Depth is defined as the distance between the surface of the river to the bottom (the river bed) this distance is measured in a straight vertical line. As you move from the source to the mouth the river depth increases. (which is shown in the Bradshaw model). This is because at the source the depth is small and shallow due to the river just starting, so theres little water in the river giving it less energy to change the depth of the channel by erosion. As you go downstream the volume/amount of water (also known as discharge) also increases because more and more tributaries join the main river. The increase of water makes the energy levels rise when theres moe energy there is more erosion (in the form of abrasion, attrition and hydraulic action). these types of erosion erode away at the river bed creating a deeper channel.
Whilst the depth varies downstream it also does so across stream. For example as you move from bank to bank the cross sectional area is curved because it is shallow at the banks and deeper towards the middle. This is because the velocity of the river. The water flows faster and causes more erosion, thus creating a deeper channel. When the velocity is higher and theres more friction deposition happens making the channel more shallow.
The depth across stream can also be affected by meanders, meanders are mainly found at the middle course of a river but can be found at the lower course as well. Meanders are affected by this is because on the outside bend of a meander the water flows faster. Due to the high velocity more erosion takes place creating a river cliff. Also forming a very deep river channel . On the inside bend water flow is much slower, the water can't support