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# Boundary conditions

## Transmission and Reflection of a Pulse at a Boundary

What happens when a pulse travelling in one medium finds that medium is joined to another?

### Investigation : Two ropes

Find two different ropes and tie both ropes together. Hold the joined ropes horizontally and create a pulse by flicking the rope up and down. What happens to the pulse when it encounters the join?

When a pulse is transmitted from one medium to another, like from a thin rope to a thicker one, the nature of the pulse will change where it meets the boundary of the two media (i.e. where the two ropes are joined). Part of the pulse will be reflected and part of it will be transmitted. Figure 1 shows the general case of a pulse meeting a boundary. The incident pulse is the one that arrives at the boundary. The reflected pulse is the one that moves back, away from the boundary. The transmitted pulse is the one that moves into the new medium, away from the boundary. The speed of the pulse depends on the mass of the rope; the pulse is faster in the thinner rope and slower in the thick rope. When the speed of the pulse increases, the pulse length will increase. If the speed decreases, the pulse length will decrease.

Consider a pulse moving from a thin rope to a thick rope. As the pulse crosses the boundary, the speed of the pulse will decrease as it moves into the thicker rope. The pulse will move slower, so the pulse length will decrease. The pulse will be reflected and inverted in the thin rope. The reflected pulse will have the same length and speed but will have a smaller amplitude. This is illustrated in Figure 2.

When a pulse moves from a thick rope to a thin rope, the opposite will happen. As the pulse crosses the boundary, the speed of the pulse will increase as it moves into the thinner rope. The pulse in the thin rope will move faster, so the pulse length will increase. The pulse will be reflected but not inverted in the thick rope. The reflected pulse will have the same length and speed but will have a smaller amplitude. This is illustrated in Figure 3

### Pulses at a Boundary I

1. Fill in the blanks or select the correct answer: A pulse in a heavy rope is traveling towards the boundary with a thin piece of string.
1. The reflected pulse in the heavy rope will/will not be inverted because

.
2. The speed of the transmitted pulse will be greater than/less than/the same as the speed of the incident pulse.
3. The speed of the reflected pulse will be greater than/less than/the same as the speed of the incident pulse.
4. The pulse length of the transmitted pulse will be greater than/less than/the same as the pulse length of the incident pulse.
5. The frequency of the transmitted pulse will be greater than/less than/the same as the frequency of the incident pulse.
2. A pulse in a light string is traveling towards the boundary with a heavy rope.
1. The reflected pulse in the light rope will/will not be inverted because

.
2. The speed of the transmitted pulse will be greater than/less than/the same as the speed of the incident pulse.
3. The speed of the reflected pulse will be greater than/less than/the same as the speed of the incident pulse.
4. The pulse length of the transmitted pulse will be greater than/less than/the same as the pulse length of the incident pulse.

## Reflection of a Pulse from Fixed and Free Ends

Let us now consider what happens to a pulse when it reaches the end of a medium. The medium can be fixed, like a rope tied to a wall, or it can be free, like a rope tied loosely to a pole.

### Reflection of a Pulse from a Fixed End

#### Investigation : Reflection of a Pulse from a Fixed End

Tie a rope to a wall or some other object that cannot move. Create a pulse in the rope by flicking one end up and down. Observe what happens to the pulse when it reaches the wall.

When the end of the medium is fixed, for example a rope tied to a wall, a pulse reflects from the fixed end, but the pulse is inverted (i.e. it is upside-down). This is shown in Figure 4.

### Reflection of a Pulse from a Free End

#### Investigation : Reflection of a Pulse from a Free End

Tie a rope to a pole in such a way that the rope can move up and down the pole. Create a pulse in the rope by flicking one end up and down. Observe what happens to the pulse when it reaches the pole.

When the end of the medium is free, for example a rope tied loosely to a pole, a pulse reflects from the free end, but the pulse is not inverted. This is shown in Figure 5. We draw the free end as a ring around the pole. The ring will move up and down the pole, while the pulse is reflected away from the pole.

#### Tip:

The fixed and free ends that were discussed in this section are examples of boundary conditions. You will see more of boundary conditions as you progress in the Physics syllabus.

#### Pulses at a Boundary II

1. A rope is tied to a tree and a single pulse is generated. What happens to the pulse as it reaches the tree? Draw a diagram to explain what happens.
2. A rope is tied to a ring that is loosely fitted around a pole. A single pulse is sent along the rope. What will happen to the pulse as it reaches the pole? Draw a diagram to explain your answer.

The following simulation will help you understand the previous examples. Choose pulse from the options (either manual, oscillate or pulse). Then click on pulse and see what happens. Change from a fixed to a free end and see what happens. Try varying the width, amplitude, damping and tension.

Figure 6
Phet simulation for transverse pulses

## Superposition of Pulses

Two or more pulses can pass through the same medium at that same time. The resulting pulse is obtained by using the principle of superposition. The principle of superposition states that the effect of the 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 Figure 7.

Definition 1: 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 Figure 8. In general, amplitudes of individual pulses add together to give the amplitude of the resultant pulse.

Definition 2: Destructive interference is when two pulses meet, resulting in a smaller pulse.

### Exercise 1: Superposition of Pulses

The two pulses shown below approach each other at 1 m·s-11m·s-1. Draw what the waveform would look like after 1s1s, 2s2s and 5s5s.

#### Solution

1. Step 1. After 1s1s :

After 1s1s, pulse A has moved 1m1m to the right and pulse B has moved 1m1m to the left.

2. Step 2. After 2s2s :

After 1s1s more, pulse A has moved 1m1m to the right and pulse B has moved 1m1m to the left.

3. Step 3. After 5s5s :

After 5s5s, pulse A has moved 5m5m to the right and pulse B has moved 5m5m to the left.

### Tip:

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

### Superposition of Pulses

1. For the following pulse, draw the resulting wave forms after 1s1s, 2s2s, 3s3s, 4s4s and 5s5s. Each pulse is travelling at 1 m·s-11m·s-1. Each block represents 1m1m. The pulses are shown as thick black lines and the undisplaced medium as dashed lines. Click here for the solution.
2. For the following pulse, draw the resulting wave forms after 1s1s, 2s2s, 3s3s, 4s4s and 5s5s. Each pulse is travelling at 1 m·s-11m·s-1. Each block represents 1m1m. The pulses are shown as thick black lines and the undisplaced medium as dashed lines.Click here for the solution.
3. For the following pulse, draw the resulting wave forms after 1s1s, 2s2s, 3s3s, 4s4s and 5s5s. Each pulse is travelling at 1 m·s-11m·s-1. Each block represents 1m1m. The pulses are shown as thick black lines and the undisplaced medium as dashed lines.Click here for the solution.
4. For the following pulse, draw the resulting wave forms after 1s1s, 2s2s, 3s3s, 4s4s and 5s5s. Each pulse is travelling at 1 m·s-11m·s-1. Each block represents 1m1m. The pulses are shown as thick black lines and the undisplaced medium as dashed lines.Click here for the solution.
5. For the following pulse, draw the resulting wave forms after 1s1s, 2s2s, 3s3s, 4s4s and 5s5s. Each pulse is travelling at 1 m·s-11m·s-1. Each block represents 1m1m. The pulses are shown as thick black lines and the undisplaced medium as dashed lines.Click here for the solution.
6. For the following pulse, draw the resulting wave forms after 1s1s, 2s2s, 3s3s, 4s4s and 5s5s. Each pulse is travelling at 1 m·s-11m·s-1. Each block represents 1m1m. The pulses are shown as thick black lines and the undisplaced medium as dashed lines.Click here for the solution.
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.

## 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 m·s-12,5m·s-1. How far will it have travelled in 6s6s?
3. A pulse covers a distance of 75cm75cm in 2,5s2,5s. What is the speed of the pulse?
4. How long does it take a pulse to cover a distance of 200mm200mm if its speed is 4 m·s-14m·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. Click here for the solution.
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. Click here for the solution.
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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