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A Xiesheng series (諧聲 xiéshēng) is a set of Chinese characters sharing the same sound-based element (sometimes called the "phonetic" component, although phonological component would be more linguistically accurate). Characters belonging to these series are generally phono-semantic compounds, where the character is composed of a semantic element (or radical) and a sound-based element, encoding information about the meaning and the pronunciation respectively.
For example, the character 波 is composed of the semantic component 氵 'water' and the sound component 皮 (believed to have been pronounced something like *ba). Thus, 波 represents a word which has to do with water and was pronounced something like *ba. In fact, the word which 波 represents means 'wave' and was pronounced like *pa in Middle Chinese.
By grouping together characters with the same sound component into a Xiesheng series, one is able to compare words that were perceived to be quite similar in pronunciation at the time that the characters for these words were invented.