Any water heater looses some heat . The idea is to shut it off at times whan tou are very unlikely to need / use HW .

For instance , turn it off at nite , after your regular bed time . Turn it off early enough , before you regularly get up , to allow it to bring the water up to temp , so you can have HW for your shower .

The schedule can be made more complicated than that .

God bless
Wyr

OK, but with most things in life, the devil is in the details.
What's the temperature of the water when you turn off
the heater?
What's the temperature of the water just before you turn
it back on.
What's the capacity of the heater.
From those three numbers, you know how many BTU's
it's gonna take to get the water hot again. Time is not a
factor, it's all about the BTU's.

That's also the number of BTU's of waste heat that your
air conditioner will have to get rid of.

If the water temp changes a LOT, you need more insulation
on the water heater.
If it's not a lot, you can make a linear estimate of the
heat lost at maximum temperature vs the heat lost
at lower temperatures as the water cools...that we already
calculated.

My guess is that, for a well insulated water heater,
the difference is in the noise level.

The bottom line in all this is that it takes about the same
amount of energy to reheat the water as you lost by turning
it off. All you're saving is the difference between the
total heat lost at max temp vs the total heat lost
at an average temperature about half way between
the two temps we measured above. That's the extra
heat your air conditioner has to take out.

And that having microsecond resolution in when you
turn it on or off is inconsequential.
Hence, the question about thermal time constant.
You may discover that turning it off at all is mostly
wishful thinking.

But, it's all about the exact details of your particular
situation.

I am guessing a lot would depend on how well insulated the WH is , to begin with ? Wrap the WH in a bunch more insulation and it may not be saving a lot .

Something else , if the hubbie , wifey & crumb crunchers are away from the house in the day time , a second off cycle saves more , except on the weekend ( indicating a need for a somewhat / potentially more $$$ time clock that can be programed for a different schedule on the week ends ) .

A different thought , in an all electric house , especially with resistance heat , only , the lost heat in winter is not really wasted . It helps heat a small part of the house . ( If the WH is not out in the garage . )

God bless
Wyr

Actually, building the 7 day programmable WH timer isn't all that expensive. Get a contactor with a 120V coil and 240V load side. Get a plastic project case to contain the contactor. Get the #95205 digital timer from harbor freight to operate the contactor. At this point, you might be in for $30, but you have an 8 On/Off digital timer.

I don't have crumb crunchers, but I am running an idea I found on waterheatertimer.org It uses an old Decora switch sized Intermatic 7 day programmable digital timer I bought from my local Habitat ReStore. I'm only using 3 of the 7 on/off cycles available. I still don't understand why most water heater timer manufacturers still use mechanical timing devices?

The HF digital timer I suggested above is actually more sophisticated than my Intermatic timer. The HF timer will do groups in addition to individual days, groups can be: MTWRFSS, MTWRF, MWF, TR, SS (M=Monday, T=Tuesday, etc). The Intermatic doesn't do groups of MWF, TR. I really need to get another one and gut the mechanical timer out of my sprinkler timer. I'm using that HF timer to cycle the dehumidifier in my garage.

Has anyone made a 220V timer out of a 110V timer? That would be an awesome write up.