Wave Front

Wave Front

Tie one end of a rope to a pole, hold the other end, and shake up and down. You get a wave travelling along the rope. The velocity of the wave v, the frequency f, and the wavelength λ, are related by

v = fλ.

In the case of the rope wave, v could be the velocity of a crest on the wave, f is the frequency that a point on the rope moves up and down, and λ could be the distance between two adjacent crests. These quantities are simple enough to visualise.

What about, say, water wave?

Imagine throwing a stone into a shallow pond. This creates circular ripples that move out from the point where the stone hits the water. Lets start by asking what is the wavelength in this case. The water surface goes up and down along the ripple, so there are crests and troughs. Isn't the wavelength just the distance between adjacent crests?

Yes and no. The problem is - which distance? You see, a crest in the ripple forms a circle. So adjacent crests are concentric circles. Distance between them can mean distance from any point on one circle, to any point on the other. So there are many possible distances. The simplest choice is to take the smallest distance. We could think of it as just the spacing between the two circles, which is fine.

What if the waves are not circular ripples but have other shapes? If I hold a metre rule horizontally and push it in and out of the water surface, I would generate a wave that travels away from the ruler. The crests would form straight lines parallel to the ruler. The circle in the ripple above and the straight line here are called wavefronts. A wavefront is a line joining points on a wave at the same phase. Same phase means same part of the oscillation cycle - in this case the crest. Using this idea, we can also say that the wavelength is the distance between adjacent crests, along a line that is perpendicular to to the wavefront.

Finding the velocity of the water wave is also a problem. Unlike the rope wave which can only travel along the rope, water waves can travel in many directions on the surface of the pond. If we want to know the velocity of a ripple, which direction should we choose? The answer follows the same idea as the wavelength above. The wave velocity is the velocity of a crest along the direction that is perpendicular to the wavefront of that crest. For the circular ripple, this would be along a radius of the circle. For the straight wave generated by the ruler, this would be perpendicular to the straight wavefront.

I have suggested how the wavelength and wave velocity can be found with the help of wavefronts on the water surface. This is in two dimensions. What about sound wave and light wave in air, which is in three dimensions? In either case, the wavefront is a two dimensional surface. For example, if you give a shout, you generate a pressure wave. The crests of this wave expands outwards from your mouth in concentric spheres. These are the wavefronts. The wavelength and wave velocity are taken along a line that is perpendicular to a wavefront, just as for the water wave above.

The Physics Notes