DOPPLER QUADRANGLE

Doppler Quadrangle: There are very special quadrangles whose side lengths are determined by the travel time of the signal.

Doppler Quadrangles is used to understand how the (c+v)(c-v) mathematics between two reference systems, which are both in motion, occurs. The positions of the reference systems at the moment when the signal is emitted and when it arrives form the vertexes of the quadrangle.

EXPLANATION:

In the figure, we see two planes moving in different directions and at different speeds. The planes send signals towards each other from B and C. When the planes reach A and C, the signals also reach the planes simultaneously. 

The lines that connect ABDC points form a Doppler Quadrangle. 

Firstly, let’s hide the auxiliary Red and Blue lines by using the button “Auxiliary Lines”. In this way, the quadrangle will be seen more clearly. We see that the side lengths of the quadrangle are different from each other. A parallelogram could have been formed here as well, but the fact that the directions and the speeds of the planes are different lead to the formation of a trapezoid quadrangle. 

Now, by using the slider bar, we can observe the movement of the planes and the signals. The flow of events is as follows:

• At the beginning of the event, the planes at B and C send signals towards each other. 
• While the signals cover the lines CA (d₁) and BD (d₃), the plane on the top covers CD(d₂) and the one on the bottom covers BA (d₀).
• When the plane on the top arrives at D, the signal reaches it. When the plane at the bottom arrives at A, the signal reaches it. 

Now we move on to (c+v)(c+v) Mathematics. 
It is true that the signals cover CA (d₁) and BD (d₃) lines, but this is only true for an observer who is watching the event from the outside. Now, let’s make the auxiliary lines visible and watch the event again.
We see that Red and Blue lines are connected to the reference systems of the planes. Additionally, we also see that Red lines are parallel and equal in length to d₄ and that the Blue lines are parallel and equal in length to d₅. 

Let’s look at the figure and write the flow of events again:
Relative to the reference systems of the planes;

• The Red line which is connected to the plane shows the incoming direction of the signal that comes towards it. 
• The Blue line which is connected to the plane shows the outgoing direction of the signal that the plane sends. 
• The signal coming towards the plane covers the Red line that belongs to it at c speed. (INCOMING signal speed is always c.). Therefore, the line that determines the travel time of the signal is “d₄”. d₄ is the line that represents the distance between the two planes at the moment the signals are sent. The travel time of the signal: t_Δ=d₄/c
• The signal that is sent from the plane reaches the other plane by following the Blue line that belongs to it. As the travel time of the signal is t_Δ=d₄/c, relative to the reference system of the plane, the speed of the signal going towards the other plane on the Blue line is d₅/t_Δ=(c±v). d₅ line represents the distance between the two planes at the moment the signals arrive. 

"±" symbol in (c±v):
If d₅<d₄, "v" gets a negative value: (c-v) 
If d₅>d₄, "v" gets a positive value: (c+v)

The distance that the planes will cover within the travel time of the signal
d₂=u₂.t_Δ

d₀=u₁.t_Δ

Now, we can write the speed of the signal relative to the observer on the ground.
The speed of the signal that is sent from the plane at the top to the plane at the bottom:
(c±v₁)=d₁/t_Δ
If d₄<d₁, "v₁" gets a positive value: (c+v₁) 
If d₄>d₁, "v₁" gets a negative value: (c-v₁)

The speed of the signal that is sent from the plane at the bottom to the plane at the top:
(c±v₂)=d₃/t_Δ
If d₄<d₃, "v2" gets a positive value: (c+v₂) 
If d₄>d₃, "v2" gets a negative value: (c-v₂)
As you can see, we obtained three different v values. 
Relative to the reference systems of the planes and relative to the reference system of the observer. 

Consequently, there are the following equations for the Doppler Quadrangle seen in the figure above. 

Travel time of the signal:

t_Δ=d₁/c

Side Lengths:

d₀=u₁.t_Δ

d₁=(c±v₁).t_Δ

d₂=u₂.t_Δ

d₃=(c±v₂).t_Δ

Diagonals:

d₄=c.t_Δ

d₅=(c±v).t_Δ

 

 

EXPLANATION:

Here we see two planes that move in different directions and another Doppler Quadrangle. We can do similar calculations for this quadrangle as well. But please be careful. The calculations above are not valid for the Doppler Quadrangle here. 
Let’s do an analysis by using auxiliary lines. Here we see in the figure that the line that determines the travel time of the signal is “d₁”. From here, we can calculate the other lines. 

Travel time of the signal: 
t_Δ=d₁/c

Side lengths:
d₀=u₁.t_Δ

d₁=c.t_Δ

d₂=u₂.t_Δ

d₃=(c±v).t_Δ

Diagonals:
d₄=(c±v₁).t_Δ 

d₅=(c±v₂).t_Δ 

(c±v): The speed of the signal that the plane sends to the other plane relative to the plane 
(c±v₁): The speed of the signal going towards the plane at the bottom relative to the observer on the ground 
(c±v₂): The speed of the signal going towards the plane at the top relative to the observer on the ground 

As a general rule;
The line that connects the positions of the two reference systems at the moment the signal is sent is the line that determines the travel time of the signal. As this line is valid and mutual for both the reference systems, travel time of the signal is the same for both the reference systems. The signal reaches its target by covering a distance that is at the same length with this line at c speed. Therefore, travel time is independent of the speeds and directions of reference systems. 

The speed and direction of the reference system and the travel time of the signal altogether determine which point the signal will arrive at. 

By using the line that connects the arrival points of the signal and the travel time, relative to the reference system that sends the signal, we can find the speed of the signal that is sent to another reference system. This value is the same for both reference systems. 

In the animations here, this topic is covered and how this event takes place is explained in detail by using auxiliary lines.

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