Pardon my ignorance on everything naval, but isn't a defined stretch of water before the hull to get at the underwater hull, a rather narrow target? Also wouldn't bulging and other anti-torpedo protection do something to minimize the effect of a hit?

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Depends a lot on the distance or actually on the decent angle of the shell. At 45 degrees its as wide as the ship is deep at lower angles it's wider. Note that said shells are still designed to be effective if they hit head on, although as noted there is something of a decrease in effectiveness. A shell that can penetrated a battlships belt armor isn't going to be effected too much by the TDS or bulges. A further note is that once the angle of decent gets low enough the shell may skip rather than "dive" so the band dissapears at that point.

The Bismarck had two seetakt 53cm wavelength radars on board mounted on the faces of the primary and secondary main battery rangefinders. These radars would give a reasonable range to the target but were essentially useless in bearing data being too wide beamed to be able to sufficently resolve bearing to a useful level. Additionally, they were not tied into the fire control system so their data had to be manually entered. They do make a useful backup to the optical rangetakers but that is about it.

The Yamato did not get a useful fire control radar until early in 1944 when some of the first production 2 Gô 2 Gata 4 Kai S sets became available. This set operated at 10 cm and gave excellent range data and fair bearing data. But, like the Seetakt set it was not tied into the ship's fire control system so any use of it had to be manually input. An additional problem was that the radar operator was not co-located with the optical rangetakers meaning that communications and coordination would be more difficult.

So, essentially both battleships would be relying on optical rangetaking for fire control with radar playing a role of giving a second set of range data to back up the optical inputs.

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On the Yamato there was a deliberate attempt to minimize the diving shell threat in the torpedo defense system. This was done by carrying the belt down to the double bottom of the ship and slanting it inward to create a fairly deep defense system. The Japanese erred in leaving this space void instead of liquid loaded however. With the Bismarck because the ship has a fairly wide beam the torpedo defense system is quite deep and there is an internal bulkhead at the back of the system that is fairly substancial. The Germans also have their system liquid loaded.
In both ships the system used is sufficently deep that most diving shells should fail to penetrate the defense system. In the Prince of Wales' case the one hit on her by Bismarck below the waterline failed to penetrate her relatively shallow defense system and instead caused flooding in the torpedo defense system itself.
The main reason the shells lack a large amount of penetration is simply that their velocity decreases extremely rapidly upon entering the water. Even following a reliable underwater trajectory, the Yamato's shells will not travel all that far after entering the water and will lack the penetrative power of an above waterline hit.

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first, i believe it was the fuso that emerged from the straight only to be destroyed by US BBs in classic line formation. the yamashiro was torpedoed while making the passage.

now this is where my knowledge of radar-assisted fire ends. beyond visibility during the day and at night, can fire controll radars detect the report from the last shot? it could be a direct hit, a dud, or a miss, producing your 150-foot water fountain. i want to know since, if you don't see the report on the first shot, you can't fire a second.

I believe you have it reverse there mac. Fuso was sunk by torpedoes from USS Melvin (so the popular story goes) inside the Straight, Yamashiro was hit by virtually everyone as she attempted to emerge from the East side.

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i'll take your word for it for now. the "turkey trots to water" article HAD it in reverse. and yes, the maryland managed to pick up west virginia's shell splashes on her radar.

This weekend I'll hook up a new scanner I got. I will then put up some pictures of what targets and shell splashes look like on the US Mk 8 FC radar scope. Unfortunately, I don't have any for Seetakt or the 2 Gô 2 Gata sets but I know these used A scopes rather than a PPI type display.

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Here's a link to a good account of the ends of the Fuso and Yamashiro.
FUSO
World war two radar in it's most advanced state could detect shell splashes well beyond 30,000 yards but I believe could "detect" hits only by the absence of the splashes.
There's an analysis of the US BB gunnery during that action at:
Performance of US Battleships at Surigao Strait
that's also worth taking a look at.

AS for diving shells apparently it's a lot more complex than I thought. See the thread at:
Type 91 diving shell

There are a lot of 'fans' here of Bismarck here, which I believe (innocently, of course) is clouding the judgement of a few people.

Bottom line? Yamato and Musashi were designed with the intent of engaging entire squadrons of battleships simultaneously. That was the whole point of the 'super battleship' - to combine the equivalent performance characteristics of smaller ships into one very large juggernaught.

Were this question 'USS Iowa vs. Yamato,' I would lean a very different direction, but the fact of the matter is that Bismarck and Tirpitz, despite being 'newer' ships, were hardly the triumphs of design that Scharnhorst and Gneisenau were. Germany's two behemoths were basically just an enlarged, modernized Bayern-class.

All things being equal, Bismarck and Tirpitz were precisely the type of 'old design' foes that Yamato and Mushashi had been built to engage. As pointed out, Bismarck was heavily reliant on optical (as opposed to radar directed) range finding, and her basic armor scheme, muzzle velocity, shell size and weight were all adequate (but hardly outstanding) for their day. The legend of Bismarck and the havok that chasing her across the Atlantic caused far outweigh the real military value of the ship itself.

Had Bismarck caught, say, HMS Nelson alone, Bismarck would have likely won, should she have chosen to fight. British gunnery, other than that of Jelicoe's Grand Fleet, was notoriously bad since the advent of Dreadnought. This is the type of opponent that Bismarck could have caught and killed (although, honestly, this is highly unlikely in of itself. Bismarck was built as a commerce raider, capable of breaking up convoys and outrunning older battleships. She [or 'he,' as the Germans preferred], would have likely run even from an elderly Revenge-class ship simply to avoid the innevitable - however light - damage).

Bismarck vs. Yamato? This one easily goes to Yamato. I am a massive fan of German warship design, the fighting spirit of German crews and the traditionally high accuracy of German gunnery, but the sad fact of the matter is that Yamato was out of Bismarck's class, and barring a catostrophically lucky shot (and the ensuing magazine explosion), Bismarck is pounded to a crisp, charred pile of twisted metal before she can respond with enough weight of shell to inflict similar damage to Yamato.

Here are a few more interesting lineups of 'what if vs. what ifs':

SMS Derfflinger vs. HMS Queen Elizabeth (Vintage WWI. The Derfflingers were better gun platforms, faster, had armor and internal subdivision that essentially made them fast battleships, had reliable shells, and probably shot better. The QEs were larger and had far bigger guns.)

Bismarck vs. HMS Vanguard(On paper, very similar ships)

Yamato vs. U.S.S. Iowa (Again, Yamato is larger, but this time is opposed by an American with absolutely superior range-finding, and an all-or-nothing armor scheme that lets him likely remain in the fight longer than Bismarck could have, and larger guns)

OpenDNS vs. OpenDNS (Wow. This is a fanboy's playground)

I don't disagree with most of what you say, but I think you're wrong on this one. I'm currently reading Friedman's new book "Firepower", which focuses mainly on battleship fire control systems. This includes a comparative analysis of the performance of Bismarck and the RN ships which engaged her, and concludes that Bismarck's shooting was not as good. I haven't reached the end of the book yet, but the impression I'm forming was that the then state-of-the-art RN system was the best surface fire control system in service (but not all of the British battleships had the latest system).

Unfortunately, the RN's high-angle AA fire control systems were markedly inferior to the US Mk 37 director, but that's another matter.

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Yamato, with bigger guns, and superior armor would decimate Bismark hands down. KaiserWilhelm makes a good point, Yamato and Musashi were constructed to fight and destroy other battleships, while Bismark was more suited to a surface raiding role due to her rather small guns 15in. which were totally outclassed by the American 16in/50 rifled gun and the massive 18in. type 91. Also Yamato would out range the Bismark by several thousand yards, so she could get off several shots before Bismark could even get in range, and if one of those shots hit the Bismark then she would have lit up like a Christmas tree. If the Bismark was not already destroyed when she got in range of the Yamato even if she hit the Yamato she could never penatrate the Yamato's turret armor. Japanese radar was reliable and duralbe, and withstood shock well beacause it was cruder and lacked as many sensitive advanced parts. Therefore, even if the ship the Yamato somehow was sinking, it could still probally fire because the fire contorl would probably not have been taken (there is also the fact that the fire control was way up on the massive pagoda structure).
I vote Yamato

While I agree Yamato is going to win this one most of the time.I don't think I'd call 15" small. There were advantages to the German guns. Against earlier 16" rounds they actually look pretty comparable. It's only when you start comparing them to the US "super heavies" and of course Yamato's 18" rounds that they start to look a touch week.
The range of the guns was more or less irrelevant. Neither had a fire control system that would allow them to have a good chance of hitting at over 30,000 yards. There was also a lot of Bismark that wasn't particularly critial area so while an 18" round would hurt it wouldn't necessarily "light" her up.
There's a lot more to Yamato than turret armor. It's also possibe for Bismark to peneterate said turret armor. The turret face is theoretically impossible for her to penetrate but even there those calculations often don't include things like multiple hits and edge effects.
Where did you get the data on the reliability of Japanese radar. I would think, by the way, that being high up in the sturcture would in some ways be more problematic.