Water, Wakes, Mountains, Trees and Sky Pools
June 30, 2014
On a recent trip to Glacier National Park I took a tour on Lake McDonald aboard the vessel DeSmet. I was particularly fascinated by the distorted reflections in the wake and waves produced by the boat. The reflections were of distant mountains on one side and of the nearer trees on the other. The overall wave pattern produced at the rear of the boat is often referred to as the Kelvin ship wave pattern, named after William Thomson, Lord Kelvin, who, in the 1880’s, was among the first to investigate them mathematically. They’re not to be confused with Kelvin waves which have to do with the Earth's Coriolis force.
A duck swimming in deep water produces a similar Kelvin wedge, or V-shaped angle of approximately 39 degrees. These patterns consist of two main components: the feathery oblique waves angled at approximately 35 degrees to the arms of the V, and the transverse waves that directly follow the boat (or duck). The similarity of a Kelvin wedge to a Mach cone in acoustics is not coincidental. The ship wave motif is a consequence of the fact that water waves are dispersive that is, their wave speed is wavelength-dependent. In addition, the speed at which their energy travels, known as the group speed, is half the wave speed.
The continuously changing curvature of the surface of the water produces fascinating distorted reflections of the features surrounding the lake, and is explained in part by geometrical optics. In the picture above, the scene shows a distant mountain on the east side of the calm lake. The pictures at left show parts of the scene reflected in the surface undulations associated with the ship waves. Corresponding reflections of trees on the much nearer western side are shown in these two photos. Very similar optical considerations arise in describing sky pools as shown here also on the surface of Lake McDonald.
Photo details: Top - Camera Maker: OLYMPUS IMAGING CORP.; Camera Model: SP570UZ; Focal Length: 4.6mm; Aperture: f/5.6; Exposure Time: 0.0080 s (1/125); ISO equiv: 64.