# Physics of waves under an ice road

As wikipedia writes,

While easier to drive across in the winter than land, roads over water present a great danger to anyone using them. Speeds are typically limited to 25 km/h (16 mph) to prevent a truck's weight from causing waves under the surface.

and also

It is advised to avoid the range of 25 and 40 km/h (16 and 25 mph) due to danger of creating resonance in the ice layer (i.e. vehicle speed and water wave speed being the same or nearly, resulting in a large wave under the ice that breaks the ice).

However, the speed of sound in ice is $4\cdot 10^3\text{m/s}\simeq 14.4\cdot 10^3\text{km/h}$.

What is the physical mechanism behind the formation of these waves, and why is the dangerous-velocity range so much different from the speed of sound?

• The wave is in the surface of the water, not in the body of the ice. Just like a boat makes a wave, or jumping on a floating dock makes a wave, the moving vehicle makes a wave. If the wave speed and vehicle speed match, then the wave gets bigger and bigger until the ice fractures. – Daniel Griscom Jan 24 '16 at 20:02

The wikipedia article specifically mentions water waves, not sound waves. The speed of a wave in water is approximately $v = \sqrt{\frac{g\lambda}{2\pi} \tanh \left(2\pi\frac d \lambda\right)}$, where $g$ is the acceleration due to gravity, $d$ is the depth of the water, and $\lambda$ is the wavelength of the wave. For shallow water ($d \ll \lambda$) this simplifies to $\sqrt{gd}$. In the range of 25 to 40 km/h for water, this corresponds to depths between 5 and 12.5 meters deep, which are very typical depths for the roads over frozen rivers and lakes.