Not Every Reference Frame is an Enclosed Compartment.

It is a premise of the Galilean principle of relativity that every reference frame behaves mechanically like an enclosed compartment at rest.  As a consequence of this premise it is presumed to be mechanically impossible to discern the motion of any reference frame by observing experiments conducted within that reference frame.  Material objects in flight within an enclosed compartment will manifest a particular velocity that arises from momentum transfer through physical contact with the compartment walls.  Objects in flight outside of the compartment will exhibit essentially the same behavior via contact with the external physical structure of the moving compartment.    However, a sound wave in flight through an enclosed compartment where the air has no wind currents in it will manifest one particular velocity while a sound wave propagating through the still air outside the compartment will manifest some other velocity — in a moving enclosed compartment the contained air’s velocity is the same as the compartment’s velocity and would add to or subtract from the sound wave’s propagation velocity.   There is then a difference in the mechanical behaviors of material objects and sound waves when they are moving through any particular medium based on whether that medium is within or outside of a moving enclosed compartment.  Under certain conditions an observer in a stationary or moving reference frame may not have to apply the principle of addition of velocities from the Galilean or Lorentz transformation equations to the propagating sound wave.  Not every reference frame is an enclosed compartment.

THE NON-APPLICABILITY OF THE PRINCIPLE OF ADDITION OF VELOCITIES TO PROPAGATING SOUND WAVES

On a windless day a train of length L travels along a level straight section of track at the constant velocity v.  An observer in the caboose has a clock and a light source with which she will send a signal to the engineer at the front of the train.  Upon seeing this signal, he will blow the whistle which will send out a sound wave that has the constant velocity c through the still air/medium.

At this short distance a light signal is effectively instantaneous so upon sending the light signal she also starts the clock that she has.  When the sound wave reaches the caboose observer’s ear she will stop her clock.  She should then measure approximately the Newtonian universal time interval t between the departure and arrival events of the sound wave in the train reference frame.

The caboose and the engine are at a fixed distance apart.  They form a tandem which is moving through the still air at the single velocity v each endpoint maintaining their distance of separation. The sound wave and the caboose begin their journeys at the endpoints of L and will meet at some location in space between the original locations of the endpoints along their adjoining line.  The sound wave travels the distance ct rearward towards the caboose and the caboose travels the distance vt forward towards the sound wave during the same interval of time t (distance = speed × time).  Adding these two distances should equal L.  Thus, all the variable values are available from within the train reference frame:

♦ L = ct + vt ; t = L / (c + v )

This formula (similar to the Michelson-Morley experiment) could be used by both the train observer (in the train reference frame) and an observer that she need not communicate with at rest on the nearby platform (in the platform reference frame).  The train observer might assume the train to be in motion and would thus measure with her clock an interval of time that would indicate that the sound wave has travelled at the unchanged velocity c for a lesser or greater distance than when the train is at rest.  This is a result of the consideration that the air molecules pass easily through the porous conceptual walls of any reference frame that is not an enclosed compartment.  Alternatively speaking, the train moves through a cloud of stationary air molecules which are not carried along by the train reference frame so that there will be no addition to or subtraction from the velocity of the sound wave but merely a change in the distance the sound wave travels.  The train observer will thusly not have to apply the principle of addition of velocities from the Galilean or Lorentz transformation between the two reference frames that are in relative motion.

Let, L = 1000 meters; c = 343 meters/second; assume v = 30 meters/second:

♦not, t = L / c, (train, air, and platform at relative rest) = [1000 m] / [343 m/s] = 2.92 s

♦but, t = L / (c + v ), (train in motion through air) = [1000 m] / ([343 m/s] + [30 m/s]) = 2.68 s