Which equation expresses the velocity of shallow-water waves as the square root of gravity times depth?

Which equation expresses the velocity of shallow-water waves as the square root of gravity times depth?

• A. C = sqrt(g/L)
• B. C = sqrt(gd)
• C. C = L/T
• D. C = gd

Answer: (B)

In shallow-water waves the restoring force is gravity and the moving mass is the entire water depth acting together, so the wave speed depends on gravity and how deep the water is. In this regime the speed is c ≈ sqrt(g × depth). Since depth is d, the velocity becomes sqrt(gd). This matches the idea that speed increases with both gravity and how much water there is above the bottom, and it stays constant with wavelength when the water is shallow. Why the other forms don’t fit: sqrt(g/L) would have the wrong dimension and would imply dependence on wavelength, which isn’t the case in the shallow-water limit. L/T is simply a velocity with no reference to gravity or depth. gd has units of length squared per time squared, not a velocity. The correct expression, sqrt(gd), has the proper units of velocity and incorporates both gravity and depth.

In shallow-water waves the restoring force is gravity and the moving mass is the entire water depth acting together, so the wave speed depends on gravity and how deep the water is. In this regime the speed is c ≈ sqrt(g × depth). Since depth is d, the velocity becomes sqrt(gd). This matches the idea that speed increases with both gravity and how much water there is above the bottom, and it stays constant with wavelength when the water is shallow.

Why the other forms don’t fit: sqrt(g/L) would have the wrong dimension and would imply dependence on wavelength, which isn’t the case in the shallow-water limit. L/T is simply a velocity with no reference to gravity or depth. gd has units of length squared per time squared, not a velocity. The correct expression, sqrt(gd), has the proper units of velocity and incorporates both gravity and depth.