Discussion
Consider the following partial differential equation for \(u(x, y)\) with the constant \(c \gt 1\):
\(\frac{\partial u}{\partial y}+c\frac{\partial u}{\partial x}=0\)
Solution of this equation is
Answer: B
The solution is u(x, y) = f(x - cy).
Try \(u = f(\xi)\) with \(\xi = x - cy\). Then \(\partial u/\partial y = -cf'(\xi)\) and \(\partial u/\partial x = f'(\xi)\), so
\[ \begin{aligned} \frac{\partial u}{\partial y} + c\frac{\partial u}{\partial x} &= -cf' + cf' \\ &= 0 \end{aligned} \]
for any differentiable f. This is the one-way wave equation: u is constant along the lines x - cy = constant.
f(x + cy) gives \(cf' + cf' = 2cf'\), not zero. f(cx + y) gives \(f' + c^{2}f'\), and f(cx - y) gives \(-f' + c^{2}f'\), which vanishes only if c = 1, excluded because c is greater than 1.
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