D'Alembert solution for the wave equation: question about the speeds

[jk22]

The solution for the wave equation with initial conditions $$u(x,0)=f(x)$$ and $$u_t(x,0)=g(x)$$

Is given for example on wikipedia : $$u(x,t)=(f(x+ct)+f(x-ct)+1/c*\int_{x-ct}^{x+ct}g(s)ds)/2$$

So a vibrating string, since there is no conditions on $g$ (like $\sqrt{1-g(x)^2/c^2}$), could move transversally at speed bigger than c, while the movement along the rope is at speed c ?
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[jasonRF]

Perhaps an analysis of the simple wave equation $\partial^2 f/\partial t^2 = c^2 \partial^2 f/\partial x^2$ indicates that, but you need to pay attention to the assumptions made when deriving the equation for waves on a string. The book "a first course in partial differential equations" by Weinberger begins with a detailed derivation, and one of the assumptions made is that the transverse speed is small compared to $c$.

Also, there are conditions on $g$. For one thing, the displacement from equilibrium must be very small for that simple linear PDE to be useful approximation.