Waves Tutorial 4 - Standing Waves

Sometimes waves appear to be standing still, i.e. the crests and the troughs appear to stay in the same place.  We can see them in water, especially water surrounded by walls.  We call them standing waves or stationary waves.  Any kind of wave can form a standing wave.

 

Musical instruments depend on standing waves:

Stationary waves are formed when two progressive waves are superposed:

If we send an incident wave down a string, which is fixed at the end, the wave is reflected at the fixed end and undergoes a phase change of p radians or 180o.  There is no phase change at the free end.

Question 1

What are the conditions needed for a standing wave?

Answer

If we send a continuous stream of waves down the string, they are reflected and a standing wave gets set up.  The frequency will be the same, the amplitude very nearly the same and the speed will be the same.  The directions are opposite.  The phase change of p radians causes cancellation at the fixed end.  This region of zero displacement is called a node.

In a progressive wave, points X and Y would be in antiphase, p radians out of phase.  However, because the wave is reflected, the phase is changed by p radians.  So they are now 2p radians out of phase, which means that they are in phase.  Superposition is constructive.   The amplitude is now at a maximum, and this is called an antinode.

Notice:

Question 2

What is meant by a node and an antinode?  A standing wave loop in a string is 66 cm long.  What is the wavelength in metres?

Answer

 

 

We can set up a standing wave using 3 cm wave apparatus.

We move the probe between the transmitter and the reflector and we detect maximum readings and minimum readings with the probe, which is connected to a microammeter.  The maximum readings coincide with the antinodes; the minimum readings with the nodes.

 

 

Comparing Standing Waves with Progressive Waves

For all standing wave patterns, these two points are true:

 

  1. The amplitude varies according to position from zero at a node, to maximum at an antinode.  The amplitude of a given point is always the same.

  1. The phase difference between two particles is zero if the points are between adjacent noted.  It is 180o if they are either side of a node.

 

If the points are a separated by an even number of nodes, they are in phase.

 

 

Therefore P and Q are in phase with each other (as are R and S, and T and U).  P and S are in antiphase, but P is in phase with U.

 

Property

Stationary Waves

Progressive Waves

Frequency

All particles vibrate at the same frequency, except at the nodes where there is no vibration

All particles vibrate at the same frequency throughout the wave.

Amplitude

Amplitude varies between zero at the nodes and maximum at the antinodes.  S will vibrate at a bigger amplitude to R.

The amplitude is the same for all particles.

Phase difference between two points.

Phase is np rad, where n is the number of nodes between the two points.