Once the trigonometry of flight MH370 has been figured out, it is possible to do a few calculations to assess the effect of the airplane's speed on the frequency of the signals it used to communicate with the Inmarsat-3F1 satellite. I have assumed that MH370 flew for 6.83 hours along a great circle route from its last known position to a point over the debris field. I have also assumed that it flew at a constant airspeed of 850 kilometers per hour. It is possible to calculate the velocity of MH370 (both speed and direction) at every point along this route. It is then possible to divide this velocity into two components, one along the line-of-sight to the Inmarsat and the other at right angles to this line-of-sight. The following graph shows the component of MH370's velocity along the line-of-sight. At the start of its silent flight, MH370 was flying at a speed of 219.3 kph (kilometers per hour) directly towards the satellite and 821 kph directly perpendicular to the line-of-sight. Therafter, the airplane flew in a direction further and further away from the satellite, and the component of the velocity along the line-of-sight decreased. By the time the airplane reached the debris field, it was travelling radially away from the satellite at a speed of 500.5 kph and directly across the satellite's field-of-view at a speed of 687 kph.
We have all heard the Doppler effect. It affects radio waves in pretty much the same way as it affects sound waves. The Doppler effect arises when there is relative motion between the source of the wave (a fire truck or MH370, for example) and the receiver (you or the satellite, respectively). The Doppler effect does not change the speed at which the waves travel. Instead, it changes the apparent frequency. The frequency of the wave transmitted by the source is not the same as the receiver heard by the receiver. When the source and receiver are travelling towards each other, the apparent frequency received is higher. The degree to which the apparent frequency is raised or lowered depends on the relationship between the speed with which the wave travels through the material between and source and receiver and the relative closing speed between the two. In the case at hand, the speed of the wave is the speed of light: 1,080,000,000 kph. The relative closing speed (above graph) ranges from 219.3 kph at the start of the silent flight to negative 500.5 kph (negative closing speed resulting in a lowering of the frequency) at the end. Expressed as a percentage of the speed of light, these two closing speeds are 219.3 / 1.08E9 = 0.00002% and -500.5 / 1.08E9 = -0.00005%.
These are very small shifts in frequency. Big passenger airplanes use at least four frequency ranges, called "bands", in their communications. When the pilots talk to ground control, they use frequencies a little bit above 100MHz, just above your FM radio. The "Primary Channel Worldwide" for the ACARS reporting system is 131.55MHz. Much has been heard in the aftermath of MH370 about the use of satellites to relay the ACARS information, but the bulk of ACARS messages do not pass through satellites. When airplanes fly over land, the ACARS communications go over a land-based, cheaper network. The transponders about which we have heard so much are sent at frequencies near 1030MHz. When airplanes are far out over the sea, and communicate their ACARS data through satellites, they use even higher frequencies. The Inmarsat-3F1 satellite transmits at 1630MHz and receives at 1530MHz. One assumes that the Inmarsat study into the Doppler effect used data exchanged at these higher frequencies. Since the airplane's speed is such a small fraction of the speed of light, the percentage shift in frequency will be almost proportional to the relative speed percentages described above. The following graph shows the frequency shift (in Hz) which would be experienced by a continuous radio wave transmitted at 1600MHz by MH370 to the satellite during its silent flight.
These frequency shifts range from an increase of about 325Hz to a decrease of about 750Hz. I will say this: the Inmarsat people must be very good engineers to detect such small changes in a 1600MHz carrier, particularly when their radio equipment was probably designed to overcome and ignore such noise.