# At what speed does heat flow in materials?

When we heat one end of an iron rod on the stove, it takes a considerable time for the other end to be heated to the same temperature as the first end. On the other hand, as we know, the atomic vibrations propagate with the speed of sound through materials considering whether the wave is longitudinal or transversal.

However, it seems to me that heat is transferred with a speed much smaller than the speed of sound of any kind in a specific material like iron. Why is this so?

If there are different waves with different speeds in materials other than sound waves, what are their mechanisms of propagation?

• en.wikipedia.org/wiki/Thermal_conduction Feb 7 '20 at 20:55
• The main difference is that sound waves are directed. But random thermal motions are just that. Random. So the vibration spreads more like diffusion. Feb 7 '20 at 21:35
• @FellowTraveller You mean that randomness can decrease the speed of propagation in a specific direction in material? Feb 7 '20 at 21:42
• No. I mean to say that the movement of the higher kinetic energy particle is not in any specific direction. It spreads evenly. Feb 7 '20 at 22:02

Thermal vibrations are random in position and time within a hot chunk of material. This means that the laws of thermal diffusion apply and not those of sound transmission, where the particle motions possess a common direction of travel.

Thermal diffusion processes in solids have a propagation speed of order ~tens of centimeters per hour whereas the sound propagation speed in the same solid might be of order ~thousands of meters per second.

Thermal vibrations move at the speed of sound, but since they are move randomly, bouncing from atoms and dislocations, they spread out according to the (heat)diffusion equation: $$\frac{\partial T}{\partial t} = K\frac{\partial ^2T}{\partial x^2}$$ where $$K=k/\rho C$$ is the ratio of heat conductivity to the density and specific heat capacity. This does not have constant velocity solutions.

Borrowing from Carslaw and Jaeger 1959, the solution for an infinitely long rod initially at zero temperature with $$x=0$$ held at temperature $$V_0$$ for $$t>0$$ is: $$T(x,t) = V_0 \mathrm{erfc}\left(\frac{x}{2\sqrt{K t}}\right).$$ Here is a plot of some solutions for $$V_0=1$$: Note how the point where the temperature is 1/2 (or any other level) at first moves fast to the right, and then slows down. This is because diffusion tends to move things a distance growing as $$\sqrt{Kt}$$ rather than at a linear speed.

An interesting special case is when there is a solid where the surface is heated by a periodic temperature $$T(t)=\sin(\omega t)$$. Then the temperature in the solid is a damped oscillation where the waves move with velocity $$\sqrt{2K\omega}$$. In a sense you can get very fast heat signals by having a large $$\omega$$, but the damping increases exponentially with the frequency so in practice rapid heat waves do not penetrate deeply into the solid.

In summary, heat moves with variable speed.

• It might be worth mentioning that, according to this (correct) solution, the temperature at the far end of the rod does change a slight amount almost immediately. Feb 8 '20 at 13:04