How does the amount of water flowing over a rectangular weir change if the water depth above the weir doubles?

A rectangular weir is a structure placed across an open channel to measure or control the flow of water. It consists of a vertical plate with a rectangular opening over which water flows. The amount of water flowing, known as the flow rate (Q), over a rectangular weir is directly related to the water depth (H) above the weir crest. The weir crest is the bottom edge of the rectangular opening over which the water passes. This relationship is defined by the weir formula, which, in its simplified form for a rectangular weir, states that the flow rate is proportional to the weir length (L) and the water depth (H) raised to the power of 1.5 (or 3/2). Specifically, the formula is Q = C L H^(3/2), where C is a constant that includes the discharge coefficient and gravitational acceleration, and L is the effective length of the weir. The water depth, H, refers to the vertical distance from the weir crest to the free surface of the water upstream of the weir.
When the water depth above the weir (H) doubles, the change in flow rate (Q) can be calculated using this relationship. Let the initial water depth be H1 and the initial flow rate be Q1. So, Q1 is proportional to H1^(3/2). If the water depth doubles, the new water depth, H2, becomes 2 H1. The new flow rate, Q2, will then be proportional to H2^(3/2), which means Q2 is proportional to (2 H1)^(3/2).
Mathematically, this expands to Q2 being proportional to 2^(3/2) H1^(3/2).
The term 2^(3/2) can be expressed as 2 raised to the power of 1.5, which is equal to 2 multiplied by the square root of 2. The square root of 2 is approximately 1.414. Therefore, 2^(3/2) is approximately 2 1.414 = 2.828.
This means that if the water depth above the weir doubles, the new flow rate (Q2) will be approximately 2.828 times the original flow rate (Q1). The flow rate does not simply double; it increases by a factor of approximately 2.828 due to the exponential relationship between flow rate and water depth (H to the power of 1.5).