Your objective is to reach the Summit of Arkngthamz. If you have not yet gotten Katria's Journal, the ghost will tell you to get it and you will receive the objective:

(Optional) Retrieve Katria's Journal

Again, this journal in the platform at the center of the Abyss. On the lower lever, there are some more dead Falmers and also a chest, so feel free to explore this cavern further if you want some more loot.

(If you are not at this point yet, than see the previous part)

After making sure you have looted Katria's body for her journal, proceed on the ramp to the west (where lays the body of a dead Falmer) and through a low tunnel with some steaming pipes. The ghost of Katria follows you. Be ready for combat, as you will be attacked by some Dwarven Spider Workers that drop on you from above. Soon you reach a room flooded with water. Dive into the water to find a chest with some treasure and potions.

Emerge at the southwest corner of the room and climb the gate,  then climb the pipe and go north, west, and north, and jump onto the grates below. Go north and wait for gate to open. You will see some devices, and Katria will tell you that these are Kinetic Resonators. Just hit them and they will do whatever they are supposed to do. Hitting the Kinetic Resonators in the room will cause the gate to open

Proceed south through the unlocked gate, and up the stairs. Turn left (to the east) and up another flight of stairs, fighting some falmers and a Falmer Nightprowler along the way. Go south in the chamber and past a falmer tent, then go east along the edge of the wall. You will pass through a locked door on the side, but don't worry about it now. Fight some Falmer Skulkers in the corner, and keep going around on the ramp along the edge of the cavern. You will soon arrive at a small cave with some Chaurus Hunters and fledglings.

How to get Zephyr Bow

Exit through a narrow tunnel to the east and arrive at an area with some trees. Wait for Katria, who mentions that this is the spot she fell from. Notice a narrow tree log to the west. There is a bow on it. Walk on the log, careful not to fall (or use telekinesis if you have it), and pick up the unique bow.

See pictures below to help you find the location of Zephyr.

It is in the cave with this tree in it:
Zephyr Bow Location

near it you will find a log that looks like this.
Skyrim Zephyr Bow Log

You receive:

Zephyr

Fires 30% faster than a standard bow.

The Zephyr bow is a pretty powerful bow that fires faster than a normal bow. Note that the Zephyr cannot be disenchanted, so do not attempt to disenchant the Zephyr

Go north and then downwards through some stone arches into an area with water pouring onto a body of water below.  The quest marker should point to a spot on the floor, go stand there and wait for Katria. She tells you to hold up, she wants to talk to you. She tells you about the Tonal Lock. Simple, and very, very deadly.

This is a PUZZLE. See the next part of the walkthrough for the Arkngthamz Resonator Doors Puzzle Solution:
Lost to the Ages Guide - Tonal Lock Doors Combination