Mach number limit of electron holes in bounded beam-plasma system

The upper limit of the electron hole Mach number under the presence of an electron beam is discussed theoretically by solving the Poisson equation. When the beam component exists in plasmas, a new hole solution appears owing to shielding effects of the beam electrons. The hole speed can greatly exceed the critical Mach number M₀ equivalent to 1.3. For the beam component ratio above approximately 0.22 the upper limit of the Mach number intrinsically vanishes when the beam temperature is equal to that of the bulk. The transverse effect due to a finite radial wave number K strongly modifies the parameter regimes for the hole solution.