The　D.　J.　Korteweg-G.　De　Vries　nonlinear　dispersive　wave　equation　can　be　readily　extended　by　adding　a　second　derivative　term　so　as　to　take　account　of　the　dissipative　effect.　It　then　leads　to　the　Korteweg-de　Vries-Burgers　equation.　This　paper　studies　the　evolution　process　of　the　shock　structure　connected　with　the　K-dV-Burgers　equation　(including　the　KdV　equation).　It　examines　the　initial　value　problem　by　numerical　methods　with　the　assumption　that　the　initial　disturbance　has　a　shock-type　profile　starting　at　an　initial　value　and　smoothly　decreasing　to　the　final　value　over　a　narrow　region.　When　the　dissipative　effect　is　included,　the　nonstationary　oscillatory　shock　front　tends　to　a　steady　oscillatory　shock　structure,　which　strongly　indicates　that　the　solution　of　the　unsteady　(time-dependent)　K-dV-Burgers　equation　asymptotically　approaches　its　steady　state　solution.