Shallow Water Waves on a Viscous Fluid—The Undular Bore

Abstract

A theory is presented for the surface profile above a fully developed Poiseuille channel flow. Small disturbances to this flow are examined, and it is shown that if the (channel depth)/(wavelength) ratio is small (shallow waves), and the Reynolds number large enough, these disturbances initially travel at the classical dynamic (Burns) wave speeds. However, by introducing appropriate far‐field coordinates it follows that the disturbance eventually travels at a different wave speed—the kinematic wave speed. To confirm this, the dynamic waves are shown to decay by using standard boundary layer techniques. This general result (of decay) agrees with previous one‐dimensional theories. The profile close to the kinematic wave front is examined and shown to satisfy an equation of the form ${\eta }_T{\mathrm{\hspace{0.17em}+\hspace{0.17em}\eta \eta }}_X{\mathrm{\hspace{0.17em}+\hspace{0.17em}\eta }}_{\mathrm{XXX}}{\mathrm{\hspace{0.17em}=\hspace{0.17em}\Delta \eta }}_{\mathrm{XX}}$ , where $\mathrm{\eta (X,\hspace{0.17em}T)}$ is the surface profile. This equation is called the Korteweg‐de Vries‐Burgers equation. The form of the steady solution of this equation exhibits all the characteristics of the undular bore. A bound on $\Delta$ agrees with stability requirements found by other authors using different methods.