2}{x}{y} = \frac{1}{v^2}\npderiv{2}{t}{y}\;.
   6 \end{equation}
   7 \end{problem} % generalized form of Problem 13.12
   8
   9 \begin{solution}
  10 Taking the partial derivatives with respect to space
  11 \begin{align}
  12   \pderiv{x}{y} &= \pderiv{z}{f} \cdot \pderiv{x}{z}
  13                  = \pderiv{z}{f} \cdot \pderiv{x}{}\p({x \pm vt})
  14                  = \pderiv{z}{f} \\
  15   \npderiv{2}{x}{y} &= \pderiv{x}{}\p({\pderiv{z}{f}})
  16                  = \npderiv{2}{z}{f} \cdot \pderiv{x}{z}
  17                  = \npderiv{2}{z}{f} \cdot \pderiv{x}{}\p({x \pm vt})
  18                  = \npderiv{2}{z}{f}\;,
  19 \end{align}
  20 where we have used the chain rule
  21 \begin{equation}
  22   \pderiv{x}{}\p({f(z(x))}) = \pderiv{z}{f}\cdot\pderiv{x}{z}
  23 \end{equation}
  24 with $z(x) = x \pm vt$.
  25
  26 Taking the partial derivitives with respect to time
  27 \begin{align}
  28   \pderiv{t}{y} &= \pderiv{z}{f} \cdot \pderiv{t}{z}
  29                  = \pderiv{z}{f} \cdot \pderiv{t}{}\p({x \pm vt})
  30                  = \pm v \pderiv{z}{f} \\
  31   \npderiv{2}{t}{y} &= \pderiv{t}{}\p({\pm v \pderiv{z}{f}})
  32                  = \pm v \npderiv{2}{z}{f} \cdot \pderiv{t}{z}
  33                  = \pm v \npderiv{2}{z}{f} \cdot \pderiv{t}{}\p({x \pm vt})
  34                  = (\pm v)^2 \npderiv{2}{z}{f}
  35                  = v^2 \npderiv{2}{z}{f}\;.
  36 \end{align}
  37 So
  38 \begin{equation}
  39   \npderiv{2}{x}{y} = \npderiv{2}{z}{f} = \frac{1}{v^2}\npderiv{2}{t}{y}
  40 \end{equation}
  41 which is what we set out to show.
  42 \end{solution}