Discussion
Consider the one-dimensional wave equation \(\frac{\partial u}{\partial t} + \frac{\partial u}{\partial x} = 0\) for \(-\infty \lt x \lt \infty\), \(t \ge 0\).
For an initial condition \(u(x,0) = e^{-x^{2}}\), the solution at \(t = 1\) is
Answer: A
The solution at \(t = 1\) is \(u(x,1) = e^{-(x-1)^{2}}\).
The equation \(u_{t} + u_{x} = 0\) is the linear advection (one-way wave) equation with unit speed. Along the characteristics \(x - t = \text{constant}\), \(u\) does not change, so the general solution is \(u(x,t) = f(x - t)\): the initial profile moves to the right at speed 1 without changing shape.
\[ \begin{aligned} u(x,0) &= f(x) = e^{-x^{2}} \\ u(x,t) &= e^{-(x-t)^{2}} \\ u(x,1) &= e^{-(x-1)^{2}} \end{aligned} \]
The option with \((x+1)\) moves the profile left, which would be the solution of \(u_{t} - u_{x} = 0\). \(e^{-x^{2}}\) leaves the profile where it was, and \(e^{-1}\) is a constant, which is not a translated Gaussian at all.
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