= Doppler frequency along a trajectory
{c}
{title2=$f_o(t)=f_s(c-\mathbf n\cdot\mathbf v_o(t))/(c-\mathbf n\cdot\mathbf v_s(\tau))$}
In the same uniform stationary sound model, let $\tau$ be emission <time>, $t$ reception <time>, and $\mathbf x_s(\tau)$ and $\mathbf x_o(t)$ source and receiver positions. The direct-path propagation condition and its direction are
$$
c(t-\tau)=R,\qquad R=|\mathbf x_o(t)-\mathbf x_s(\tau)|>0,\qquad \mathbf n=\frac{\mathbf x_o(t)-\mathbf x_s(\tau)}{R}.
$$
The <unit vector> $\mathbf n$ points from the source's old emission position to the receiver's reception position. With <velocities> $\mathbf v_s=d\mathbf x_s/d\tau$ and $\mathbf v_o=d\mathbf x_o/dt$, taking a <derivative> of the propagation condition gives
$$
c\left(1-\frac{d\tau}{dt}\right)=\mathbf n\cdot\mathbf v_o-\mathbf n\cdot\mathbf v_s\frac{d\tau}{dt},\qquad f_o(t)=f_s\frac{d\tau}{dt}=f_s\frac{c-\mathbf n\cdot\mathbf v_o(t)}{c-\mathbf n\cdot\mathbf v_s(\tau)}.
$$
Here the <dot products> project onto the actual propagation direction. The last equality counts source <wave phase> cycles at reception for constant emitted <frequency> $f_s$. It assumes both speeds are below $c$, a unique direct arrival and smooth motion; interpreting a measured short-window <frequency spectrum> additionally requires little <frequency> change within that window. The source <velocity> must be evaluated at emission, not at reception. This delayed geometry matters when the direction changes.
For an original pass-by model, a source follows $\mathbf x_s(\tau)=(v\tau,0)$ and a stationary receiver is at $(0,b)$ with $b>0$. The source passes its closest position at emission <time> $\tau=0$. Then
$$
R=\sqrt{v^2\tau^2+b^2},\qquad t=\tau+R/c,\qquad f_o(t)=f_s\frac{c}{c+v^2\tau/R}.
$$
These parametric equations give received <frequency> against reception <time>. For $\tau<0$ the shift is positive, for $\tau>0$ negative, and the unshifted emission at $\tau=0$ reaches the receiver at $t=b/c$. Thus zero shift occurs after the source's geometric closest passage in the material's <reference frame>. Increasing $b$ spreads the transition over a longer interval even though $v$ is unchanged. Far before and after passage, the <frequency> approaches $f_sc/(c-v)$ and $f_sc/(c+v)$ respectively.
\Image[/doppler-receiver-frequency-trajectories.png]
{title=Receiver offset changes the shape of a passing source's <Doppler shift>}
{id=doppler-receiver-frequency-trajectories}
{description=Original model predictions for $c=343\,\mathrm{m\,s^{-1}}$, $v=20\,\mathrm{m\,s^{-1}}$, $f_s=500\,\mathrm{Hz}$ and receiver offsets $b=2\,\mathrm m$ and $10\,\mathrm m$. The upper panel shows the shared straight source <trajectory> and the two receiver positions; the lower panel uses reception <time> $t=\tau+R/c$. Dots at $t=b/c$ mark zero shift. Dashed horizontal limits are the one-dimensional approach and recession values. No curve is measured data.}
Back to article page
