  The addition of higher order asymptotic corrections to the Korteweg-de Vries
equation results in the extended Korteweg-de Vries equation. These higher order
terms destabilise the dispersive shock wave solution, also termed an undular
bore in fluid dynamics, and result in the emission of resonant radiation. In
broad terms, there are three possible dispersive shock wave regimes: radiating
dispersive shock wave (RDSW), cross-over dispersive shock wave (CDSW) and
travelling dispersive shock wave (TDSW). While there are existing solutions for
the RDSW and TDSW regimes obtained using modulation theory, there is no
existing solution for the CDSW regime. Modulation theory and the associated
concept of a Whitham shock are used to obtain this CDSW solution. In addition,
it is found that the resonant wavetrain emitted by the extended Korteweg-de
Vries equation with water wave coefficients has a minimal amplitude. This
minimal amplitude is explained based on the developed Whitham modulation
theory.
