The word "sh" (or "shh") is an exclamation for silence:
Shh! They're listening...
Etymonline only mentions a date (1847) and the common practice of "putting a finger to the lips." Does anyone else have more information about its origin?
Many words which mean "silence, please" have the digraph 'sh'. E.g. hush and shush.
The origin of all these words however, was the Middle English word huisst(pronounced "wheesht"), which originated in round about 1350–1400A.D. Huisst as expected, meant "silence, peace".
The Scottish plea for silence also includes the 'sh' digraph: wheesht.
The sound for signalling a desire for quiet has long been associated with the sound 'sh'.
It's been a while since I read the book, but Fredrik Lindström's "Jordens smartaste ord" (ISBN: 9789100580360) discusses the word "shh" in depth and posits it's the only word common to all human languages. It's even present in languages that don't use the sound.
Since it's been so long since I read the book, I don't know his sources.
An answer for which I do not have a source, but is worth consideration, is that the sound "shhh" is soothing to babies, perhaps because it imitates the sounds in the womb.
Taking an educated guess here:
If you want to silence somebody by means of sound, you usually want to make a sound that is audible to a close addressee but not to somebody further away (whom you do not want to hear you).
The latter excludes low-frequency vowels and voiced consonants as they contain low-frequency sounds which are carried well over distance and barriers (there is a reason why you devoice when whispering). It also excludes high-frequency vowels and plosives (stop consonants) as they stand out against ambient sounds (at least, if you want them to be audible by the addressee). For the same reason, you also want to exclude using more than one sound/phoneme.
When restricting ourselves to the English phoneme repertoire, this leaves us with fff, hhh, sss, shh and thh (IPA: f, h, s, ʃ, θ). Of these, hhh is confusable with abnormal breathing and (in my personal opinion) fff, sss and thh stand out more against ambient sound than shh, which is acustically similar to white noise.
Thus, shh seems the most logical sound to use for such a purpose.
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