The Wave Equation for Plane Waves

• From Newton's Second Law, we found
  $$\ensuremath{\frac{\partial {p}}{\partial {x}}} = -\rho \ensuremath{\frac{\partial {v}}{\partial {t}}}.$$

• From Hooke's Law, we found
  $$p = - K \ensuremath{\frac{\partial {\xi}}{\partial {x}}}.$$

• If we differentiate the first equation once with respect to $x$ and differentiate the second twice with respect to time $t$, noting that particle velocity $v = \partial {\xi}/\partial {t}$, we find
    $$\ensuremath{\frac{\partial^{2} {p}}{\partial {t}^{2}}} = \frac{K}... ...ial {x}^{2}}} = c^{2} \ensuremath{\frac{\partial^{2} {p}}{\partial {x}^{2}}},$$ (6)
  which is the one-dimensional wave equation for lossless pressure plane waves.