In　this　paper,　we　investigate　the　asymptotic　behavior　of　global　classical　solutions　to　the　mixed　initial-boundary　value　problem　with　small　BV　data　for　linearly　degenerate　quasilinear　hyperbolic　systems　with　general　nonlinear　boundary　conditions　in　the　half　space　{(t,x)|t　＞=　0,　x　＞=　0}.　Based　on　the　existence　result　on　the　global　classical　solution,　we　prove　that　when　t　tends　to　the　infinity,　the　solution　approaches　a　combination　of　C　(1)　traveling　wave　solutions,　provided　that　the　C　(1)　norm　of　the　initial　and　boundary　data　is　bounded　and　the　BV　norm　of　the　initial　and　boundary　data　is　sufficiently　small.　Applications　to　quasilinear　hyperbolic　systems　arising　in　physics　and　mechanics,　particularly　to　the　system　describing　the　motion　of　the　relativistic　string　in　the　Minkowski　space-time　R　(1+n)　,　are　also　given.