杏吧原创

Making waves

Question: Why does a boat鈥檚 wake last so long in a fluid as 鈥渞unny鈥� as water?
If you go to the Lake District and climb down Cat鈥檚 Bells, you can look down on
Derwent Water and watch the ferries. Each wake is many times the length of the
boat producing it and lasts for a long time.

Answer: The fact that water is a 鈥渞unny鈥� (not viscous) liquid does not
prevent it from forming a moving wave. Because low viscosity means little
friction and consequently little damping, it is in fact neccessary for a wave to
be able to propagate. When a medium is viscous, damping is too strong and a wave
moves slowly. Moreover, its amplitude decreases quickly and so the wave dies out
fast. If the viscosity is very high, then the wave that appears is almost
stationary and disappears quickly鈥攜ou see this in honey if you move a
teaspoon through it.

When the surface of water is hit, say by a stone thrown into a pond, a
circular wave front starts spreading at constant speed from the point of
contact. As the ship is moving and hitting more and more points on the water鈥檚
surface, it is continuously producing new circular wavefronts from points which
are outside the previous circular waves.

These circles intersect, and in accordance with Huygens鈥檚 principle (which
says the resulting wavefront is an envelope of the elementary circular
wavefronts), the resulting wave adopts a V-shape. The original circular waves
are closely packed in this V. This resulting V-shape wave is dragged behind the
ship and thus moves forward at the same speed. It is able to exist for so long
because it is quite high when it is formed from the numerous elementary circular
waves which contribute to its amplitude at the points where they intersect.

Radek Pelc

Hostivice u Prahy, Czech Republic

Answer: The longevity of boat wakes is surprising only because we are used to
seeing waves formed by pebbles thrown into water. These are the circular waves
on the surface of water that move away from the point where the water was
disturbed. They appear to die away relatively quickly. This is because circular
waves increase in length as they travel, so the energy of each crest has to be
shared over a greater and greater length as the wave spreads.

A boat creates a pair of plane waves as it moves through water, one each side
of the boat and moving away from it. A plane wave travels in one direction, at
right angles to the crest. Because water is runny, the rate of conversion of
kinetic energy to heat energy is slow. So as the wave moves, its amplitude
decreases relatively slowly. The decay of amplitude with time is approximately
exponential.

Michael Brimicombe

Aylesbury, Buckinghamshire

Wakey wakey

Question: From casual observation, it seems the angle of a boat鈥檚 wake is the
same regardless of the vessel鈥檚 speed and size. So what determines the angle of
the wake?

Answer: Almost anything from a swan to an oil tanker moving with constant
velocity over deep water produces a disturbance known as the Kelvin ship wave
pattern. Irrespective of the speed of the object, the wake is wholly contained
within a wedge of semi-angle 19.5 掳. Within this wedge, the structure of the
wake depends on the shape and size of the hull and the ship鈥檚 speed.

The wave pattern at any one time is created by the interference of all the
waves produced at previous points on the ship鈥檚 path. If all of these waves are
added together, taking account of their direction and phase, it is found that
they interfere constructively only within the wedge of the Kelvin pattern.
Outside the wedge they cancel each other out, leaving calm water.

You may wonder why the wedge angle does not depend on the speed of the ship
in the same way that the shock wave of a supersonic plane takes a more acute
angle as the speed of the plane increases. The different types of pattern arise
from the fact that the speed of sound does not depend on wavelength, while the
speed of waves on deep water increases with increasing wavelength. In deep water
a ship creates waves across the whole spectrum of wavelengths (and therefore
speeds) and it cannot outstrip them in the same way that a supersonic plane can
outstrip the sound that it creates. In shallow water the waves behave more like
sound waves, and the angle of the wake sharpens at higher speeds. You can
demonstrate this effect by dragging your finger through a shallow layer of water
on a tray.

Peter Macgregor

Greenock, Strathclyde

Many readers have pointed out that the wake angle appears to change
dramatically when a speedboat accelerates rapidly. It is important to realise
that the Kelvin pattern occurs only in deep water at constant velocity. A fuller
explanation provided by the above author, showing the derivation of the Kelvin
angle and the varying wake angle produced with 鈥渟ound-like鈥� waves, can be found
at www.newscientist.com/ns/19990605/lastword.html鈥抬诲.

We have a problem

Question: While watching the film Apollo 13 recently I began to
wonder what system of coordinates was used for navigation in space journeys
between the Moon and the Earth. Terrestrial concepts such as north and south
must be irrelevant, but all the reference points such as the Earth, Moon and Sun
are moving relative to each other. So how do astronauts know they are on
course?

Answer: On a trip to the Moon, astronauts navigated the same way Christopher
Columbus and the early explorers did: by the stars. Although the stars move in
their own local orbits, they are so far away that for all intents and purposes,
they are fixed. By triangulating off a few stars the astronauts could accurately
determine the spacecraft鈥檚 orientation and heading.

However, because the stars are so far away, and a spacecraft鈥檚 motion so
small, the stars could not be used to determine the absolute position. This had
to be determined by 鈥渄ead reckoning鈥�: using time and speed to calculate
distance. NASA also used the spacecraft radio signal to determine the angle from
the ground receiver to the spacecraft and the timing of that signal for distance
determination.

Stephen Claeys

Niagara Falls, New York

This week鈥檚 question

Weird spheres: What are the metal spheres, about 20 centimetres in diameter,
that are fixed at regularly intervals along high-voltage power lines in
France?

Vivienne Thomas

London