# Why is transfer of heat very slow as compared to transfer of sound in solids?

If I heat a one-meter long iron rod at one end, the heat takes several seconds to reach the other end, while sound takes fractions of a second. Why is it so?

• It should be noted that velocity of sound in a given material is constant but velocity of transfer of heat depends on temperature difference. I guess you could make heat move as fast as sound by applying a large enough difference of temperature - although the iron rod might be turning into plasma while transmitting heat.
– Pere
Commented Aug 4 at 12:02
• @Pere, no speed of heat transfer doesn't depend on temperature difference, magnitude does. Heating the rod more will increase how quickly the rest of the rod increases by some fixed amount of temperature, but won't affect how quickly it reaches equilibrium (absent effects that change its conductivity constant). Commented Aug 4 at 23:13
• @Acccumulation - Point taken, but "the heat takes several seconds to reach the other end" is ambiguous enough to understand it as time taken to raise the temperature in the other end by a fixed amount, or by a noticeable amount, just as measuring sound arrival is not about equilibrium.
– Pere
Commented Aug 5 at 15:50
• Heat transport in solids is described by the heat transport equation, which shows that temperature change does not transport as a wave, but as expanding diffusing cloud. This means that actually, the change of temperature reaches the other end very quickly; the heat equation model says infinitely fast, but in reality, the actual speed has a limit close to the speed of sound. Commented Aug 6 at 15:15
• However, temperature change at the other end, after such a short time, is very weak, and hard to detect. It takes much more time for the temperature change at the other end to build up to a value that can be reliably detected by a thermometer. So the observed speed of "signal" made of temperature variation depends on how accurately we can measure the temperature at the other end, and can be up to the speed of sound. It is the rate of heat transfer that is slow, but this is not due to slower speed of some wave, it is because heat does not move like a wave. Commented Aug 6 at 15:19

Here is why.

In the case of sound propagation, the vibrations of the molecules in the bulk are pointed in the same direction, and hence propagate as a wave at the sonic velocity in the solid.

Heat transfer is instead diffusive- the vibrations of all the molecules in the bulk are in random directions and so do not propagate through the solid as a coherent wave. Diffusive transport is inherently slow in solids- far slower than the sonic velocity.

• I think this is just restating the question in different terms -- why is diffusive transport so much slower than the sonic velocity in solids? Commented Aug 5 at 6:54
• The answer points out that the direction of motion is linear/directed in one case, and random in the other. This basic geometric observation seems to explain the question very well to me. If you take a large crowd of people, and start pushing in a common direction from one end, the whole crowd will move relatively quickly in that direction. If the people at the beginning mill around randomly, it will take much longer for this movement to translate through the whole crowd. @IanH
– AnoE
Commented Aug 5 at 12:02
• This really doesn't make sense when you consider that there is a cosine component of any individual atom motion vector, as the dot product with any given unit direction vector, that is usually nonzero. I suspect that the true answer is that most motions of individual atoms engaged in Brownian motion on average cancel out, so there is not much to propagate, except at the edges of the "heat wave", where there is a heat differential. Commented Aug 7 at 7:52

If you hit one end of a rod, you push iron atoms at the end. The bonds between atoms are like stiff springs. Pushing an atom compresses the spring and pushes on the next atom. This continues through the layers. Because the springs are very stiff, there is little time delay between movement of one layer and the next. The speed of sound is high.

After the sound pulse passes, the atoms return to their starting position with little change to their random thermal motion. This is much like throwing a rock in a pond. As the wave passes, water swirls with increased energy. But the energy travels with the wave. As the wave passes, atoms return to normal.

Atoms vibrate in a random motion around an equilibrium position. One atom shakes its neighbors, which shake their neighbors. An atom can push, pull, or yank sideways on its neighbors. At any given instant, some atoms have more energy, others less. An atoms with above average energy will most likely bump into neighbors with less. Most likely the above average atoms will lose energy and the neighbors will gain. At equilibrium, all atoms vibrate with equal energy on the average. The temperature is the same everywhere.

Interactions between an atom and its neighbors occur at the speed of sound. But this doesn't mean that energy from an above average atom will propagate across the rod in the direction the atom happens to push. The energy added to an atom depends on how all its neighbors happen to push or pull on it. They may add energy to a high energy atom. Typically the excess energy of a high energy atoms tends to spread among its neighbors. If one of those neighbors winds up with high energy, it typically will share with all its neighbors. The path of energy in the rod is a random walk.

At equilibrium, energy walks in all directions equally. As much goes left as right. On average, it doesn't go anywhere.

If you put one end of the rod in contact with a hot object, atoms at the end are in contact with atoms that vibrate with more energy. These have more energy to share than their colder neighbors. More energy randomly walks toward colder atoms than randomly walks back. The result is that heat propagates up the rod at the speed of a random walk.

• Why is the connections between atoms "stiff" when you push the rod and "loose" when the atoms wiggle randomly by heat? Commented Aug 6 at 8:16
• @Falco - It is stiff in both cases. When one atom pushes on another, the "spring" doesn't deform much and there is little time delay before the other atom moves. It makes the frequency of vibration high. Commented Aug 6 at 13:16
• Thank you for the reply. Maybe I can illustrate it differently: When I push the rod the Atoms to the left immediately move because the atoms on the right are pushed. If the atoms on the right move though heat, the atoms on the left do not move directly - there seems to be some wiggle room, the atoms can move without pushing the neighboring atoms. Why is this wiggle-room not present when one pushes the rod? Commented Aug 6 at 13:41
• @Falco - Atoms are connected together with atomic bonds. This is much like connecting them with springs. This allows atoms to jiggle. Some of the jiggling is sound. Some is heat. When you push on one end, there is some jiggling involved to get the far end moving. See Is it correct to say that it is theoretically impossible for perfect rigid bodies to exist? Commented Aug 6 at 14:25

The Core Explanation is that Sound Energy is not stored , while Heat Energy is stored.

When we make Current flow though a wire , the electrons generally do not get stored along the wire , hence the flow is comparable to speed of light.

When we make Light (or Heat) go through vacuum , the Photons (or Electro-Magnetic Waves) do not get stored in that vacuum , hence the flow is very quick.

When Heat Energy flows though a metal rod , that Energy will get stored all along that rod. Hence the flow is quite slow.
Each tiny volume will get gently heated up & then gradually pass the heat along.
Very slow !

When Sound Energy flows though a metal rod , that Energy/vibration will not get stored all along that rod. Hence the flow is quite fast.
Each tiny volume will quickly get the Energy/vibration & then quickly pass that along.
Very fast !

Similar Phenomena occur where "passing without storing" is faster while "transporting while storing" is slower.

• See my answer - based on yours. Commented Aug 5 at 7:26
• Tuning fork has entered the chat... Commented Aug 6 at 2:23
• "When we make Current flow though a wire , the electrons generally do not get stored along the wire" Real wires have inductance and capacitance, which do store electrons, or alternatively, the energy in fields. This is why the velocity of propagation is, for a typical cable, only around 1/2 the speed of light. Electromagnetic fields certainly exist in a vacuum, the impedance of freespace is finite, and the speed of light is finite. Commented Aug 6 at 7:30

Heat and sound are both transferred via vibrations in a material, and both are transferred at the speed of sound. The mistake you are making is that you are comparing the speed of sound (in m/s) with the rate at which one end of a rod heats up (°C/s). The units are different: what heating rate (°C/s) do you need to be "more" than a given velocity (m/s)? These are not directly comparable.

If you're asking why the rod doesn't reach thermal equilibrium as fast as the speed of sound, then it is because the vibrations that carry heat are scattered from defects in the material. So, the effective distance these vibrations need to cover to reach the other end is much, much greater than the length of the rod. If it weren't for this, you would have near infinite thermal conductivity.

In addition to this, heat will also flow back the other way as the cool end will also have thermal vibrations: this is why the heating rate (in °C/s) is related to the temperature difference between the hot and cold ends of the rod.

Sound, on the other hand, is carried primarily by long-wavelength vibrations which scatter less. These also exist in heat transport, but their low frequencies mean they carry much less energy so contribute relatively little to the heating of the other end.

All of this ignores contributions from electrons, which are important in metals. But even there the basic idea is the same.

This adds (hopefully) to Prem's answer, which is a clear and good explanation.

When "heat" is applied to a thermally conducting rod (high or low or other) heat energy "enters the rod and causes the atoms to be thermally excited. This takes an inflow of energy, which is stored as molecular vibration, and temperature rise and spread is dependant on the thermal mass and thermal conductivity of the material. Heat energy is stored and will transfer along the rod at a rate and causing a temperature rise both dependant on the above mentioned properties. Heat energy can leave the sides of the rod by convection and radiation. These in turn depend (simplistically) in the case of convection on boundary layer effect, surface properties and the surrounding fluid properties, and in the case of radiation on the 4th power of temperature and surface emissivity. Take all these effects together (and some others) and you get a temperature spreading model. If ay of these models "just happened" to be different (eg radiation changing with 5th or third powers of temperature then we would be dealing with "other Physics" and the world would work differently than it does now.

Sound transfer also depends on models and parameters - but they are not the same mechanisms as for temperature. In the case of sound transfer the primary effect is mechanical compression causing a physical wave to be transferred to adjacent material. Just as boundary layer effects to not (primarily) affect radiation and we do not expect convection to follow fourth power temperature effects, so too the physics of what happens is relevant to the mechanisms involved. There is energy storage, but because of the tensile properties, modulus of elasticity, density and much more and of resonant effects (which can be extremely significant*) based on the bars topography the "sound transmission" the stored energy tends to mainly transfer in a compression wave, with mechanical laws are quite different from the thermal transmission laws. ie there are many physical laws at work in a body and we see some of them manifested under some forms of excitation, and another (possibly overlapping) set with another excitation. Here we do not see the bar melt or vapourise or spin rapidly, as it might with other or more extreme excitations.

*Physical laws can lead to interesting results.
I have a small sheet of metal with some holes cut in it - maybe a 400 x 250 mm rectangular plate maybe 3 or 4 mm thick with a few randomly spaced 25 to 50mm holes - the product of some long past unknown process. When thrown on a hard surface this is the LOUDEST object I have ever met. It emits a "clang" so loud that it is very painful to my aging ears and those with excellent hearing will often exclaim at the pain and decry my stupidity in demonstrating my amazing toy to them. I do not know what properties make it such a superb sounder. Nothing else comes close. It's just that the various "laws" at work have been optimised to make a noise maker.

• Your toy might be useful for riot crowd dispersal, instead of fire hoses, rubber bullet and tear gas. Together with the sound of scraping nails over a blackboard and microphone interference squeal over a loudspeaker :-P Hey kids! We are going to see "mad " uncle Russel today. Better take some ear plugs!
– KDP
Commented Aug 6 at 1:08
• @Kdp Mad Uncle Russell? :-) . It's almost as if you know me :-). bit.ly/oldgreyguy || 35 countries, 73.7 years (so far), lots of fun. || 1.5 litre Pepsi bottles at about 180 psi are MUCH louder. Commented Aug 6 at 8:38

# Statistics

If you apply a large force to one end of your rod, how many different ways can the atoms in the rod respond to the force? Since the force is applied nearly uniformly, the atoms can basically only respond in the direction that the force was applied. In particular, the number of microstates corresponding to force propagation is relatively small, so no matter which one of them occur, you will see the force travelling down the rod.

If you instead apply a large heat source to one end of your rod, how many different ways can the atoms in the rod respond to the heat? Well, the answer is: a ridiculous number. The atoms can start bouncing around willy-nilly in every direction, over a large range of velocities. Some of those velocities will add up to a net migration of heat towards the cold end, while others will add up to no movement, and yet more will add up to some heat moving in the "wrong" direction. The phase space for the heated rod is ridiculously larger than for the struck rod. A large number of states will correspond to time evolution of the atoms bouncing around but not actually propagating the heat. And so when you average over all possible states, the thermal conductivity mostly corresponds to the number of states in which the heat actually migrates to the colder end of the rod. With random velocity vectors, there are many ways for the rod to respond to the heat source without actually propagating the heat, unlike the strike, which only admits a smaller number of states consistent with a unified force vector over all the atoms.

# Entropy

Another way to view it is that striking the rod is an extremely low-entropy energy input, while heating it is a very high-entropy input. Getting any behavioral pattern to quickly travel down the rod is a low-entropy state change. Therefore, you must inject a large amount of low-entropy energy to make such a change occur. This is why you can't use a blowtorch to make your rod into an interesting musical instrument. Fire is a high-entropy energy source because most of the energy is disordered.

Now, the reason we can use fire to move vehicles down a road at decent speed is because we highly constrain the dimensions of the fire, and the walls of a cylinder/turbine impose an order on the energy flow that effectively reduces the entropy of combustion, turning it into a unified semi-linear movement (which then gets turned into circular movement, which then gets turned back into linear movement). Of course, we pay a large cost for eliminating this entropy, which is seen in the high temperature of the exhaust. We can recover that lost energy in something like a Stirling engine, but then we give up a lot of speed and power to do so. There's no free lunch, after all. Also note that most vehicles don't move anywhere close to the speed of sound of air, let alone concrete, asphalt or iron. And vehicles that do travel at the speed of sound in air do so at enormous energy expenditures.

You are perfectly right that both heat transfer and sound transfer happen via vibrations of atoms/molecules which bump into each other via the electromagnetical force. However, there is a core difference: Sound waves have macroscopic wavelengths, while heat vibrations have much, much higher frequencies. So much so that it does not even make sense to talk about wavelengths anymore when talking about thermal vibrations. The consequence is, that atoms/molecules scatter thermal phonons (quasi-particle that transmits vibrational energy), but not sound waves.

What's more, the high frequencies of thermal vibrations put so much energy into single phonons, that thermal phonons can be absorbed/emitted by molecules. That allows molecules to change phonon direction pretty much randomly!

So, while a slab of uniform material is fully transparent to sound waves, it behaves more like a cloud to light for heat transfer: The energy bounces around wildly without any aim, and only moves through the medium in a statistical fashion. Maybe, after zillions of collisions, does some energy make it through to the other side, but its path is just a long random walk in a dark forrest.

Sound propagates as a coherent mechanical wave. If $$x$$ is the direction of propagation of the wave, all the atoms/molecules with the same $$x$$ coordinate will oscillate in the same manner, transferring momentum to their neighbors. In the case of a solid medium, you can approximate the material as a collection of beads (the atoms/molecules) connected by springs (the atomic/molecular cohesion forces). The stiffer the springs, the faster the propagation of mechanical energy. This is why sounds propagates faster in steel than in water or air, where the cohesion forces are much weaker.

Heat, on the other hand, is "disordered mechanical energy". When you heat up a region of a material, its atoms/molecules will start to move faster and to collide more and more with each other. Also in this case, the atoms/molecules will transfer mechanical energy to their neighbors via the interatomic/intermolecular forces. The stronger these forces, the faster the energy transfer: as sound, heat propagates faster in steel than in water or air. However, since the motion of the atoms/molecules is random, sometimes the atoms/molecules will move in one direction, sometimes in the opposite one. This incoherent motion results in a much slower energy transfer than the one achieved in sound propagation (see here for a cartoon representation of heat transfer).

I should specify that the heat transfer mechanism I described is referred to as conduction. For simplicity, I have not discussed heat transfer by convection or radiation.

Heat conduction is an anharmonic effect. If all the atoms were perfect harmonic oscillators, then there would be no heat conduction at all. In this sense, heat conduction should be slow in the first place as atoms can be well approximated as harmonic oscillators.

On the other hand, sound waves can be well characterized by a harmonic oscillator model.

The frequency related explanation seems to me also adequate, but there is also dispersion to consider. Heat will disperse outside the medium, sound will travel mainly along the faster medium. Group velocity for sound does not change much, it does for red frequencies. Just another angle.