Shahi snan is not a Vedic idea. It was recognized in between 1400AD to 1650AD when numerous Akharas (Group of Sadhus) came up in north India to lecture, ensure Hindu religion against Mughal attacks and so forth. In the end there wound up numerous Akharas and all gatherings of Sadhus needed to bathe at the promising waterway bank spots on the Ganges and different religious pioneer spots (Nasik, Ujjain, Varanasi, Prayag) on propitious days.
Commonly an engagement broke out between Akharas regarding who will take a plunge first as there were contrasts in philosophies inside the Akharas; sadhus were permitted to convey weapons and that govern is as yet appropriate. This issue was generally settled amid the reign of Shahjahan in his court and Akharas were given their numbers when they would get the chance to shower in a groupp.
The Akhara who got the 'firman' from the Emperor to take the primary plunge on uncommon days called it a 'Shahi snan'(bath because of Emperor's requests or because of Baadshah). In the end numerous Akharas got such 'firman' and till today take after similar tenets of showering succession on uncommon days in Sinhasta mela and Kumbh mela. That is the means by which it moved toward becoming Shahi snan.
There is nothing like a non-shahi snan it’s just a few dates are been decided according to the Hindu mythology which makes several baths the specific and intensive one. I personally think every time an individual takes a dip in Ganga it’s divine and impeccable.
“All the best”
