Doppler effect in 3D

The position of a wave source in three dimensions is described by a vector $\vec{r}_s(t)$. This source is emitting waves of frequency $f_s$. These waves travel at speed $c$ to an observer. The position vector of the observer is $\vec{r}_o(t)$. Because of the relative motion between the source and the observer, the observer detects a frequency $f_o$ that may be different from $f_s$.

A peak of the wave leaves the source at $t_0$ when the source is at $\vec{r}_s(t_0)$. This peak arrives at the observer at time $t_1$ when it is at $\vec{r}_o(t_1)$. The sound travels from $\vec{r}_s(t_0)$ to $\vec{r}_o(t_1)$ in a time $t_1 -t_0 = |\vec{r}_2(t_1) - \vec{r}_1(t_0)|/c$.

The next wave peak leaves the source at $t_0+T$ when the source is at $\vec{r}_s(t_0+T)$. Here $T$ is the period, $T=1/f_s$. This peak arrives at the observer at time $t_2$ when it is at $\vec{r}_o(t_2)$. The sound travels from $\vec{r}_s(t_0+T)$ to $\vec{r}_o(t_2)$ in a time $t_2 -t_0 -T= |\vec{r}_o(t_2) - \vec{r}_s(t_0+T)|/c$. The frequency at the observer is $f_o=1/(t_2-t_1)$.

The equations for the Doppler effect can be summarized as:

 

$\large |\vec{r}_o(t_1) - \vec{r}_s(t_0)|=c(t_1 -t_0)$,

$\large |\vec{r}_o(t_2) - \vec{r}_s(t_0+T)|=c(t_2-t_0-T)$,

$\large f_o=\frac{1}{t_2-t_1}$.

 

$f_o$

$t$

$\vec{r}_s(t)=$ $\hat{x} +$ $\hat{y} +$ $\hat{z}$ [m].

$\vec{r}_o(t)=$ $\hat{x} +$ $\hat{y} +$ $\hat{z}$ [m].

$f_s=$  [Hz] $c=$  [m/s]

$f_o$ from $t=$  to $t=$ .

At time $t=$  s, the observer hears a frequency of  Hz.

At time $t=$  s, the observer hears a frequency of  Hz.

$|\vec{r}_s-\vec{r_o}|$

$t$

$\large \frac{d|\vec{r}_s-\vec{r_o}|}{dt}$

$t$

At every time $t_1$, the nonlinear equation,

\[\begin{equation} \sqrt{(r_{ox}(t_1)-r_{sx}(t_0))^2+(r_{oy}(t_1)-r_{sy}(t_0))^2+(r_{oz}(t_1)-r_{sz}(t_0))^2}=c(t_1 -t_0), \end{equation}\]

is solved for $t_0$ using a binary search method. Using this value of $t_0$ the equation,

\[\begin{equation} \sqrt{(r_{ox}(t_2)-r_{sx}(t_0+T))^2+(r_{oy}(t_2)-r_{sy}(t_0+T))^2+(r_{oz}(t_2)-r_{sz}(t_0+T))^2}=c(t_2 -t_0-T), \end{equation}\]

is solved using a binary search for $t_2$. The frequency that is observed is $f_o=1/(t_2-t_1)$.

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