# Standing waves

You know that we can describe a travelling wave with a mathematical form like this:

The equation above represents a wave travelling to which direction?

Q:  The wave travels which way?

RIGHT                  LEFT

So an identical wave travelling in the OTHER direction will have an equation like this:

What happens when these two waves meet? Is there some way to add them together? Perhaps you might remember this little trigonometric identity ...

Q:  Use the trig identity to add together the two waves
y1(x, t) and y2(x, t).

The result should simplify quite a bit.

When two identical waves meet, travelling in opposite directions, they add up in a special way. The summed waves appears to stand still; the pattern doesn't slide to the right or the left. We therefore call it a standing wave.

Image courtesy of LucasVB and wikimedia.org.

#### Nodes

If you look at a standing wave

you can see that there are some locations at which the displacement is always zero. We call these locations nodes.

These nodes must occur whenever the position satisfies a particular condition:

Q:  When does sin(kx) = 0?

Q:  At what values of x will we find a node?

Right.

Hey, wait a minute. The wave number k is related to the wavelength λ, right? So we should be able to express the locations of nodes in terms of the wavelength, too.

Q:  How is the wavelength λ related to wave number k?

Q:  Write the locations of nodes in terms of wavelength.

#### Antinodes

Let's look again at a standing wave:

This time, focus on the places where the displacement reaches a maximum value. We call these locations antinodes.

These nodes must occur whenever the position satisfies a different particular condition:

Q:  When does sin(kx) = +/- 1?

Q:  At what values of x will we find an antinode?

Right.

Hey, wait a minute. The wave number k is related to the wavelength λ, right? So we should be able to express the locations of antinodes in terms of the wavelength, too.

Q:  Write the locations of antinodes in terms of wavelength.