# Levitation of frogs and pyramids?

Where mystery exists, often strange theories emerge to fill the void. Levitation is regularly cited as a skill the ancients possessed, somehow lost in the mists of time. They apparently used this skill to effortlessly lift large objects. Various people have even suggested that the huge stones that form the Egyptian pyramids were positioned using some form of levitation.

Some have attempted to explain these mysterious phenomena as an acoustic effect.  In his essay “Anti-gravity and the World Grid” Bruce Cathie recounts the story of a scientist, Dr Jung, who travelled to Tibet in 1939 where he was shown levitation by Buddhist Monks. By all accounts a stone was first placed on a bowl-shaped stone. Then, after much trumpeting, drumming and chanting the stone was seen to fly 500 metres along and 250 metres up onto a platform. Cathie has found various similar bowl-shaped stones in Egypt and so supposes that the Tibetans learned this skill from the ancient Egyptians. This and other similar stories seem to have cemented the idea that the ancients used acoustic levitation in ways that have now been forgotten. The recent emergence of acoustic levitation as a practically useful tool for positioning small objects, only seems to have fuelled these ideas.

The physics of acoustic levitation are now well known, so it seems like a good idea to use them to explore the possibility of levitation of more massive objects. The force on a flat reflecting object due to an incident sound wave is given by

$F_{A}=A(\frac{1+\gamma}{2})(\frac{p^{2}}{\rho c^{2}})$

Where the most interesting variables are, A, which is the area of the object (in m2) facing the incoming wave and, P, is the acoustic pressure amplitude (in Pascals). The other variables, γ, ρ and c are physical constants (γ =1.4=ratio of heat capacities for air, ρ =1.2kg/m3=density of the air and c=330m/s=speed of sound of air). If we are to levitate an object then we must overcome gravity and so the sound needs to supply a force greater than

$F_{G}=mg$

Where, m, is the mass of the object and g, is the acceleration due to gravity. Now we have the key equations for force, we can combine them and rearrange for the acoustic pressure

$P=\sqrt{\frac{m}{A}\times\frac{2g\rho c^{2}}{1+\gamma}}$

So, with this equation, we can work out how loud a sound needs to be to levitate an object of a certain mass and size. Usually loudness is expressed on a Decibel scale, relative to an agreed reference standard (20×10-6Pa), where it is called Sound Pressure Level (SPL). This is just

$SPL=20\log_{10}{\left(\frac{P}{20\times 10^{-6}}\right)}$

The power associated with this wave (in Watts) can then be found from

$W=A\left(\frac{P^{2}}{\rho c}\right)$

That’s the final piece in the puzzle. We can now plot the two most important things we need to know about the sound: how loud it is and how much energy is associated with the wave. The graph below shows the values we get if we choose a selection of objects and plug their mass and area into the equations.

But what do these numbers mean? In terms of loudness, the loudest ever rock concert was measured at 139dB, which is not enough even to levitate a small expanded polystyrene ball.  In terms of unamplified musical instruments, drums and cymbals can reach similar levels, when heard at very close range. Again, not enough to levitate a small polystyrene ball. Interestingly the threshold of pain ranges between 120dB and 140dB, meaning that sounds louder than this will not just be obnoxiously loud, but they will result in physical pain.  The loudest ever sound from a loudspeaker was reported to be from a Wyle WAS3000 which was measured at a colossal 165dB.  We can see from the graph, that had the inventors of this loudspeaker desired, they could have levitated frogs. You might have already heard of frog levitation, which won Andrei Geim and Michael Berry the Ig Nobel prize in 2000. However, they used a very powerful magnetic field to induce diamagnetic levitation. Going beyond the loudest loudspeaker, sounds can still exist, but the sources of these loud sounds are either explosions or other extreme natural events such as volcanic eruptions.  Indeed, the explosion of Krakatoa in 1883, which was heard clearly 5000 km away, has been estimated as the loudest sound ever. Amazingly, over the coming days, this explosive eruption event was monitored on sensitive pressure measuring equipment as it travelled around the world multiple times. But even this sound would not have been quite enough to levitate one of the 2.5 tonne stones that were used to make the great pyramid in Giza.

The power needed to achieve pyramid stone levitation is worth thinking about. And, in doing so, we need to remember that loudspeakers are not that efficient. For example, a well-designed loudspeaker is only 5% efficient, meaning that we would need 160 MW of input electrical power to get our required 8 MW of acoustic power output. This would be enough to power 160,000 homes in the UK and require a small power station all of its own. So, what does all this mean for the possibility of the ancients using acoustic levitation to build the pyramids? Well, there is no conceivable way they could have achieved the necessary sound levels with any combination of chanting, musical instruments and bowl shaped stones.

One way to perform silent acoustic levitation is by using ultrasound, waves so high in frequency (or pitch) that we, as humans, can’t hear them. This may seem like a cunning way around the deafening loudness required for levitation. But, very loud ultrasound is known to be dangerous in other ways. Exposure to 160dB ultrasound for a prolonged period is thought to result in death. The cause of death being by a heating effect.  And that’s the problem, as ultrasound is absorbed more strongly by air than audible frequencies, so a strong heating effect goes hand-in-hand with very high power ultrasound. There are other difficulties as, similar to the situation for audio speakers, the maximum ultrasound levels ever achieved in air were around 165dB. To get to the loudness levels needed to levitate large objects, new technology would be needed. And, the use of ultrasound does nothing to help the power problem. So, the use of ultrasound does not really help. Ultrasound might be quieter, but the heating effects are much more violent than for audible sound, so the threat to the health of those near the loudspeakers remains. In any case, levitation of large objects needs new ultrasonic loudspeaker technology to make it possible.

But, if you want to build your own Giza-style pyramid with acoustic levitation today, the physics suggest it is possible. You first need access to a small power station and then to develop a new loudspeaker technology that can achieve the required loudness. Then you just need to clear the surrounding region as the sound levels near the speaker would probably result in death.