Hodograph

Hodograph

[hod-uh-graf, -grahf]
A hodograph is a diagram that gives a vectorial visual representation of the movement of a body or a fluid. It is the locus of one end of a variable vector, with the other end fixed. The position of any plotted data on such a diagram is proportional to the velocity of the moving particle. It is also called a velocity diagram. It appears to have been used by James Bradley, but its practical development is mainly from Sir William Rowan Hamilton, who published an account of it in the Proceedings of the Royal Irish Academy in 1846.

Applications

It is used in physics, astronomy and fluid mechanics to plot deformation of material, motion of planet or any other data that involves the velocities of different part of a body.

See Swinging Atwood's machine

Meteorology

In meteorology, hodographs are used to plot winds from sounding of the Earth's atmosphere. It is a polar diagram where wind direction is indicated by the angle from the center axis and its strength by the distance from the center. In the figure to the right, at the bottom one finds values of wind at 4 heights above ground. They are plotted by the vectors vec V_0 to vec V_4. One has to notice that direction are plotted as mentioned is the upper right corner.

With the hodograph and thermodynamic diagrams like the tephigram, meteorologists can calculate:

  • Wind shear: The lines uniting the extremities of successive vectors represent the variation in direction and value of the wind in a layer of the atmosphere. Wind shear is important information in the development of thunderstorms and future evolution of wind at these levels.
  • Turbulence: wind shear indicate the possible turbulence that would cause a hazard to aviation.
  • Temperature advection: change of temperature in a layer of air can be calculated by the direction of the wind at that level and the direction of the wind shear with the next level. In the northern hemisphere, warm air is to the right of a wind shear between levels in the atmosphere. The opposite is true in the southern one (see thermal wind). So in the example hodograph, the wind vec V_3 from southwest meet the right side of the wind shear which means a warm advection and thus warming of the air at that level.

Further reading

  • Feynman's Lost Lecture — The Motion of Planets Around the Sun by David L. Goodstein & Judith R. Goodstein (ISBN 0-393-03918-8, W.W.Norton & Company: New York, 1996). In this book the hodograph is used to geometrically derive elliptical (Keplerian) orbits from Newton's laws of motion and gravitation.

See also

  • Visual Calculus A related approach useful in solving a variety of integral calculus problems.

External links

References

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