Newton's Second Law for Plane Waves

Figure 2: An infinitesimal section of an enclosed fluid with pressure variation.

• We can evaluate Newton's Second Law for the infinitesimal section of fluid diagrammed in Fig. 2, which is assumed to be subjected to a differential pressure along its length.

• If the fluid has a mass density $\rho$ and cross-sectional area $A$, the section mass is given by $\rho A dx$.

• From Newton's Second Law ($F = ma$), we then have:
  $$\begin{eqnarray*} F &=& A \left[ p - (p + (\ensuremath{\frac{\partial {p}}{\part... ...\ &=& \rho A dx \ensuremath{\frac{\partial {v}}{\partial {t}}}, \end{eqnarray*}$$
  where $v$ is particle velocity.

• From this expression, we find:
    $$\ensuremath{\frac{\partial {p}}{\partial {x}}} = -\rho \ensuremath{\frac{\partial {v}}{\partial {t}}}.$$ (1)

• Converting to an expression for volume velocity ($u = A v$), we find:
    $$\ensuremath{\frac{\partial {p}}{\partial {x}}} = -\frac{\rho}{A} \ensuremath{\frac{\partial {u}}{\partial {t}}}.$$ (2)