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**WAVE MOTION**

**Stationary Waves**

intro |

__Introduction__

Stationary or Standing waves have become very important in physics in the last hundred years or so.

Understanding them has not only given insights into sound but many other important topics: e.g. A.C. circuit theory, quantum mechanics, nanotechnology.

__Formation of Stationary Waves __

The conditions for standing waves are:

1. two waves travelling in __opposite directions__ along the __same line of travel__ and in the __same plane __

2. the waves have the __same speed__

3. the waves have the __same frequency__

4. the waves have the __same approximate amplitude__

As a result of **superposition** (waves adding/subtracting), a resultant wave is produced.

Now, depending on the phase difference between the waves, this resultant wave appears to move slowly to the right or to the left or disappear completely.

It is only when the phase difference is exactly zero, that is when the two waves are exactly in phase, that 'standing/stationary waves' occur.

Description of the numbered waveforms above:

1. Two waves having the same amplitudes approach eachother from opposite directions.

2. The two waves are 180^{o} out of phase with eachother and therefore cancel out(black horizontal line).

3. The phase difference between the two waves narrows. The resultant grows but is not in phase with either of the two waves.

4. The phase difference between the two waves is narrower still. The resultant is larger but is still out of phase with the two waves.

5. The phase difference between the two waves is now zero. The resultant has its maximum value and is in phase with the two waves.

These 'in phase' waves produce an amplitude that is the sum of the individual amplitudes, the region being called an **antinode**.

Between two antinodes is a region where the superposition is zero. This is called a **node**.

When the phenomenon is demonstrated with a horizontal vibrating string, the antinode areas appear blurred.

To observe the motion of the string moving up and down a **strobe** lamp is used.

__Properties of Stationary Waves __

The diagram shows how a standing wave moves up and down over time.

Points of interest:

1. separation of adjacent nodes is half a wavelength *(λ*/2)

2. separation of adjacent antinodes is also *λ*/2

3. hence separation of adjacent nodes and antinodes is *λ*/4

3. the maximum amplitude is 2a (twice that of a single wave)

4. a standing wave does not transfer energy(its two components however, do transfer energy in their respective directions)

__Stationary Wave Theory __

Consider two waves, R and L, travelling in opposite directions. Their displacements * y_{R}* and

**are given by*:**

*y*_{L}

where,

*derivations to follow at some future time

When the two waves are superposed, the resultant displacement * y_{T}* is given by:

From double angle trigonometry, using one of the '**Factor Formulae**' :

Comparing this with the expression for * y_{T}* , it is apparent that

**and**

*C*= 2*πft***D=**.

*kx*

Therefore,

If we now make,

Then * y_{T}* can be rewritten in a form similar to that of a simple sine wave

*y*=*a*sin(*2πf*)

The term** A **takes on the significance of being the

__vertical displacement of the standing wave__.

From the expression for** A** it can be seen that the magnitude of

**depends on the lateral position**

*A***.**

*x*

Consider the magnitude of * A* at different horizontal displacements (

**) along the standing wave.**

*x*

* A *=

**0**at a node,

*=*

**A***at an antinode*

**± 2a**

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