 Solution

ID: past-exam-of-the-waves-and-compressible-flow-course-of-the-university-of-oxford/2023/2/a/ii/solution

For

w = e^{α z /2} W

w_{zz} - α w_{z} = e^{α z /2} (W_{zz} - \frac{α ^{2}}{4} W), w_{xx} = e^{α z /2} W_{xx} .

\partial_{t}^{2} (\nabla^{2} W - \frac{α ^{2}}{4} W) + αg W_{xx} = 0, W (x, 0, t) = W (x, h, t) = 0 .

The positive exponential factor does not change where the vertical velocity vanishes.

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