Find the displacement  of an infinite string using Fourier transform method given that the string is initially at rest and the initial displacement is , .

To find the displacement \( u(x, t) \) of an infinite string using the Fourier transform method, we start with the wave equation for the string:

\[ \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} \]

where \( c \) is the speed of the wave on the string.

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