by blue line with phase velocity marked by a triangle) at 30 MHz , (c) with an additional controlling wave at $60$ MHz and phase velocity given by grey circle and dotted line.[]{data-label="fig:085"}](Fig9c.eps "fig:"){width="7.5cm" height="7.2cm"}

![Beam velocity distribution functions at the output of the TWT for $s = 1.27$ : (a) test beam ($I_b=2$ nA) without electrostatic wave, (b) with helix mode (trapping domain shown by red line with phase velocity marked by a square) and beam mode (trapping domain shown by blue line with phase velocity marked by a triangle) at 30 MHz, (c) with an additional controlling wave at $60$ MHz and phase velocity given by grey circle and dotted line.[]{data-label="fig:127"}](Fig10a.eps "fig:"){width="7.5cm" height="7.2cm"} ![Beam velocity distribution functions at the output of the TWT for $s = 1.27$ : (a) test beam ($I_b=2$ nA) without electrostatic wave, (b) with helix mode (trapping domain shown by red line with phase velocity marked by a square) and beam mode (trapping domain shown by blue line with phase velocity marked by a triangle) at 30 MHz, (c) with an additional controlling wave at $60$ MHz and phase velocity given by grey circle and dotted line.[]{data-label="fig:127"}](Fig10b.eps "fig:"){width="7.5cm" height="7.2cm"} ![Beam velocity distribution functions at 