energy over change in time is given by the equation:where water mass and gravitational acceleration are constant. We can use the channel slope and the stream velocity as a stand-in for : the water will lose elevation at a rate given by the downward component of velocity . For a channel slope (as measured from the horizontal) of :where is the downstream flow velocity. It is noted that for small angles, . Rewriting the first equation, we now have:Remembering that power is energy per time and using the equivalence between work against the bed and loss in potential energy, we can write:Finally, we know that mass is equal to density times volume. From this, we can rewrite the mass on the right hand sidewhere is the channel length, is the channel width (breadth), and is the channel depth (height). We use the definition of dischargewhere is the cross-sectional area, which can often be reasonably approximated as a rectangle with the characteristic width and depth. This absorbs velocity, width, and depth. We define stream power per unit channel length, so that term goes to 1, and the derivation is complete. Various Forms (Total) Stream power Stream power is the rate of energy dissipation against the bed and banks of a river or stream per unit downstream length. It is given by the equation:where Ω is the stream power, ρ is the density of water (1000 kg/m3), g is acceleration due to gravity (9.8 m/s2), Q is discharge (m3/s), and S is the channel slope. Total Stream Power Total stream power often refers simply to stream power, but some authors use it as the rate of energy dissipation against the bed and banks of a river or stream per entire stream length. It is given by the equation:where Ω is the stream power, per unit downstream length and L is the length of the stream. Unit (or Specific) Stream power Unit stream power is stream power per unit channel width, and is given by the equation:where ω is the unit stream power, and b is the width of the channel. Normalizing the stream power by the width of the river allows for a better comparison between rivers of various widths. This also provides a better estimation of the sediment carrying capacity of the river as wide rivers with high stream power are exerting less force per surface area than a narrow river