 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2019/paper-1/2/c/solution

The slowest surviving mode has

n = 1, m = 0

. Hence

T (x, y, t) \sim \frac{4 a ^{2}}{π ^{3}} sin \frac{π x}{a} e^{- λ π^{2} t / a^{2}} (t \to \infty) .

It is independent of

y

: insulation allows the constant transverse mode, while every nonconstant transverse mode decays faster. The next possible relative decay rates are

8 λ π^{2} / a^{2}

from

n = 3, m = 0

and

16 λ π^{2} / b^{2}

from

n = 1, m = 4

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