 Solution

Put $\xi =x+vt$ and $F(\xi )=A{e}^{-{\xi }^2/x_0^2}$, with $x_0\ne 0$. The chain rule givesFor a nonzero pulse the wave equation on a string therefore requiresIt is not a solution for arbitrary $v$; the zero pulse $A=0$ is the trivial exception. Here $\mu$ is linear mass density, and $T>0$ is tension. With $v>0$, the maximum lies at $x=-vt$, so the Gaussian function translates towards decreasing $x$ by $v(t_2-t_1)$ between the two times. Its amplitude and width stay constant.