What I expected to happen:
The primed TNT would stay within the enclosed/blocked off cannon and blow it up.
What happened:
The primed TNT was blasted out THROUGH the cobblestone and out of the cannon.
How to replicate:
Build cannon with steps shown below, fill the two dispensers which are facing each other with TNT, fill the one other dispenser with TNT minecarts, flip the lever until all the minecarts have been dispensed onto the rails within the device, push the button.
Evidence:
The video attached (poor quality), the same video in better quality: https://www.youtube.com/watch?v=_3YaQI2lQqE
Related issues
is duplicated by
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These screenshots don't show the bug that is described. It only shows the contraption that is used. Please use a screenshot that ACTUALLY SHOWS the bug being performed.

Now I have an image of the glitch taking place,

I'm sorry, but the image that was posted doesn't clearly show the bug taking place, as the TNT just exploded. Please make a screenshot showing the bug taking place before the TNT explodes.
EDIT: Found the issue. For the other blocks, you're using a powered rail, a sign, a ladder, and water. The powered rail, sign, and ladder act as a block, but the water does not. Therefore, the primed TNT will glitch because those things act as a block.
This works as intended.

I believe there has been a misunderstanding, minecart tnt explodes, forcing primed tnt through a cobblestone block as shown in the video

I apologize for the poor quality of the video yet do not know how to improve it as the file plays perfectly on my computer yet not as an attachment here.

here is the URL for the youtube video which shows the footage with greater clarity : https://www.youtube.com/watch?v=_3YaQI2lQqE&hd=1

Checked the video, and it appears that the cannon works perfectly fine.
Invalid.

That's the whole point!! It shouldn't, it is SEALED.

Chali, I don't know what your talking about. All I said was that the cannon has no issues, and the primed TNT is supposed to "glitch" through the block due to it being fired by a cannon.
Intended/Invalid.

No, the point is, it is a glitch because the cannon is entirely encased in cobblestone yet the tnt still fires out of it

It being the cannon.

Tested and confirmed.
I have tested this cannon in multiple versions and the problem only exists in Minecraft 14w17a.
I have also tested this cannon ussing different materials and they all have the same result.
BTW why is it so important that you can seal the cannon without the TNT phasing though?

It is important because the tnt takes longer to fly in an arc than straight, therefore when loading the cannon with a greater charge of tnt minecarts, the tnt is propelled in a larger arc which it can't complete before it explodes. If however the tnt cannot phase through blocks, then on the real version of the cannon with an open ended barrel, the tnt is forced straight out taking a shorter time to cover the same distance. Thus when adding a greater charge, the tnt can go further if it does not glitch through blocks. I have done this in the past and when loading 27 minecarts with tnt into it, the tnt shot out the barrel (two blocks above ground) and flew 257 blocks before exploding on the ground. On the other hand now that the tnt glitches through, the cannon's max range is under 100 blocks as the larges number of tnt minecarts you can load it with is around 11 before the tnt explodes in the air.

Oh... Okay.
Thanks for letting me know.

Cannot confirm for 15w36d but please recheck

Related if not even duplicated by MC-12730

Can't reproduce
Cannot reproduce either; it destroys the cannon completely. As it's intended.

I see this in 1.12.2 with half slabs sometimes.

I can confirm this in 20w12a

Confirmed in 21w10a