Time domain representation of a waveform from an accelerating source (doppler effect) Consider the scenario where a stationary observer observes a source moving in 3 dimensions, emitting a signal $s(t)$.  I would assume only the distance between the source and the observer, $x(t)$, has any effect on the frequency scaling.
The only thing I have ever learned (or been able to find online) is the frequency scaling equation $f=(1+\frac{\Delta v}{c})f_0$.  Is there a more general formulation that, using $s(t)$ and $x(t)$, would allow me to model the signal recieved at the observer?
I would imagine that a model exists for radar, but I haven't been able to find it.  I expect it can be solved with the Fourier transform but I don't know where to begin.
 A: 
I would assume only the distance between the source and the observer, $x(t)$, has any effect on the frequency scaling.

Wave Dispersion
There are multiple effects that matter here if the medium is dispersive (i.e., the wave frequency depends upon the wave number or another way of stating this is that the phase speed depends upon the wave frequency and/or wavenumber).  For the example of radar, the Earth's atmosphere is not dispersive until one reaches large altitudes (i.e., few to several 10s of km) and only for specific ranges of frequencies.  So we can ignore dispersion for the moment.
Time of Flight
Since the observer is at a finite distance, $\mathbf{x}\left(t\right)$, from the reflecting object that is moving at a velocity, $\mathbf{v}\left(t\right)$, relative to the observer, one may naively expect that some signals would arrive at different times than others.  Since radar is a free mode of electromagnetic radiation, mostly radio waves, and the medium is not dispersive nor ionized, all frequencies propagate at the same speed, the speed of light, $c$.  If you are far away from the reflecting object (i.e., $\lvert \mathbf{x}\left(t\right) \rvert \gg L$, where $L$ is the size of the reflecting object), then there should be no noticeable time of flight effects.  Meaning, you should not receive frequency one at a different time than frequency two since they both share the same propagation distance.
Doppler Effect
The effect that can give you information about the objects velocity is known as the Doppler effect, which is mathematically defined as (assuming a constant source frequency/wavenumber):
$$
\omega_{obs}\left(t\right) = \gamma \left( \omega_{src} + \mathbf{k}_{src} \cdot \mathbf{v}\left(t\right) \right) \tag{1}
$$
where $\omega_{obs(src)}$ is the frequency in the observer's(source's) rest frame, $\mathbf{k}_{src}$ is the wavenumber of the source, $\gamma$ is the relativistic Lorentz factor (one can assume $\gamma \approx 1$ for most cases in Earth's atmosphere), and $\mathbf{v}\left(t\right)$ is the velocity of the source relative to the observer.  Note that $\mathbf{k}_{src}$ and $\mathbf{v}\left(t\right)$ are both represented here as 3-vectors, not scalars.
Answer

Is there a more general formulation that, using $s(t)$ and $x(t)$, would allow me to model the signal recieved at the observer?

Yes, you can use the Doppler effect in Equation 1 to determine the velocity of the source if you know $\omega_{src}$ and $\mathbf{k}_{src}$, which you would if this were the radar example to which you elude at the end of your question.  Using multiple radar source stations (and/or multiple observations over a given time interval), one could derive a numerical estimate for $\mathbf{v}\left(t\right)$.
Again, in the case of the radar example, you know when a signal was emitted and you know the speed at which it was emitted (i.e., $c$), then you can determine the magnitude of the distance to the source using the delay time, $\Delta \tau\left(t\right)$, between the emitted and received signal.  This would be given by:
$$
\lvert \mathbf{x}\left(t\right) \rvert = \frac{c \Delta \tau\left(t\right)}{2} \tag{2}
$$
Most radar systems send a beamed signal (i.e., narrow angular focus) in one direction at a given time and rotate the emitter to sweep across the entire sky.  This provides directional information, since you know the azimuthal angles, $\phi$, during which you observed the reflecting source.  Thus, you could use this information to define a 2D direction for the magnitude calculated from Equation 2.  The third directional component could be estimated if the radar system had finite poloidal receiving bins or if one used multiple radar transmitters to triangulate the signal.  This would all be integrated with the estimate for $\mathbf{v}\left(t\right)$ (may be a 2D or 3D estimate, depending on the radar system) to determine the full 3-vector distance, $\mathbf{x}\left(t\right)$.  Numerical derivatives can be used to determine full 3-vector velocity of the reflecting object, $\mathbf{v}\left(t\right)$.
Update
I forgot to mention that if $\omega_{obs}\left(t\right)$ increased/decreased in time, it would represent an acceleration of the source, i.e., $\mathbf{v} = \mathbf{v}\left(t\right)$.
A: I believe I found the solution for my question, it was actually fairly simple (no calculus or explicit use of the Doppler effect!).  In the received signal $r(t)$, the value at any time $t$ is equal to the sum of the perturbations that would reach the observer at time $t$.  Assuming that the speed of the source is significantly lower than the speed of a wave in the medium, there is only one single perturbation that reaches the observer at time $t$.  This perturbation occurred some time $\tau$ before $t$.  The value of $\tau$ at any time $t$ is solved by:
$\tau(t) = \frac{x(t-\tau(t))}{v} \tag{1}$
where $x(t)$ is the distance of the source from the observer at time $t$, and $v$ is the speed of a wave in the medium.
Therefore, $r(t)$ can be described by:
$r(t) = s(t - \tau(t))  \text{  where  } \frac{x(t-\tau(t))}{v} - \tau(t) = 0 \tag{2}$
