Fourier's wine cellar Joseph Fourier (1768-1830) was an expert on heat conduction. To find the perfect depth to build his wine cellar, he solved the following problem on subsoil temperature variations:

(∂/∂t)u = α² (∂²/∂y²) u , 0 < y < [infinity],

, subject to the boundary conditions:

u(y, t) bounded as y → [infinity]
u(t) = uₒ e^(iωₒt)

Here ωₒ is the frequency of the temperature variation at the surface, ωₒ = 2π/(1 day) for daily variations and ωₒ = 2π/(1 year) for seasonal variations. u is the temperature of the soil. y is the depth underground, positive downward, with y = 0 being the surface. For soil, the thermal conductivity is α² = 0.01 cm²/s. We use the complex exponential instead of cos ωₒt because it makes the algebra easier. One can always take the real part of the solution afterwards.
(a) Solve the problem (i. e., find u(y, t)) by assuming that your solution can be written in the form:
u(y, t) = Φ(y)e^(iωₒt)

(b) For daily variations, uₒ = 5°C and ωₒ = 2π/(1 day). Find the depth below which the magnitude of temperature variation is less than 2°C.