1. ENPh257: Thermodynamics 00: Thermal waves
© Chris Waltham, UBC Physics & Astronomy, 2018

2. Thermal (diffusive) waves
The partial differential equation 𝜕𝑇 𝜕𝑡 = 𝑘 𝑐𝜌 𝜕 2 𝑇 𝜕 𝑥 2 has solutions that are damped, diffusive, dispersive waves. Part of the lab task is to look for these waves. The steady-state form of the wave solutions is: 𝑇 𝑥,𝑡 = 𝑇 0 exp − 𝑘 𝜔 𝑥 cos 𝜔𝑡− 𝑘 𝜔 𝑥 + 𝑇 Here, the frequency of power modulation (𝜔/2𝜋) is chosen by you, 𝑇 0 is determined by the power amplitude, and 𝑇 is the mean temperature of the rod. The wave number 𝑘 𝜔 = 𝑐𝜌𝜔 2𝑘 and the wave (phase-) velocity is 𝑣=𝜔/𝑘 𝜔 = 2𝑘𝜔 𝑐𝜌 (not constant). The damping distance 1/𝑘 𝜔 , which for brass, is 3 mm for 1 Hz, 3 cm for 0.01 Hz etc. The wavelength 2𝜋/𝑘 𝜔 , which for brass, is 19 mm for 1 Hz, 19 cm for 0.01 Hz etc. Because the wavelength and damping distance are linked, observation is a trade-off between the two.
© Chris Waltham, UBC Physics & Astronomy, 2018

3. The math
This may not make sense until you have had more math courses. Postulate a wave solution to the thermal diffusion equation: 𝑇 𝑥,𝑡 = 𝑇 0 exp {𝑗 𝜔𝑡− 𝑘 𝜔 𝑥 } + 𝑇 Test in the diffusion equation: 𝑗𝜔= 𝑘 𝑐𝜌 𝑘 𝜔 2 𝑗 = ±(1+𝑗)/ 2 , so 𝑘 𝜔 has an oscillatory part and a exponential decay part (+ is non-physical).
On a sunny day soils are hottest at noon on the surface, and hottest at midnight ~ 30 cm down.
How deep would you find the soils warmest in mid-winter?
© Chris Waltham, UBC Physics & Astronomy, 2018

4. The math
On a sunny day soils are hottest at noon on the surface, and hottest at midnight ~ 30 cm down.
How deep would you find the soils warmest in mid-winter?
𝑗𝜔= 𝑘 𝑐𝜌 𝑘 𝜔 2 This is straightforward scaling problem: Frequency is the inverse of the period. Wavenumber is proportional to the inverse of the any characteristic length (e.g. wavelength). Multiply the period by 365, multiply all lengths by Answer: 5.7 m Temperature benefit will be the same as for diurnal changes; the x- axis of the graph on the right is just stretched, i.e. a few % of the surface amplitude.
© Chris Waltham, UBC Physics & Astronomy, 2018