We report on p-i-n waveguide photodetectors with a ${{\rm Ge}_{0.92}}{{\rm Sn}_{0.08}}/{\rm Ge}$ multiple-quantum-well (MQW) active layer on a strain-relaxed Ge-buffered silicon substrate. The waveguide-photodetector structure is used to elongate the photo-absorption path and keeps a short photo-generated carrier transmission path. In addition, the double-mesa structure with a low substrate doping concentration is implemented, which minimizes the parasitic capacitance. As a result, a high responsivity of 119 mA/W at ${-}{1}\;{\rm V}$ and a high bandwidth of more than 10 GHz at ${-}{7}\;{\rm V}$ were achieved at a 2 µm wavelength. Compared with the surface-illuminated photodetector, the responsivity was improved by ${\sim}{8}$ times at a 2 µm wavelength, while keeping the comparable bandwidth.