Standing Waves and Boundary Conditions, Grade 10 Physics
1/ Reflection of a Transverse Wave from a Fixed End

We have seen that when a pulse meets a fixed endpoint, the pulse is reflected, but it is inverted. Since a transverse wave is a series of pulses, a transverse wave meeting a fixed endpoint is also reflected and the reflected wave is inverted. That means that the peaks and troughs are swapped around.

2/ Reflection of a Transverse Wave from a Free End
If transverse waves are reflected from an end, which is free to move, the waves sent down the string are reflected but do not suffer a phase shift as shown in Figure 6.4.

3/ Standing Waves
What happens when a reflected transverse wave meets an incident transverse wave? When two waves move in opposite directions, through each other, interference takes place. If the two waves have the same frequency and wavelength then standing waves are generated. Standing waves are so-called because they appear to be standing still.

Activity :: Investigation : Creating Standing Waves
Tie a rope to a fixed object such that the tied end does not move. Continuously move the free end up and down to generate firstly transverse waves and later standing waves.
We can now look closely how standing waves are formed. Figure 6.5 shows a reflected wave meeting an incident wave

Figure 6.5: A reflected wave (solid line) approaches the incident wave (dashed line).
When they touch, both waves have an amplitude of zero:

Figure 6.6: A reflected wave (solid line) meets the incident wave (dashed line).
If we wait for a short time the ends of the two waves move past each other and the waves overlap. To find the resultant wave, we add the two together.

Figure 6.7: A reflected wave (solid line) overlaps slightly with the incident wave (dashed line).
In this picture, we show the two waves as dotted lines and the sum of the two in the overlap region is shown as a solid line:

The important thing to note in this case is that there are some points where the two waves always destructively interfere to zero. If we let the two waves move a little further we get the picture below:

Again we have to add the two waves together in the overlap region to see what the sum of the waves looks like.

In this case the two waves have moved half a cycle past each other but because they are out of phase they cancel out completely.

When the waves have moved past each other so that they are overlapping for a large region the situation looks like a wave oscillating in place. The following sequence of diagrams show what the resulting wave will look like. To make it clearer, the arrows at the top of the picture show peaks where maximum positive constructive interference is taking place. The arrows at the bottom of the picture show places where maximum negative interference is taking place.

As time goes by the peaks become smaller and the troughs become shallower but they do not move.

For an instant the entire region will look completely flat.

The various points continue their motion in the same manner.

Eventually the picture looks like the complete reflection through the x-axis of what we started with:

Then all the points begin to move back. Each point on the line is oscillating up and down with a different amplitude.

If we look at the overall result, we get a standing wave.

If we superimpose the two cases where the peaks were at a maximum and the case where the same waves were at a minimum we can see the lines that the points oscillate between. We call this the envelope of the standing wave as it contains all the oscillations of the individual points. To make the concept of the envelope clearer let us draw arrows describing the motion of points along the line.

Every point in the medium containing a standing wave oscillates up and down and the amplitude of the oscillations depends on the location of the point. It is convenient to draw the envelope for the oscillations to describe the motion. We cannot draw the up and down arrows for every single point!
Standing waves can be a problem in for example indoor concerts where the dimensions of the concert venue coincide with particular wavelengths. Standing waves can appear as ‘feedback’, which would occur if the standing wave was picked up by the microphones on stage and amplified.

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Physics 10.V Transverse Wave

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