It's an interesting problem isn't it?
I have asked this of several people with varying mathematics ability and nobody can seem to derive an answer. It also seems to be beyond my personal mathematical abilities.
While it may be discouraging to tell you that I do not know the answer, let me offer you some encouragement.
It is conspicuously obvious that there can always only be one answer for x for any given values of, (my), y and z.
Nice idea redefining y and z to simplify the process, I never thought of that, (and you should see some of the monster formulae I've got). But of course, one would have to split them up again to simplify for x.
Since you display an interest in this problem, and have the ability to derive that first step, I'll tell you how I arrived at this problem in the first place.
When I was in high school science class in about 1986, the teacher demonstrated how earthquake epicentres are triangulated by working out the distance the shock wave travelled before arriving at each seismic station.
Plotted on a common time scale you would get three sets of shock waves like this:
Earthquakes contain both primary waves, (P), and secondary waves, (S), that travel at different speeds, so the distance a set of shock waves have travelled can be directly derived from the delay between both those waves, taking into account the composition of the material they are travelling through.
Thus, it is a simple matter to plot a circle around each station at the distance each detected, and by plotting three stations triangulate the epicentre, sometimes resulting in a "
triangle of error". The epicentre need not be inside the triangle formed by the stations to be triangulated.
This got me thinking. Why do you need the S-waves at all? There should be sufficient available information to derive the epicentre without the added benefit of this information, and the problem boils down to the geometric problem I presented.
There is indeed only one value for x when y and z have specific values. The information we seek appears lost to us initially, because the first inkling we get of an earthquake is at time=0 when it hits the first station and we have no idea how long it took to get there. But knowing that the missing time/distance, (x), is a component of the other two readings, (x+y), and (x+z), gives us the information we need in order to leave x with a single unique value for each location, and value of y and z.
Obviously we can instantly figure out that if the earthquake occurred directly underneath station A then we would get:
y = 10
z = 10
and therefore x=0.
Or if the earthquake occurred directly in the middle of an equal lateral triangle, it would hit all stations at the same time, and therefore:
y=0
z=0
Therefore x must equal the distance from the apex to the centre-point.
Now obviously A wont always be the first station to detect the earthquake, so you might get:
x=14
y=27
z=0
In this case the answer you would be after is z, and the answer in this case lies outside the equal lateral triangle also. But obviously the formula would be the same except you would change the x and z terms around.
Now obviously we have a couple of options:
1. We can plug random values of x into the formula and try to home in on the answer that way.
2. We can do the calculations prior and put them in a look up table.
3. We can work out the formula and derive the exact answer to a potentially infinite number of decimal places for any point extending off to infinity anywhere on that plane.
Having access to fast computers might make one opt for the lazy option 1 or 2, but that still doesn't solve the problem.
And here's the best bit of all. This formula has endless practical value.
Imagine a security system with 3 microphones or seismic detectors that can instantly triangulate your position in a building by the noises you make and your footsteps.
Better yet imagine putting this concept in 3 dimensions, (a tetrahedron of sensors), and being able to instantly work out the location of any object emitting a particular type of energy/radiation, at any range, (with decreasing accuracy over distance in the real world).
Now I don't want to get all geeky on you, but can you say, "Tricorder" boys and girls?
Better yet, you could add a different type of sensor in another tetrahedron within the same space, say you have seismic, and radiation.
Going straight to the logical conclusion you could have an entire sphere of tetrahedrons within each other all in the same space, each slightly offset, (the individual orientation of each one obviously doesn't matter).
The size of those tetrahedrons, (distance sensors are apart), would have a bearing on the accuracy also.
The accuracy with which they were placed in their positions would also have a bearing if you were relying on the tetrahedron being perfect.
However, if a formula could be devised for a "non-equal lateral" tetrahedron, then that is simply a matter of calibration, perhaps even self-calibration if additional sensors were added and/or mathematical derivation used so the device could triangulate the location of it's own sensors in relation to one another.
Ta-da!
Time for the
Mark II?