# What is the frequency distribution of the electromagnetic wave emitted by a charge accelerating along a given trajectory?

Charged particles radiate when accelerated. From a few other Physics.SE answers like What is the frequency of a photon emitted by an accelerated charged particle? and How electromagnetic waves are created? it is understood that the accelerated charge emits a frequency spectrum, not a single frequency.

Most physics literatures on classical electrodynamics solve the problem only for synchrotron radiation or bremsstrahlung.

What is the general procedure for finding the frequency spectrum if the particle moves in a given trajectory r(t)? For example, if the charged particle is accelerated in a straight line (constant acceleration along x-axis) then how can we calculate the frequency distributions of the EM waves?

This is in standard texts. For example, in J.D. Jackson, Classical electrodynamics, third edition, Eq. 14.67 $$$$\frac{d^2I}{d\omega d\Omega} = \frac{e^2\omega^2}{4\pi^2 c} \left | \int_{-\infty}^\infty \hat n \times (\hat n \times \vec \beta) e^{i\omega(t-\hat n \cdot \vec r(t)/c)} dt \right |^2$$$$ as the intensity distribution as a function of frequency and direction. Here $$\vec r(t)$$ is the particle trajectory, $$c\vec \beta = \frac{d\vec r(t)}{dt}$$, and $$\hat n = \hat x \sin\theta \cos\phi+\hat y \sin\theta\sin\phi +\hat z \cos\theta$$ is the unit vector in the direction of radiation. Jackson's result is in Gaussian units. Jackson discusses when this is valid. Essentially when the radiation is localized in space and time (for example, no radiation before the big bang. Your example of a constant acceleration is a well known badly behaved trajectory.)
A standard way to derive the result is to start with an arbitrary localized in space and time current and charge density. Plug these into the Lorenz gauge retarded coulomb's law equations for the vector and scalar potentials. Fourier transform the potentials in time to get their frequency components, take the large distance limit to separate the radiation fields which go like $$r^{-1}$$. The result is that the radiation fields are proportional to the component of the space-time Fourier transform of the current density evaluated at $$\vec k = \frac{\hat n\omega}{c}$$, and $$\omega$$ which is transverse to the radiation direction. Calculating the time integral of the Poynting vector then gives the intensity (i.e. the energy radiated per unit frequency and solid angle). You can see that the integral in Jackson's result is proportional to the Fourier transform of the current density at this $$\vec k$$ value. That is $$$$\begin{split} \vec J(\vec k,\omega) &= \int d^3r' dt' ec\vec \beta \delta^3(\vec r'-\vec r(t')) e^{-i \vec k\cdot \vec r'+i\omega t' } \\ &= ec \int dt' \vec \beta(t') e^{-i\vec k \cdot \vec r(t') +i\omega t'} \end{split}$$$$ with $$\vec k = \hat n \frac{\omega}{c}$$. Taking $$- \hat n \times (\hat n \times \vec J(\vec k,\omega))$$ gives the transverse component.