So, on the planet of Qualia, a tiny island of the coast of Da’Tora called Saluk has disobeyed Tora VII the Slaughterers rule, which is never a good idea, as you can guess from the name.
He stationed his fleet around the island, and decided to sink it under the sea, to show his enemies that he means serious business. Saluk is 7 miles by 3 mile, or 11 kilometers to 4 kilometers. The Qualians have tech equivalent to 1960s earth.
So, is there a way to sink an island under the sea?
Soil liquifaction. They use timed detonations on the sea floor, probably nuclear, to shake the island. This causes the island's sandy soil (heavily saturated by the annual monsoon season) to begin behaving as a liquid. As it settles and sloughs off it is reclaimed by the sea.
Excavating it with explosives is probably not a viable option. In 1965, the US Navy conducted a series of tests called Operation Sailor Hat. These were massive-scale conventional explosive detonations that were meant to mimic the blast effects of a nuclear warhead (only without all that pesky radiation). Each test consisted of 500 (short) tons of TNT, which for technical reasons was considered equivalent to a 1-kiloton nuclear bomb. You can see one of the resultant craters here:
It's big, but it's hardly on the scale of obliterating an island. Of course, 1 kT is not all that big as nukes go, but you begin to see the problem. Physically destroying the island would require an immense amount of explosives, whether conventional or nuclear. Digging it out by hand or machinery runs into a similar problem. If your island is 11km x 4km and you need to remove an average of 5m of material, that's 220 million cubic meters of stuff (mostly sand and dirt, but it could have some bedrock if you're unlucky). Compare that to the Sailor Hat detonations, which dug out around 30,000 cubic meters each.
If it's near shore and you want to reshape the map to demonstrate your power, there's always Alexander the Great's solution: make it a peninsula. Here's an aerial photo of the city of Tyre, ca. 1934:
Prior to Alexander's siege in 332 BC, that sandy area on the right didn't exist; Tyre was an island. Alexander's forces built a land bridge out to reach it, and the rest of the peninsula was gradually built up by deposition.
He raises the sea level!
The island in question is inside an inland sea! The level of the island is lower then that of the outside sea! When you make a hole in the land bridge you would sink it!
As shown bellow!
Depending on how fearsome of a reputation he wants and if the enemies of his people are downwind of the island, he can cause the island to disappear beneath the waves with 1960 era technology.
Not those piddly little nuclear devices mentioned in other answers, but this:
Castle Bravo
Castle Bravo was a thermonuclear device which, due to a misunderstanding of how lithium reacted under the staggering conditions in the secondary, achieved a yield of 15Mt, the largest nuclear device ever created by the United States. The device destroyed almost all the surface vegetation and structures of the surrounding atoll, and excavated a crater 1.8km in diameter.
Oh, wait. The island is 11km by 4km. Maybe the Russians can help:
Tsar Bomba "on the waaaaaay"
This little beauty achieved a yield of 50Mt, without its third stage (the tamper was made of lead rather than Uranium). The fireball alone was five miles in diameter, so an airburst will essentially vapourize anything on the surface of the island. I haven't found any reliable estimate as to the size crater a ground burst would create, but it would likely cover a fair fraction of the island. IF you were particularly bloody minded, use the Uranium tamper and bring the yield up to 100Mt.
"But that's hardly elegant", you protest. Well, the ever inventive Americans may have discovered ways to reach even higher yields (gigaton) without the gargantuan "steampunk" like design of the Tsar Bomba:
But given that high efficiency is tied to high yields — and relatively high weights — it’s clear that the innovations that allowed for the placing of warheads on MIRVed, submarine-launched platforms are still pretty impressive. The really magical range seems to be for weapons that in the hundred kiloton range (more than 100 kilotons but under a megaton), yet under 1,000 kilograms. Every one of those dates from after 1962, and probably involves the real breakthroughs in warhead design that were first used with the Operation Dominic test series (1962). This is the kind of strategic miniaturization that makes war planners happy.
Or, to take another tack, and returning to the initial impetus for me looking at this topic, we know that the famous “Tsar Bomba” of the Soviet Union weighed 27,000 kilograms and had a maximum yield of 100 Mt, giving it a yield-to-weight ratio of “only” 3.43 kilotons/kilograms. That’s pretty high, but not for a weapon that used so much fusion energy. It was clear to the Atomic Energy Commission that the Soviets had just scaled up a traditional H-bomb design and had not developed any new tricks. By contrast, the US was confident in 1961 that they could make a 100 Mt weapon that weighed around 13,600 kg (30,000 lb) — an impressive 7.35 kiloton/kilogram ratio, something well above the 6 kt/kg achieved maximum. By 1962, after the Dominic series, they thought they might be able to pull off 50 Mt in only a 4,500 kg (10,000 lb) package — a kind of ridiculous 11 kt/kg ratio. (In this estimate, they noted that the weapon might have an impractically large diameter as a result, perhaps because the secondary was spherical as opposed to cylindrical.) So we can see, without really knowing much about the US had in mind, that it was planning something very, very different from what the Soviets set off.
http://www.projectrho.com/public_html/rocket/spacegunconvent.php
Still not good enough?
To eradicate any trace of the island (or indeed any nearby continents), the most powerful single "plausible" device was outlined by Anthony Zuppero, who was apparently asked to do the calculation for a device which would wipe out the Soviet rocket fields in Siberia in a two minute window. He evidently calculated a massive 5Gt weapon launched by an ORION nuclear pulse drive...
Doomsday ORION weapon, artwork by William Black
Saluk is 7 miles by 3 mile, or 11 kilometers to 4 kilometers. The Qualians have tech equivalent to 1960s earth.
So, is there a way to sink an island under the sea?
Yes, but it all depends on what the island is made of.
Solid rock: you'd need an impossibly large number of nuclear explosions. Much better at that point to simply blast the island's surface with three megaton-size explosions.
Silt and sand: Saluk is not an "island" but rather a "char". A sufficiently prolonged, sufficiently large series of explosions can cause soil liquefaction.
Ancient volcanic caldera: you would need the caldera to be already cracked obliquely, in such a way that a reasonably small (< 20) number of megaton-size blasts would trigger a flank collapse.
If the "sea" is an inland sea and the conditions allow it, it might be possible to stage a spectacular apocalypse on the scale of the Zanclean flood (also on xkcd).
Couple of concepts to work with here... the Slaughterer would probably have gotten that name by being brutal, and drowning everyone would certainly work for that name, however does he want to sink the island or the people. or does the would sink need to be meant literally. If he were to "sink" the island, but the island be fine, it has the same end result and still has the same effect but means he can have the island as well.
Tidal Wave
A tidal wave would do it, the island could be washed away, the people drowned and infrastructure ruined, seems pretty close to sinking to me. Now these are not easy at all, but with the right geography around the island, its still possible even with 1960s tech, they could either cause a cliff face on a neighboring island to collapse, (research Lituya Bay Alaska mega tsunami) or if the was a fault line nearby, then maybe, i stress maybe a carefully placed nuke on this fault line would cause it to shift enough that it could generate a tidal wave.
Visit Saluk, and explore our miles of caves and tunnels
If the island was riddled with underground caverns and tunnel systems then a lot of the work would already be done for the bombs, a single nuke, or some well placed conventionally explosives in various points throughout the cave system would likely make the ground unstable and cause it to collapse, bringing it below sea level and then the water would just need to flood in.
for example: Son Doong Cave, Vietnam is a cave passage, which means it is a single "room" of more than 5 kilometres (3.1 mi) long, 200 metres (660 ft) high and 150 metres (490 ft) wide. if the island had something like this, maybe it is already an underwater cave, but has other caves around the island all interconnected then maybe it would be enough, the entire island wouldn't disappear completely underwater, but enough that what was left could be nothing more then a bunch of rocky islets
How Much does the baddie want to destroy
The Final thing to think about is what does the Slaughterer want to actually sink?
Saluk is 7 miles by 3 mile, or 11 kilometers to 4 kilometers.
Is Saluk like Manhatten? in that the entire island is built up and a city within itself, if so then he could destroy them through starvation before anything else, and it makes actually sinking it harder... but if that is not the case then, all he has to do is sink the main town, still digging or blowing up enough bedrock is going to be hard, but a lot easier then sinking the entire island.
Just use hydraulics. Drill into the "core" of the island and pump high pressure sea water in. The pressure will expand and extend any fractures in the rock until the water finds a way out.
Just repeat this enough and there will not be enough solid stone to support the island and it will slide of into the sea in massive land slides. These land slides will cause tsunamis but same is true of all ways to sink an island. Just be careful to sell all of your nearby beach front properties before sinking islands, ok?
If you want the island to vanish in a single cataclysm rather than a series of land slides, you are going to need good data on the geology of the island and precise modelling and planning of the process. You can combine this with soil liquefaction mentioned by other answers for simpler control of the collapse timing and extent.
The drilling and hydraulics can be masked as oil exploration. So with enough planning this could be done discreetly. "Who claims I am not the Chosen of God? My enemies lie in the bottom of the sea with their very island shattered by Gods hand. Who claims I am not the Chosen of God?"
If the island does not happen to be situated above a geological feature that can be exploited (volcano, water reactive(but still dry) minerals, steep and deep cliff), they'll just have to start digging: The island is 44km², so at an average height of 100m, that's 4.4km³ that need to be removed to make the water just lap over the whole thing. That volume can be removed in about 5 years, if copious slave labor is available, which i bet is the case... https://en.m.wikipedia.org/wiki/White_Sea–Baltic_Canal
Rather than raise the ocean to sink the island, you could employ some weather manipulation to hit the island with a series of large hurricanes to wash it away.
In the 1950's scientists studied the use of nuclear weapons to steer or even destroy hurricanes.
http://atomic-skies.blogspot.com/2013/09/nuking-hurricanes.html
This would definitely be in line with 1960's era technology. In our world the treaty banning aboveground nuclear testing in 1963 ended this line of research, but it is still considered highly likely to work.; with possible unintended consequences including; making the hurricane stronger rather than weaker, stearing the storm in the wrong direction causing a worse landfall location, dissipating one storm only to have a larger follow up storm, and of course the addition of some radioavtive contamination to the area.
A hurricane level storm can be already be devestating to an island. A series of them all steared to hit the island in succession could literally uproot all vegetation, knock down all the structures, and wash it all into the sea.
