Standing Waves

What forms when you pluck a guitar string? What forms on the rope when you play jumping rope? Standing waves as it is called are said to be waves that are standing still. These are created due to the interference of waves.

When a traveling wave is blocked, it will be reflected back and forth, thus creating interference and then, standing waves are created. Standing waves are two waves having the same frequency, the same amplitude but are moving in opposite direction. It has two areas – constructive and destructive. The areas that are not moving are called destructive, causing the formation of nodes, and the constructive is the moving area forming antinodes. The antinodes move to maximum amplitude.

The distance from one node to the other is half the wavelength, this means that the length of the string is equal to \(\frac {\lambda}{2}\) and the wavelength is twice the length of the string, that is \(\lambda=2L\).


Example 1.

Calculate the wavelength of the standing wave as shown below.

Solution.

Given the length of the string \(5.0\;m\) which is half the wavelength. We use the equation \(\lambda=2L\) to solve for the wavelength.

\( \lambda=2 \times 5.0\;m=10.0\;m\)


Example 2.

A standing wave with 4 nodes has a wavelength of \(50\;m\). What is the length of the string?

Solution 1:

The length of the string is the distance from node to node. Given that the length of the string is half the wavelength, \(L=\frac{\lambda}{2}\), having 4 nodes means 3 string lengths, hence, the length of the wave is given as

\(L=\frac {\lambda}{2}(3)=\frac {3}{2}\lambda\)

thus,

\(L=\frac {3}{2} (50\;m)=75\;m\).