I have two small wooden tubs. Unfortunately I do not remember who they are from. I thought it was Historic, but I cannot now find them on their site. One is used for dishes and the other for general washing of bodies and clothes. Though as Bob says, they are too small to sit in. These are wax lined, so be careful not to leave soapy water standing in the tubs. If you do, the soap is very searching and will soak through the wax and push it off of the wood. The wood takes a long time to dry and looks horrible with the spots of flaking wax.All the illuminations and drawings I have seen have shown Tubs being used for bathing rather than cut apart barrels. Both are made by a Cooper, but a barrel is rather expensive to cut up. Especially in an age when they were very necessary. I cannot imagine a barrel wearing out to the point of non-usability, but if it did, it would also be useless for bathing.
A tub is round with straight sides (easier and less expensive to make). A barrel is round with curved sides to aid in moving and providing strength. Some tubs are pictured as very shallow, water is poured over the person who is bathing. Other tubs are shown as quite deep and the people are soaking in them. In most cases servants or at least assistants are present to fetch more water, pour water and generally help out. Keep in mind that most people went to a bath house or Stews. I separate the two as a bath house is used to get clean. A stews is used to err meet women of an accommodating nature. Some are pictured as eating while in tubs. And not just one couple, but rather one long board across the top of a number of tubs serving as a table top for the food. A couple of people per tub are in attendance, naked except for their hats and jewelry. If a stews is doing this well a business, then they can afford to purchase less expensive tubs rather than cutting down worn-out and near worthless barrels. But that is just my 'take' on the situation.
What would you want for sitting in to wash at an event? Well weight consideration would dictate a half tub. That is to say - not a soaker tub. It would need be a foot or so deep. If you wish to sit on a low stool, then the water should at least come to your waist. This also has the advantage of using much less water. I also suggest a one person tub unless you have help, are really strong and have the room to transport it. So a 36 inch diameter circle would do well. You could cheat and use plywood for the bottom. I would suggest melting wax and giving the inside a coating to stop the leaking. To actually cooper the tub to the point of not leaking would be a difficult job. The bottom of the boards should be rabbeted to the bottom boards. Make a spacer for the top to aid in assembly. I don't know what sort of banding would have been used. However you could cheat and use metal stapping band at the top and bottom. Draw a plan full sized, with the center point radiating lines out to the boards. This will give you the angle to set your table saw blade at. Then just rip your lengths of wood that way for a better fit. Otherwise it is just time and a hand plane to angle the boards individually, after they have been quarter split from your log.
Are you sure Historic does not sell bathing tubs?

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Woodcrafter
14th c. Woodworking