Using the temperature distribution of Problem 4-18, show that the time lag between maximum points in the temperature wave at the surface and at a depth x is given by

Problem 4-18
When a sine-wave temperature distribution is impressed on the surface of a semi-infinite solid, the temperature distribution in the solid is given by

where
Tx,Ï„ = temperature at depth x and time Ï„ after start of temperature wave at surface
Tm = mean surface temperature
n = frequency of wave, cycles per unit time
A = amplitude of temperature wave at surface
If a sine-wave temperature distribution is impressed on the surface of a large slab of concrete such that the temperature varies from 35 to 90ºC and a complete cycle is accomplished in 15 min, find the heat flow through a plane 5 cm from the surface 2 h after the start of the initial wave.

At=-