|MadSci Network: Physics|
What a great question. Clearly you have been spending a lot of time laying in bed thinking when you could have been sleeping. There are several assumptions in your question that we don't have any data to support either way. The first assumption is that a person traveling through time would appear instantenously. Perhaps the time traveler would just fade in gradually, one molecule at a time. But, let's go with your assumption for the sake of discussion. As an object moves into a space previously occupied by air, if it moves into that space gradually, the air molecules have time to shift and move out of the way. As the object moves into a space more quickly, the air molecules that are displaced bunch up, creating a localized higher pressure area. Whenever a material, such as a loudspeaker cone, vibrates, it alternately creates localized high and low pressure waves as the air tries to move along with the speaker cone. Air molecules near the solid object try and communicate to molecules farther away that they need to move out of the way. The information between adjacient air molecules is transmitted at the speed of sound, roughly 700 miles per hour. Let's think of the air molecules as if they were a row of dominos. When the first one tips over, it knocks the next one over, and so forth until the dominos have all fallen down. The dominos fall at a characteristic rate, determined by their size and the effects of gravity. When an object moves fast enough, air molecules are not able to transmit information to other air molecules fast enough to let them know to move out of the way. Let's say that instead of tipping over the first domino, a train comes plowing through the row. It smashes into the first domino. The first domino doesn't have time to tip over before the train smashes into the second domino, followed by the third. Pretty soon you have a row of dominos smashed up against the front of the train. A fast moving object in air will build up a pile of air molecules on the front of the object. That pile of air molecules is essentially a region of high pressure that we call a sonic boom. The number of air molecules involved in a particular sonic boom will depend upon the size of the object and the speed it is traveling. An F-16 fighter jet traveling at twice the speed of sound will not produce as large a sonic boom as the space shuttle traveling at 25 times the speed of sound. Likewise an F-16 traveling at just above the speed of sound should not creat as loud a sonic boom as one traveling two times the speed of sound since less air is displaced per unit of time. The sound from an explosion is also largely a sonic boom. In a chemical explosion a solid occupying a certain space is suddenly converted into a gas occupying essentially the same space. It is similar to having gas compressed at perhaps a million pounds per square inch and the vessel containing the gas suddenly disappears. The gas from the explosion is at a very high pressure, and as the gas expands, it produces a sonic boom in proportion to the velocity and size of the explosive. An explosive with a higher velocity will produce a higher pitched cracking sound, compared with the lower pitch from a lower velocity explosive. The size of the sonic boom, however, is still largely a function of the size of the explosion, rather than just the velocity of the explosive. So, going back to your question, let's say that an object the size of a person suddenly appears in a location previously occupied by an equal volume of air. As you suggest, the air will be suddenly displaced by 30 cm or so and should create a pretty good shock wave. However, let's consider the amount of air displaced. Say we send a 220 pound guy back through time. If this guy has a mass density of about 1 g/cc (same as water), his body occupies a volume of 100 liters or about 0.1 cubic meter. From http://wahiduddin.net/calc/ density_altitude.htm we find that the density of air is approximately 1.2250 kg/m3. The volume of air displaced by our time traveler will be approximately 0.1 cubic meter or about 0.1223 kg of air. We should be able to simulate the shock wave produced by displacing that amount of air by using an explosive that produces approximately the same amount of gas (0.1 cubic meter of gas). Since a chemical explosive essentially converts solid to gas, then a chemical explosive weighing 0.12 kg should produce a similar shock wave as produced by out time traveler. The explosion is probably similar to a good size hand grenade. So, even if our time traveler appears in an instant, the size of the shock wave he produces is still proportional to his size, rather than infinite in size. The time traveler, however, will have the outside of his body subjected to the other side of the shock wave, which should be rather unpleasant for him, unfortunately. It would almost certainly pop his ear drums and could leave him badly damaged. So, our time traveler may have something more to worry about than suddenly surprising granny and have her beat him with her umbrella. Great question. Of course something else to worry about is whether there are other types of shock waves generated by traveling through time. But that is beyond my expertise.
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