Rivers do more work shaping the land than any other agent. Two numbers describe a stream: how steep it is and how much water it carries. In Landforms Made by Water you met valleys and floodplains; here we measure the water that makes them, keeping careful track of units.
Why it works
Gradient is the drop in elevation per distance along the channel, often in meters per kilometer: steep near the source, gentle near the mouth. Discharge is the volume of water passing a point each second. Think of the water crossing a gauging line in one second: it fills a box as wide as the channel ($w$), as deep as the water ($d$), and as long as the distance the water travels in that second ($v$). So $Q = w \times d \times v$, in cubic meters per second. Discharge grows downstream as tributaries join.
Key idea
Gradient = drop ÷ distance. Discharge $Q = w \times d \times v$ (cross-section area × average speed), in m³/s.
Worked examples
Worked example: Discharge at a gauge
A river is 25 m wide and 2 m deep on average, flowing 1.2 m/s. Find its discharge.
- Area: $25 \times 2 = 50\,\text{m}^2$.
- $Q = 50 \times 1.2 = 60\,\text{m}^3/\text{s}$, enough to fill an Olympic pool in under a minute.
Answer: $60\,\text{m}^3/\text{s}$.
Worked example: Gradient
A mountain stream drops 90 m in 3 km. A large river drops 30 m in 100 km. Compare their gradients.
- Stream: $90 \div 3 = 30\,\text{m/km}$.
- River: $30 \div 100 = 0.3\,\text{m/km}$, a hundred times gentler.
Answer: 30 m/km and 0.3 m/km.
USGS gauging stations measure thousands of U.S. rivers around the clock. They record the water level (the stage) and turn it into discharge with a rating curve, built from many hands-on measurements of width, depth and speed.
Common mistake
A steep, foaming mountain stream looks faster and mightier than a big, calm river downstream. In fact the river usually carries far more water, and its average speed is often as high or higher: the stream loses much of its energy to friction against boulders in a shallow, rough bed. Check with $Q = wdv$: a wide, deep channel wins easily.