DOPPLER QUADRILATERAL

Doppler Quadrilateral: Very special quadrilaterals determined by the signal arrival time.

Doppler Quadrilaterals are used to understand how (c+v)(c-v) mathematics occurs between two moving reference systems. The positions of the reference systems at the time of signal emission and reception form the vertices of the quadrilateral.

EXPLANATION:

The figure shows two planes moving in different directions and at different speeds. The planes send signals from points B and C, respectively. When the planes reach points A and C, the signals also simultaneously reach the planes.

The lines connecting the points ABDC form a Doppler Quadrilateral.

First, let us use the Auxiliary Lines button to hide the Red and Blue auxiliary lines. This will make the quadrilateral clearer. We see that the side lengths of the quadrilateral are different from each other. A parallelogram could also have been formed, but due to the different directions and speeds of the planes, the quadrilateral became a trapezoid.

Now let us use the slider to observe the movement of the planes and signals. The sequence of events is as follows:

At the beginning of the event, the planes at points B and C send signals to each other.
While the signals travel along CA (d₁) and BD (d₃) lines, the upper plane travels along CD (d₂) and the lower plane travels along BA (d₀).
When the upper plane reaches point D, the signal reaches it. When the lower plane reaches point A, the signal reaches it.

Now let us proceed to the (c+v)(c+v) Mathematics.

It is true that the signals travel along CA (d₁) and BD (d₃) lines, but this is only true from the perspective of a ground-based observer watching the event. Now let us make the auxiliary lines visible again and rewatch the event.

We see that the Red and Blue lines are tied to the reference systems of the planes. Additionally, the Red lines are parallel to and equal in length to the "d₄" line, while the Blue lines are parallel to and equal in length to the "d₅" line.

Let us look at the figure again and rewrite the sequence of events according to the reference systems of the planes:

The Red line attached to the plane indicates the direction of the incoming signal.
The Blue line attached to the plane indicates the direction of the signal sent by the plane.
The incoming signal travels along the Red line at the speed of c (the speed of incoming signals is always c). Therefore, the line "d₄" determines the signal's arrival time. The "d₄" line represents the distance between the two planes at the time of signal emission. Signal arrival time: t_Δ=d₄/c
The signal sent from the plane follows the Blue line and reaches the other plane. Since the signal's arrival time is t_Δ=d₄/c, the speed of the signal traveling along the Blue line is d₅/t_Δ=(c±v). The "d₅" line represents the distance between the two planes at the time of signal arrival.

The "±" sign in (c±v) indicates:

If d₅<d₄, "v" is negative: (c-v)
If d₅>d₄, "v" is positive: (c+v)

The distance the planes will cover during the signal's arrival time:

d₂=u₂.t_Δ
d₀=u₁.t_Δ

Now, we can write the signal speeds according to the ground observer:
The speed of the signal sent from the upper plane to the lower plane:

(c±v₁)=d₁/t_Δ
If d₄<d₁, "v₁" is positive: (c+v₁)
If d₄>d₁, "v₁" is negative: (c-v₁)

The speed of the signal sent from the lower plane to the upper plane:

(c±v₂)=d₃/t_Δ
If d₄<d₃, "v₂" is positive: (c+v₂)
If d₄>d₃, "v₂" is negative: (c-v₂)

As can be seen, three different v values are obtained.
According to the reference systems of the planes and the observer's reference system.
As a result, for the Doppler Quadrilateral seen in the above figure, the following equations apply:

Signal arrival time:

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 another example of a Doppler Quadrilateral with two planes moving in different directions. Similar calculations can be made within this quadrilateral. But keep in mind, the calculations above do not apply to this Doppler Quadrilateral.

Let us analyze using auxiliary lines. In this figure, we see that the line "d₁" determines the signal arrival time. From this point, we can calculate the other lines.

Signal arrival time:

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 sent to the other plane according to the plane.
(c±v₁): The speed of the signal going to the lower plane according to the ground observer.
(c±v₂): The speed of the signal going to the upper plane according to the ground observer.

As a general rule;
The line connecting the current positions of the two reference systems at the time of signal emission determines the signal arrival time. Since this line is valid and unique for both reference systems, the signal arrival time is the same for both reference systems. The signal reaches its target by traveling a distance equal to this line at the speed of c. Therefore, the arrival time is independent of the speeds and directions of the reference systems.

The point where the signal will arrive is determined by the speed, direction of the reference system, and the signal arrival time.

By connecting the signal arrival points with a line and using the signal arrival time, we can find the speed of the signal sent by the reference system to the other reference system. This value will be the same for both reference systems.

The animations here explain this topic, detailing how the event occurs using auxiliary lines.