Associated with the annual variation of the weather, the...

Associated with the annual variation of the weather, the average ground surface temperature at a given location will have a maximum in summer and a minimum in winter. If you dig a hole exactly six months after the maximum occurs, how deep will you have to dig to reach the peak of the resulting temperature response, and what will be the percent decay of the amplitude? Take $\alpha=1.2 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}$ for the soil diffusivity.

Key Concepts

Phase Lag in Diffusive Systems

When a periodic signal is applied at the boundary of a diffusive medium, the temperature response at a given depth exhibits a phase delay relative to the boundary condition. This phase lag increases with depth, meaning that the maximum (or minimum) in the temperature response occurs later in time as one goes deeper.

Periodic Forcing in Thermal Diffusion

In many natural systems, such as the ground responding to changes in ambient weather, temperature at the surface varies periodically (e.g., with the seasons). The concept involves applying a periodic boundary condition to the heat equation, which results in a temperature response in the medium that both oscillates in time and decays with depth.

Exponential Attenuation of Amplitude

As heat penetrates a medium under periodic forcing, the amplitude of temperature oscillations decays exponentially with depth. This exponential decay is a hallmark of diffusive processes under periodic excitation, whereby the further from the source, the weaker the temperature fluctuations become.

Heat Conduction Equation

This is the fundamental partial differential equation that governs how heat diffuses through a medium over time. It relates the change in temperature with respect to time to the spatial temperature gradient, and it forms the basis for analyzing how temperature variations propagate into materials like soil.

Thermal Diffusivity

Thermal diffusivity is a material property that indicates how quickly heat spreads out through a medium. It essentially combines the effects of thermal conductivity, density, and specific heat capacity, determining the rate at which temperature disturbances are smoothed out within the material.

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