The missing mass spectroscopy of $Xi^{-}$ hypernuclei with the $(K^{-},K^{+})$ reaction is planned to be performed at the J-PARC K1.8 beam line by using a new magnetic spectrometer, Strangeness $-2$ Spectrometer (S-2S). A $v{C}$cerenkov detector with a radiation medium of pure water (refractive index of 1.33) is designed to be used for on-line proton rejection for a momentum range of 1.2 to 1.6 GeV/$c$ in S-2S. Prototype water $v{C}$erenkov detectors were developed and tested with positron beams and cosmic rays to estimate their proton-rejection capability. We achieved an average number of photoelectrons of greater than 200 with the latest prototype for cosmic rays, which was stable during an expected beam time of one month. The performance of the prototype in the cosmic-ray test was well reproduced with a Monte Carlo simulation in which some input parameters were adjusted. Based on the Monte Carlo simulation, we expect to achieve $>90%$ proton-rejection efficiency while maintaining $>95%$ $K^{+}$ survival ratio in the whole S-2S acceptance. The performance satisfies the requirements to conduct the spectroscopic study of $Xi^{-}$ hypernuclei at J-PARC.
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