Doppler Shifter, I Hardly Know Her!!!

Now I switch to a different venue for my thought experiment.  It will involve two automobiles traveling down a smooth straight level section of turnpike.  Each auto will set their cruise controls at a constant velocity, v, which they have agreed upon beforehand.  Each driver has fully operational digital clocks, annoyingly loud horns, and bright halogen lights on board.  It is dusk on a clear windless day.

As the convoy (a lead auto and a following auto) makes its way down the road, the pair are rolling in tandem with each other; neither accelerating, nor decelerating, relative to one another.  They can be regarded as at rest relative to one other.  However, the air is at rest relative to both autos as they are moving down the road (the air and road are both at rest relative to the autos).

So to check that they are a safe distance, L, apart (stopping distance at this speed) the driver of the following auto conjures up a test.  She makes a hands free cell phone call to the driver in the lead car.  When he answers she tells him her plan.

On a windless day they are rolling down the turnpike in tandem, thus they at rest relative to each other; but at the same time, they are in motion, at the same velocity, relative to the road and air.  She proposes to flash her headlights as a signal to her comrade‘s auto.  When he sees the light signal, he is to honk his horn.    At the same that she flashes her lights, she starts her digital clock.   Thusly, the time she measures, since the light signal is effectively instantaneous, will be for the horn sound to return to her:

The distance they are apart will not be L = ct, but rather L = ct + vt, where v is the velocity of the tandem relative to the road and still air (c is the velocity of sound).  They will begin at the distance L apart, then her auto and the sound pulse will meet somewhere within L by algebra.  This reflects the forward motion of her vehicle at the same time as the sound wave is traveling rearwards.  This accounts for all the variables and determines the distance they are apart.

The pair travels on further, after checking their safe distance.  Now, towards the end of their trip, they have reached the familiar environs near the exit for her town.  He speeds up to a new constant velocity, u, to make time without risk of losing her.  At this new velocity, he gradually pulls away from her.  He will travel on to the next town alone.  Then, she imagines a continuance for their little thought experiment.

As he is gradually separating, she realizes that the source (he) and the receiver (she) are no longer in tandem, now a Doppler effect appears because they are no longer traveling at the same speed.  An aspect of Doppler (sound waves through air) is that when the source approaches the receiver, there is a slight mathematical difference than when the receiver approaches the source (introducing a sort of wind in either reference frame).

So if she makes the measurement of the change in frequency from her friend’s auto horn while he gradually separates from her, then she will find the source sound wave to have apparently changed frequency.  She knows the frequency of the horn from when the autos are at rest relative to each other.

In other words she can determine the frequency of the sound of the auto horn while the autos are rolling in tandem (they are at rest relative to each other but moving at the same velocity relative to the air, so ff = f0 ). She can now discern whether she is in motion relative to the air, earth and his auto (which she thinks could be at rest); or whether the air (wind) , earth and his auto are in motion while she is at rest.

This can seen by a comparison of the two formulas that fit these two scenarios (the different frequencies can be used to solve for using their different velocities); if the frequency she measures is one value of f then she concludes that she is moving; if the frequency has some other value f, then she concludes she is at rest:

♦ f = [ c / (c + v_s)] f<sub>0

♦ f = [(c + vr ) / c] f0</sub>

These formulae are clearly different, just by appearance.  Therein, the nature of her motion is revealed.  Either she is stationary, with the medium in motion; or she is in motion, with the medium being stationary.  She has a mathematical means of determining this.

She can distinguish, by her thought experiment, whether the autos are in motion with the Earth stationary; or the autos are stationary, with the Earth in motion.  This is contrary to the principles of relativity which state that both situations are equivalent, or equally valid descriptions of her motion, thus they are interchangeable in a way.  But her thought experiment shows that this is unsound.

http://stargazerslounge.com/blog/1308-geryllax-vus-blog/

http://en.wikipedia.org/wiki/Doppler_effect

Galileo's Ship 2 ~ The Bellringer

In 1632, Galileo proposed his thought experiment, called “Galileo’s Ship“:

“…That is why I said you should be below decks; for if this took place above in the open air, which would not follow the course of the ship, more or less noticeable differences would be seen in some of the noted effects…”

This is the point which I wish to drive my shoulder into.  Focusing on the air as a medium for the transmission of sound waves, I want to transmutate these noticeable differences into mathematically measurable phenomena.

In Galileo’s Ship once again:

“…have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that, you discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still…”

It is this not being able to tell whether the ship is moving, or at rest, which is the critical point.  This is the foundation of Galilean invariance, or his principle of relativity.  It is reiterated by Newton, and appears in the axioms of Einstein; no experiment can be done to detect absolute motion.  The laws of physics are the same from the point of view from a reference frame, or within the reference frame.

However, it has already been observed by scientists that sound waves violate this invariance, or relativity. Sound waves seem to cross this wall of separation between two reference frames, one at rest and one in relative motion (at a constant velocity).  An observer within the ship cabin, shares the motion of the ship, along with the air molecules. This observer is at rest within the ship’s reference frame.  The laws of physics take on their simplest form.

An observer on the shore sees the ship observer and air molecules following the translatory motion of the ship as it travels through the water.  This shore observer factors this translation into a formula, but realizes that he cannot communicate any of this mathematical information to the ship observer.

If the cabin observer moved to the open air of the main deck, I think she would have a different set of experiences than she had had below decks.  If a sailor is set to ring a bell at the aft end of the ship, then the sound waves would travel to the fore end of the ship where the ship observer could be positioned.  The fore and aft positions are at rest relative to each other; but are moving in tandem relative to the still air.  This is a critical point; so she proposes to perform a thought experiment.

If she were to send a light signal -- a lantern maybe -- to a sailor at the bell, then this sailor would ring the bell (disregarding reaction times).  This light signal would effectively be instantaneous over this short distance, D.  If she started her chronometer at this exact same moment, then she would measure the time, t1, for the sound wave to return to her.  However, because the ship is in translatory motion, with the air at rest (windless day), then the formula she would use is not t = D / c, as below decks; instead, she would use t1 = (D - vt1) / c in the open still air.

This would be identical in form to the shore observer’s mathematics (from the Michelson-Morley experiment to detect the Aether).  He would simply factor in the translatory motion of the ship in his calculations.  The ship observer can safely assume that, because she knows of the violation of invariance by sound waves, that the time she measures would be associated with the formula that includes the ship’s velocity, v. This unknown can then be solved for.

This counterclaims the principles of relativity.  A certain kind of motion is revealed; it is somewhere between absolute and relative motion.  This intermediary motion spins silken threads between reference frames.

It seems counterintuitive that the mere addition of four solid walls and the introduction of very slow sound waves can produce more insight into scientific phenomena than super fast light waves.  This may be more Philosophy than Physics; what rides on a narrow gauge rail, is our perception of reality.

http://en.wikipedia.org/wiki/Galileo's_ship

http://stargazerslounge.com/blog/1308-geryllax-vus-blog/