# 6.1 Superposition of pulses

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## Superposition of pulses

Two or more pulses can pass through the same medium at that same time in the same place. When they do they interact with each other to form a different disturbance at that point. The resulting pulse is obtained by using the principle of superposition . The principle of superposition states that the effect of the different pulses is the sum of their individual effects. After pulses pass through each other, each pulse continues along its original direction of travel, and their original amplitudes remain unchanged.

Constructive interference takes place when two pulses meet each other to create a larger pulse. The amplitude of the resulting pulse is the sum of the amplitudes of the two initial pulses. This is shown in [link] .

Constructive interference
Constructive interference is when two pulses meet, resulting in a bigger pulse.

Destructive interference takes place when two pulses meet and cancel each other. The amplitude of the resulting pulse is the sum of the amplitudes of the two initial pulses, but the one amplitude will be a negative number. This is shown in [link] . In general, amplitudes of individual pulses add together to give the amplitude of the resultant pulse.

Destructive interference
Destructive interference is when two pulses meet, resulting in a smaller pulse. Superposition of two pulses. The left-hand series of images demonstrates destructive interference, since the pulses cancel each other. The right-hand series of images demonstrate a partial cancelation of two pulses, as their amplitudes are not the same in magnitude.

The two pulses shown below approach each other at $1\phantom{\rule{2pt}{0ex}}\mathrm{m}·\mathrm{s}{}^{-1}$ . Draw what the waveform would look like after $1\phantom{\rule{2pt}{0ex}}\mathrm{s}$ , $2\phantom{\rule{2pt}{0ex}}\mathrm{s}$ and $5\phantom{\rule{2pt}{0ex}}\mathrm{s}$ .

1. After $1\phantom{\rule{2pt}{0ex}}\mathrm{s}$ , pulse A has moved $1\phantom{\rule{2pt}{0ex}}\mathrm{m}$ to the right and pulse B has moved $1\phantom{\rule{2pt}{0ex}}\mathrm{m}$ to the left.

2. After $1\phantom{\rule{2pt}{0ex}}\mathrm{s}$ more, pulse A has moved $1\phantom{\rule{2pt}{0ex}}\mathrm{m}$ to the right and pulse B has moved $1\phantom{\rule{2pt}{0ex}}\mathrm{m}$ to the left.

3. After $5\phantom{\rule{2pt}{0ex}}\mathrm{s}$ , pulse A has moved $5\phantom{\rule{2pt}{0ex}}\mathrm{m}$ to the right and pulse B has moved $5\phantom{\rule{2pt}{0ex}}\mathrm{m}$ to the left.

The idea of superposition is one that occurs often in physics. You will see much, much more of superposition!

## Aim

To demonstrate constructive and destructive interference

## Apparatus

Ripple tank apparatus

## Method

1. Set up the ripple tank
2. Produce a single pulse and observe what happens
3. Produce two pulses simultaneously and observe what happens
4. Produce two pulses at slightly different times and observe what happens

## Results and conclusion

You should observe that when you produce two pulses simultaneously you see them interfere constructively and when you produce two pulses at slightly different times you see them interfere destructively.

## Problems involving superposition of pulses

1. For the following pulse, draw the resulting wave forms after $1\phantom{\rule{2pt}{0ex}}\mathrm{s}$ , $2\phantom{\rule{2pt}{0ex}}\mathrm{s}$ , $3\phantom{\rule{2pt}{0ex}}\mathrm{s}$ , $4\phantom{\rule{2pt}{0ex}}\mathrm{s}$ and $5\phantom{\rule{2pt}{0ex}}\mathrm{s}$ . Each pulse is travelling at $1\phantom{\rule{2pt}{0ex}}\mathrm{m}·\mathrm{s}{}^{-1}$ . Each block represents $1\phantom{\rule{2pt}{0ex}}\mathrm{m}$ . The pulses are shown as thick black lines and the undisplaced medium as dashed lines.
2. For the following pulse, draw the resulting wave forms after $1\phantom{\rule{2pt}{0ex}}\mathrm{s}$ , $2\phantom{\rule{2pt}{0ex}}\mathrm{s}$ , $3\phantom{\rule{2pt}{0ex}}\mathrm{s}$ , $4\phantom{\rule{2pt}{0ex}}\mathrm{s}$ and $5\phantom{\rule{2pt}{0ex}}\mathrm{s}$ . Each pulse is travelling at $1\phantom{\rule{2pt}{0ex}}\mathrm{m}·\mathrm{s}{}^{-1}$ . Each block represents $1\phantom{\rule{2pt}{0ex}}\mathrm{m}$ . The pulses are shown as thick black lines and the undisplaced medium as dashed lines.
3. For the following pulse, draw the resulting wave forms after $1\phantom{\rule{2pt}{0ex}}\mathrm{s}$ , $2\phantom{\rule{2pt}{0ex}}\mathrm{s}$ , $3\phantom{\rule{2pt}{0ex}}\mathrm{s}$ , $4\phantom{\rule{2pt}{0ex}}\mathrm{s}$ and $5\phantom{\rule{2pt}{0ex}}\mathrm{s}$ . Each pulse is travelling at $1\phantom{\rule{2pt}{0ex}}\mathrm{m}·\mathrm{s}{}^{-1}$ . Each block represents $1\phantom{\rule{2pt}{0ex}}\mathrm{m}$ . The pulses are shown as thick black lines and the undisplaced medium as dashed lines.
4. For the following pulse, draw the resulting wave forms after $1\phantom{\rule{2pt}{0ex}}\mathrm{s}$ , $2\phantom{\rule{2pt}{0ex}}\mathrm{s}$ , $3\phantom{\rule{2pt}{0ex}}\mathrm{s}$ , $4\phantom{\rule{2pt}{0ex}}\mathrm{s}$ and $5\phantom{\rule{2pt}{0ex}}\mathrm{s}$ . Each pulse is travelling at $1\phantom{\rule{2pt}{0ex}}\mathrm{m}·\mathrm{s}{}^{-1}$ . Each block represents $1\phantom{\rule{2pt}{0ex}}\mathrm{m}$ . The pulses are shown as thick black lines and the undisplaced medium as dashed lines.
5. For the following pulse, draw the resulting wave forms after $1\phantom{\rule{2pt}{0ex}}\mathrm{s}$ , $2\phantom{\rule{2pt}{0ex}}\mathrm{s}$ , $3\phantom{\rule{2pt}{0ex}}\mathrm{s}$ , $4\phantom{\rule{2pt}{0ex}}\mathrm{s}$ and $5\phantom{\rule{2pt}{0ex}}\mathrm{s}$ . Each pulse is travelling at $1\phantom{\rule{2pt}{0ex}}\mathrm{m}·\mathrm{s}{}^{-1}$ . Each block represents $1\phantom{\rule{2pt}{0ex}}\mathrm{m}$ . The pulses are shown as thick black lines and the undisplaced medium as dashed lines.
6. For the following pulse, draw the resulting wave forms after $1\phantom{\rule{2pt}{0ex}}\mathrm{s}$ , $2\phantom{\rule{2pt}{0ex}}\mathrm{s}$ , $3\phantom{\rule{2pt}{0ex}}\mathrm{s}$ , $4\phantom{\rule{2pt}{0ex}}\mathrm{s}$ and $5\phantom{\rule{2pt}{0ex}}\mathrm{s}$ . Each pulse is travelling at $1\phantom{\rule{2pt}{0ex}}\mathrm{m}·\mathrm{s}{}^{-1}$ . Each block represents $1\phantom{\rule{2pt}{0ex}}\mathrm{m}$ . The pulses are shown as thick black lines and the undisplaced medium as dashed lines.
7. What is superposition of waves?
8. What is constructive interference?
9. What is destructive interference?

The following presentation provides a summary of the work covered in this chapter. Although the presentation is titled waves, the presentation covers pulses only.

## Summary

• A medium is the substance or material in which a wave will move
• A pulse is a single disturbance that moves through a medium
• The amplitude of a pules is a measurement of how far the medium is displaced from rest
• Pulse speed is the distance a pulse travels per unit time
• Constructive interference is when two pulses meet and result in a bigger pulse
• Destructive interference is when two pulses meet and and result in a smaller pulse
• We can draw graphs to show the motion of a particle in the medium or to show the motion of a pulse through the medium
• When a pulse moves from a thin rope to a thick rope, the speed and pulse length decrease. The pulse will be reflected and inverted in the thin rope. The reflected pulse has the same length and speed, but a different amplitude
• When a pulse moves from a thick rope to a thin rope, the speed and pulse length increase. The pulse will be reflected in the thick rope. The reflected pulse has the same length and speed, but a different amplitude
• A pulse reaching a free end will be reflected but not inverted. A pulse reaching a fixed end will be reflected and inverted

## Exercises - transverse pulses

1. A heavy rope is flicked upwards, creating a single pulse in the rope. Make a drawing of the rope and indicate the following in your drawing:
1. The direction of motion of the pulse
2. Amplitude
3. Pulse length
4. Position of rest
2. A pulse has a speed of $2,5\phantom{\rule{2pt}{0ex}}\mathrm{m}·\mathrm{s}{}^{-1}$ . How far will it have travelled in $6\phantom{\rule{2pt}{0ex}}\mathrm{s}$ ?
3. A pulse covers a distance of $75\phantom{\rule{2pt}{0ex}}\mathrm{cm}$ in $2,5\phantom{\rule{2pt}{0ex}}\mathrm{s}$ . What is the speed of the pulse?
4. How long does it take a pulse to cover a distance of $200\phantom{\rule{2pt}{0ex}}\mathrm{mm}$ if its speed is $4\phantom{\rule{2pt}{0ex}}\mathrm{m}·\mathrm{s}{}^{-1}$ ?
5. The following position-time graph for a pulse in a slinky spring is given. Draw an accurate sketch graph of the velocity of the pulse against time.
6. The following velocity-time graph for a particle in a medium is given. Draw an accurate sketch graph of the position of the particle vs. time.
7. Describe what happens to a pulse in a slinky spring when:
1. the slinky spring is tied to a wall.
2. the slinky spring is loose, i.e. not tied to a wall.
8. The following diagrams each show two approaching pulses. Redraw the diagrams to show what type of interference takes place, and label the type of interference.
9. Two pulses, A and B, of identical shape and amplitude are simultaneously generated in two identical wires of equal mass and length. Wire A is, however, pulled tighter than wire B. Which pulse will arrive at the other end first, or will they both arrive at the same time?

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