T. Homma, the author and M. Takahashi proved that there is a good algorithm for recognizing the $3$-sphere $S^3$ in $3$-manifolds with Heegard-splittings of genus two and later T. Homma and the author proved that any $3$-bridge knot diagrams of the trivial knot $T$ always have waves but generally speaking there are many knot diagrams of $T$ without waves. In this paper, we define the concept of $n$-waves and $0$-waves means waves. Then it is shown that there exist knot diagrams of $T$ with no $n$-waves, where $n$ is an arbitrary non-negative integer. Furthermore we consider a method to distinguish by computer whether knot diagrams with a certain range of crossings give the trivial knot. Of course, the method does permit us to distinguish $3$-bridge knot diagrams to be trivial and so at this present a plenty of knot diagrams are distinguished to be trivial by computer.