Doors – Best way to cover bedroom doorway that has no door

Edit: Thanks for posting the photo, answer updated (see below).

Here is some general advice.

Since larger chunks of wood are missing, use wood filler to fill it in (found at any home improvement store). Use a putty knife to match the shape of the trim as closely as possible. After the filler cures, use sand paper to round it out.

You will also need to paint over the filler. Get a chip of the wood trim and take it with you to the home improvement store and get them to color-match the paint (you may also be able to pull a piece of the trim off the wall).

The other option would be to find similar trim at the store and replace the damaged piece. Some fancy trims can be tough to match, but what you have looks pretty basic.

For what it's worth, in the last place we rented, our rabbit used to gnaw on the door trim in our bedroom. We eventually trained the rabbit not to do that, but it took a while, and the damage got pretty bad. We repaired the trim using the above methods. We got our full security deposit back.

First off, a reality check; not trying to discourage you, just trying to set some expectations here:

• First off, the actual pressures you're dealing with are very small. Absolute pressure is measured in Pascals; 1Pa = .0001psi. Your 80dB sound is inducing pressure changes of approximately .2Pa = .00002psi (that's two hundred-thousandths of a psi).
• Second, the percentage changes you need to make to these small pressures are very large. The Bel scale is logarithmic, based on a "reference" pressure. One Bel - 10dB - represents a tenfold increase in pressure. So, a 90% reduction in the pressure of your 80dB sound only reduces it to 70dB; to get the sound level below 10dB (you or I would call that "silence"; in laboratory conditions the threshold for human hearing is 0dB) would require reducing the pressure differential by 99.99999%.
• This level of isolation is possible but extremely cost-prohibitive in most applications; million-dollar recording/mixing studio spaces are built with lower levels of soundproofing than this. You normally see this level of isolation in medical and sound engineering research labs, where they literally build a room within a room, suspended on vibration-dampening struts, with a near-vacuum maintained between outer and inner walls of the chamber.

The upshot of all of this is that, no, a reduction in sound of 80dB is just not possible on a $1000 budget.

Now, all that said, don't lose heart. First, you seem to have much more ability to make changes to your rented space than the average apartment dweller would have (though I'd double-check your rental agreement; if you're required to return the space to its original condition - or pay damages - when you leave, think carefully about studding in a new wall).

Second, much smaller reductions in SPL can make a very big difference. Soundproofing that reduces an outside noise by 20dB makes the noise sound only a fourth as loud. This is easily doable in your budget (add some sound-insulating material such as mass-loaded vinyl between the existing wall and an extra sheet of drywall or wall covering), add a sound-insulating curtain or three, etc and you'll start hearing results pretty quickly. Two layers of mass-loaded vinyl will reduce most sounds above 125Hz by 18dB just by itself.