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The traveling-wave form $f(x,t)=f(x-vt,0)$ implies the wave equation because applying the chain rule to compute its second-order partial derivatives reveals a linear relationship between the spatial and temporal derivatives. Differentiating with respect to $x$ yields $\frac{\partial^2 f}{\partial x^2} = \frac{\partial^2 f}{\partial(x-vt)^2}$, while differentiating with respect to $t$ introduces a factor of $-v$ twice, yielding $\frac{\partial^2 f}{\partial t^2} = v^2 \frac{\partial^2 f}{\partial(x-vt)^2}$.

Conditions: The function $f$ is twice differentiable.; The wave propagates at a constant speed $v$ in the positive x-direction.

The traveling-wave form $f(x,t)=f(x-vt,0)$ implies the wave equation because applying the chain rule to compute its second-order partial derivatives reveals a linear relationship between the spatial and temporal derivatives. Differentiating with respect to $x$ yields $\frac{\partial^2 f}{\partial x^2} = \frac{\partial^2 f}{\partial(x-vt)^2}$, while differentiating with respect to $t$ introduces a factor of $-v$ twice, yielding $\frac{\partial^2 f}{\partial t^2} = v^2 \frac{\partial^2 f}{\partial(x-vt)^2}$.

Conditions: The function $f$ is twice differentiable.; The wave propagates at a constant speed $v$ in the positive x-direction.