You have to be careful with this kind of 'I'm on' signal.
A lot of regulations for un-licensed radio frequencies (such as the ISM band, etc) stipulate a maximum duty cycle on the transmission, meaning that you are only allowed to actively transmit for a certain % of the time. This prevents blocking of a particular frequency and allows better TDM of signals.
For example, for the ISM (Industrial, Scientific, Medical) range:
Dependent on the sub-band the transmission power is limited to 10 dBm … 27 dBm. The permitted time allocation (duty cycle) also varies with the sub-band. So interferences with other 868 MHz equipment are reduced and thus a better transmission quality can be achieved.
- ISM 433/868
So whatever solution you go for you will have to create some kind of periodic pulsed beacon that sends a short burst signal at pre-defined periods rather than just a constant 'on' signal.
This should also mean that you can save power in between the pulses as you can turn the transmitter off (a lot of TX chips have a 'sleep' or 'shutdown' mode) when it's not actively being used.
As has been mentioned in some of the comments, there is most likely a set of reinforcement bars in the concrete forming a 'Faraday cage'. This blocks a large number of signals from penetrating as they hit the bars and then get grounded by them. So, you need to pick a frequency that has a wavelength that is smaller than the space between the bars. Also, concrete can absorb the signal at certain frequencies.
From what I understand this is mostly due to the water content in the concrete. Water contains hydrogen. Hydrogen resonates at 2.4GHz. Many transmissions (WiFi for example) occur at 2.4GHz, so the hydrogen in the water in the concrete absorbs the transmission.*
So, the ISM 868MHz band has a wavelength of about 0.35m - this is probably going to be way too big to fit between the bars (I'm not sure what the regulations state about bar spacing). ISM 915 takes it down to 0.33m - still too big. The 2.4GHz is 0.125m - much more realistic but may not get through the concrete. So you'd be looking somewhere in the mid-to-high 1GHz range. Personally I'm not aware of a license free frequency range in that area. You'd need to check with the RF licensing people in your area (FCC, OfCom, etc).
*(This is purely my own conjecture - please correct me as I'd like to know the truth myself)
RF signals are attenuated greatly by water, Very low frequency (VLF) is used for submarine communications but requires a huge antenna system. Therefore Bluetooth and other RF-based systems won't be suitable.
For a depth of 30 meters accoustic modems are a practical solution. I've done some software programming for a system that used a AquaComm: Underwater wireless modem and it has an RS232 port and simple to use commands. From memory the board and transducer combined would have taken up an area of around 100 x 100 x 200mm, so I'm not sure if that would meet your definition of "small" or not.
Optical communications would be another possibility, although the only shipping commercial products I could find had a much shorter range. I found an interesting paper Using Optical Communication for Remote Underwater Robot Operation where they reported 30 meters over the length of a pool, although that dropped to around 9 meters in a harbor. With optical it would depend a lot on the water quality and ambient light.
Because it sounds like your device is always tethered by a rope the simplest and cheapest and easiest way might be to replace the rope with a cable. Assuming there aren't any dire consequences to the sensor failing and long-term reliability isn't an issue maybe standard mains cable would do the job if properly sealed at each end. It sounds like this setup it for some form of experiment?
As mentioned in a comment at that depth keeping your device watertight won't be easy so generally keeping the underwater component of the system as small as possible and removing the need for battery access will make it easier. For the same reason it might also be worth considering leaving the Arduino above water and just place the sensor(s) underwater if possible.
Best Answer
To transmit power across a gap you can do it by transformer action but the efficiency will be low because the magnetic field from the power coil doesn't 100% couple to the receive coil. Depending on the gap the coupling may be only (say) 10% and this means you have to use a lot of brute force to get the power you need on the receive coil.
A significant improvement on this is to use tuned coils operating at resonant frequency.
I'm going to describe one I was involved with last year - it didn't achieve anything more than 10% power coupling but, the constraints on the application were enormous - this is why I'm using it as an example - if you can avoid these constraints (described below) then you should get easily better than 50% power efficiency.
It needed to transfer power to the end of a 500MW Power generator's turbine rotor and the shafts are quite large. In effect the receive coil was about 1.5 metres diameter: -
The red circular line is the coil sitting in a raised insulated groove above the metal of the rotor. The metal was expected to give poor coupling but at the frequency used, it wasn't too bad at sucking the power away (maybe 20% loss). The coil was a single turn of 1.6mm Cu.
The stator coil coupled to about 30º of the circumference i.e. it wasn't a full turn of 1.5m diameter. It was a 4 turn coil. Each of the 4 turns used 3 litz wires of diameter 1mm with 250 strands in each litz wire, 750 strands in total. With a full circular coil, power-coupling would be better; probably something like double.
The coil-gap was about 40mm and, despite all the constraints, the arrangement could easily produce 50VRMS on the tuned rotor coil. The power needed on the rotor was about 2W maximum. Smaller gaps are going to be more effective.
The stator coil had two tuning capacitors, one in parallel and one to bring the 600kHz AC feed in. The feed-in voltage was about 30VRMS but due to the tuning it was generating about 100VRMS across the coil. Better litz wire would improve the voltage and reduce the power losses - the coil rose about 5 to 10ºC when operating under ambient conditions. This design was also constrained with a small diameter 600kHz input power-feed cable (about 3mm for screened twisted pair) and the cable was about 10 metres long - having the 600kHz generator up close to the power coil is the way to get more efficiency in power-coupling.
Also, the efficiency wasn't that great because of the 40mm gap and the need to run at temperatures approaching 100ºC.
Summary - yes, you can transmit power magnetically across a gap and having a full loop on both the primary and secondary will couple much better than the fractional coupling described above. Having a smaller gap also couples much better. The receive loop described above had a single turn because it was also coupling data back out from the turbine mounted electronics - more turns on this winding would help - try and match the input coil profile and turns with the receive profile and turns. Use of ferrite material was impractical on this particular job but they would help.