A pressure wave supercharger for compressing fresh air for an internal combustion engine includes a cold gas housing, a hot gas housing, and a rotor casing inside which a rotatable cell rotor is disposed. The hot gas housing has a high-pressure exhaust gas duct and a low-pressure exhaust gas duct, while the cold gas housing has a fresh air duct and a charge air duct. The high-pressure exhaust gas duct, the low-pressure exhaust gas duct, the fresh air duct and the charge air duct are fluidically connected to the cell rotor. The hot gas housing has a heat exchanger which is designed in such a way that at least a first bearing for a rotor shaft can be cooled.