Wave Velocity

The velocity v of a wave is related to its frequency f and wavelength λ by

v = fλ.

How is this possible?

Think of a long rope, fixed at one end to a wall. Hold the other end, pull the rope horizontal, and shake up and down quickly with your hand. You will create a wave and see this travelling to the wall.

The wave has a repeating pattern. The highest part is called the crest, the lowest part the trough. The length of each pattern is called the wavelength. For exampe, this is the distance from one crest to the next, or one trough to the next. The wavelength is usually represented by the symbol λ.

Look at a crest, say at A. This crest moves to the right with time. So it has a velocity. Let this be v.

The point on the rope at A actually does not move the the right at all. It is like an illusion. Imagine using a pencil to make a mark at A. This mark just moves up and down. When it goes down and comes back up again to the same position, it completes one cycle. The time it takes is called a period. Let this period be T. The number of times it moves up and down in one second is called the frequency. Let the frequency be f. We can calculate the frequency using

f = 1/T.

Lets start from the time when a crest is at A.

Lets put a pencil mark a short distance away on the rope at B, and a bit further on the next crest at C. After a short time, the crest moves to B, but the pencil mark at A just moves downwards. The crest at B happens because the pencil mark there has moved up to the top. So it is the co-ordinated motion between A and B that gives the appearance of a moving wave.

At the same time, the pencil mark at C would also move down, and crest at C would move to the right. So there is no more crest at C, and the only crest between A and C is at B. As time passes, the crest at B moves and finally reaches C. Exactly how long does this take?

Lets see. For the crest at C to re-appear, the pencil mark there must have moved down and come back up again. This just means one cycle, which takes one period. So it takes exactly one period for the crest to move from A to C!

We are now ready to calculate the velocity. The distance between A and C is the wavelength λ, and the time taken is the period T. So the velocity is

v = λ/T.

Since 1/T is the frequency, we finally have

v = fλ,

which is what we set out to prove.


Copyright 2010 by Kai Hock. All rights reserved.
Last updated: 21 May 2011.