This is the familiar SHO equation, with solutions
This is the familiar SHO equation, with solutions
but the cos kx does not fit the fixed ends, y(0)=y(L)=0, so kn=n?/L, so?n=n ?v/L
The wave equation is linear, so sums of solutions are also solutions. The shape at time zero can be what we please, so we need both sin and cos in t, and then the general solutions are