Right now merfolks are totally envious of us watching a live stream video of cat(fight) via WiFi or satellite internet, ok ok I get it's not the speed but its latency... Urrrr. Nevermind that I sincerely hope that the merfolks can receive WiFi but obviously the signal doesn't penetrate far underwater, any solution to transmit a signal at least matching our WiFi in both strength and coverage without divine intervention and breaking a bank? Note: just work on the transmission of signal, my merfolks can take care of the transmitting and receiving ends themselves.
Sound is the most effective way to communicate across a vast expanse of ocean – travelling at a speed of five times greater under water than in the air - so it’s not surprising then to discover that Blue Whales have evolved the ability to communicate with sound across the water. ... Due to their solitary lifestyles, Blue Whales have evolved an exceptional way of speaking to one another across huge distances. As you would expect from the largest animal on the planet, Blue Whales have exceptionally deep voices and are able to be vocal at frequencies as low as 14 Hz - well below the ability of human hearing - with a volume greater than 180 decibels, which makes the Blue Whale the loudest animal on the planet. https://www.nationalgeographic.com.au/science/blue-whales-and-communication.aspx
Transmission of messages through a medium requires two things (1) the carrier wave and (2) the superimposed signal.
A carrier signal is a transmitted [...] pulse or wave at a steady base frequency of alternation on which information can be imposed by increasing signal strength, varying the base frequency, varying the wave phase, or other means. This variation is called modulation. With the advent of laser transmission over optical fiber media, a carrier can also be a laser-generated light beam on which information is imposed. https://searchnetworking.techtarget.com/definition/carrier-signal
Think of a child being pushed along in a buggy. The child will soon learn that if they are travelling across a cobbled or other rough surface whilst they are vocalising, the sound will 'vibrate' even if they keep singing the same note.
The merfolk gain the co-operation of Blue Whales in return for favours such as removing barnacles and parasitic fish.
Whilst on duty the whales emit bursts of sound at a constant pitch (the carrier wave) and the mers use a vibrator to superimpose the signal. This sound can travel across huge oceans and be picked up by resonators and the signal decoded.
Special acoustic modems that can successfully transmit digital data underwater have been developed. These modems convert digital data into special underwater sound signals that can be transmitted between two submerged submarines or between a submerged submarine and a surface ship. These digital signals can represent words and pictures, just as on land, allowing submarines to send and receive e-mail. Underwater acoustic modems are relatively slow compared to telephone or cable modems on land. https://dosits.org/people-and-sound/communication/how-is-sounds-used-to-transmit-data-underwater/
Is wireless streaming a requirement? If they just want to stream video from the internet, use cables. We already have a bunch of them across various oceans, and they can probably get less latency and higher speeds if they hijack one of those. WiFi routers are commonly backed by cables anyway...
Also, WiFi doesn't reach particularly far, relatively and practically speaking. A signal that reliably penetrate some 20-30m of water would replace WiFi for your merfolk. That is much more feasible than to get a wireless signal down to Atlantis to begin with. Light or low frequency sound seems like viable solutions here, though the speed might be rubbish.
If you are willing to ignore the opacity of walls then you could transmit signals via near UV frequency light. You would have a usable line of sight range of ~100s of meters (which exceeds most home WiFi units for example). For an example absorption graph have a look at this site. If the signal light is reasonably bright, and you paint your mer-house walls white or some other reflective colour then you could get around the line of sight issue as the signal could be detected after bouncing off a few walls (but would still need to keep your doors open). Do mer-people need doors in their houses? All this assumes that mer-folk don't see far into the UV.
Alternately you can use ultra-sound as a signal propagator. The information bandwidth will be much lower than with UV, but it appears that even at frequencies of 10^2-10^3 kHz, you could still transmit 10s to 100s of metes, including around corners. So as long as the mer-critters cant hear above a few 10s of kHz that shouldn't be a problem. (Do merm-men/maids use sonar?).
The idea here is that instead of moving through the water like sound or EM waves, the waves here are generated in the ocean floor in the Local Comms-Node, and come out to whomever wants to connect to the system. A simple tranciever connected to the I-Pod, Cephalopod or whatever would be dropped to the ocean floor (analogous to a Wi-Fi aerial), and you're a handshake away from an afternoon of newsfeeds, browsing Fishbook or the Deep-Trench web of minnow porn and government's dirty secrets:
The S-wave moves as a shear or transverse wave, so motion is perpendicular to the direction of wave propagation.
Propagation of a spherical S-wave in a 2d grid (empirical model):
Attribution: Wikipedia 2019 CCSAA License
There was a huge amount of research going into blue-green lasers and underwater transmission LOS with undersea repeaters for the US nuclear submarine program back in the late 80's early 90's - I knew a researcher through OGC who witnessed a keynote speaker at a conference on this topic get walked away from the podium in their way up to present and told that their research had just been classified and acquired by DOD for that programme.
The intent was to have a good, non RF comms system which was undetectable other than by intended recipients, high efficiency and efficacy, etc etc.
Maybe your merfolks have whopping big naturally-occurring Neodimium YAG lasers with one frequency doubling crystal inline grown into huge genetically engineered chambered nautilus to give off mid green beams at 2" wide and several zillion TeV...
If you have magic in this world, Æthernet connections. When I first wrote this as a comment, I was thinking in terms of our own technology, and thinking that would just be a slightly different name for wired connections. But... magic. It is what you define it to be, and it would make total sense for network connections through the Æther to be able to move around without having to worry about cables moving around within this world at least. There might not even need to be cables, as many universes use the Æther to communicate magically over long distances without any tech - but that detail is up to you.