at the wind front and propagate in both directions from the front (see § 3 and Figs 2 and 5 – 7). These waves are non-monochromatic. Figure 8 shows a typical spectrum of low-frequency waves in the wind frame. This spectrum has a maximum at the frequency $\omega_{\rm max}$ which is a few times higher than the proton gyrofrequency $\omega_{Bp}=eB_0 /m_pc\Gamma_0$ in the field of $B_0$.

At high frequencies, $\omega > \omega_{\rm max}$, the spectrum of low-frequency waves may be fitted by a power law:

$$|B(\omega )|^2\propto 
\omega ^{-\beta}\,,
\label{Bomega}$$

where $\beta\simeq 1.6$.

In the region ahead of the wind front, the mean field of low-frequency waves depends on $\alpha$, and in the wind frame, for $\alpha_{\rm cr}\lesssim \alpha \lesssim 1$ this field is

$$\langle B_w\rangle =(\langle B_z\rangle^2 + 
\langle E_y\rangle^2)^{1/2}\simeq 0.1 B_0 \simeq 0.1 B_{\rm dec}
/\Gamma_0
\label{avBw}$$

within a factor of 2 or so.

At $\alpha > 1$, the intensity of low-frequency waves is suppressed in the region ahead of the wind front (see Fig. 6) because the typical frequency, $\omega\simeq \omega_{\rm max}$, of these waves is smaller than the relativistic plasma frequency of the ambient medium, $\omega_p=
(4\pi n_0e^2/m_e\langle\Gamma_e\rangle )^{1/2}$ ([@Gallant92]). In this case, only a high-frequency 