Suppose a hot slab of thickness 2a and initial uniform temperature 1 is suddenly immersed into water

Question:

Suppose a hot slab of thickness 2a and initial uniform temperature θ1 is suddenly immersed into water of temperature θa < θ1, thereby reducing the temperature at the surface of the slab abruptly θ0 and keeping it there. Expand the temperature in the slab in a Fourier series. After some time all but the longest wavelength Fourier component of the temperature will have decayed1 and then the temperature becomes sinusoidal. After what time will the temperature difference between the center of the slab and its surface decay to 0.01 of the initial difference θ1 – θ0?

Distribution
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