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By comparing the mathematical properties of the traveling-wave solution with the physical wave equation, we can derive the wave speed. The traveling wave $f(x,t)=f(x-vt,0)$ satisfies $\frac{\partial^2 f}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 f}{\partial t^2}$.

Conditions: The wave can be written in the traveling-wave form $f(x-vt,0)$.; The physical system is a string with tension $F_T$ and linear density $\mu$.

By comparing the mathematical properties of the traveling-wave solution with the physical wave equation, we can derive the wave speed. The traveling wave $f(x,t)=f(x-vt,0)$ satisfies $\frac{\partial^2 f}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 f}{\partial t^2}$.

Conditions: The wave can be written in the traveling-wave form $f(x-vt,0)$.; The physical system is a string with tension $F_T$ and linear density $\mu$.