This chapter forms the basis of the discussion into mechanical waves. Waves are all around us, even though most of us are not aware of it. The most common waves are waves in the sea, but waves can be created in any container of water, ranging from an ocean to a tea-cup. Waves do not only occur in water, they occur in any kind of medium. Earthquakes generate waves that travel through the rock of the Earth. When your friend speaks to you he produces sound waves that travel through the air to your ears. Light is made up of electromagnetic waves. A wave is simply moving energy.

In this chapter, as well as in the following chapters, we will speak about waves moving in a medium. A medium is just the substance or material through which waves move. In other words the medium carries the wave from one place to another. The medium does not create the wave and the medium is not the wave. Therefore the medium does not travel with the wave as the wave propagates through it. Air is a medium for sound waves, water is a medium for water waves and rock is a medium for earthquakes (which are also a type of wave). Air, water and rock are therefore examples of media (media is the plural of medium).

Definition 1: Medium

A medium is the substance or material in which a wave will move.

In each medium, the atoms that make up the medium are moved temporarily from their rest position. In order for a wave to travel, the different parts of the medium must be able to interact with each other.

Investigation : Observation of Pulses

Take a heavy rope. Have two people hold the rope stretched out horizontally. Flick the rope at one end only once.

Figure 1

What happens to the disturbance that you created in the rope? Does it stay at the place where it was created or does it move down the length of the rope?

In the activity, we created a pulse. A pulse is a single disturbance that moves through a medium. In a transverse pulse the displacement of the medium is perpendicular to the direction of motion of the pulse. Figure 2 shows an example of a transverse pulse. In the activity, the rope or spring was held horizontally and the pulse moved the rope up and down. This was an example of a transverse pulse.

Definition 2: Pulse

A pulse is a single disturbance that moves through a medium.

Pulse Length and Amplitude

The amplitude of a pulse is a measurement of how far the medium is displaced momentarily from a position of rest. The pulse length is a measurement of how long the pulse is. Both these quantities are shown in Figure 2.

Definition 3: Amplitude

The amplitude of a pulse is a measurement of how far the medium is displaced from rest.

Figure 2: Example of a transverse pulse

Investigation : Pulse Length and Amplitude

The graphs below show the positions of a pulse at different times.

Figure 3

Use your ruler to measure the lengths of aa and pp. Fill your answers in the table.

Table 1

Time	a a	p p
t=0t=0 s
t=1t=1 s
t=2t=2 s
t=3t=3 s

What do you notice about the values of aa and pp?

In the activity, we found that the values for how high the pulse (aa) is and how wide the pulse (pp) is the same at different times. Pulse length and amplitude are two important quantities of a pulse.

When a pulse moves through a medium, there are two different motions: the motion of the particles of the medium and the motion of the pulse. These two motions are at right angles to each other when the pulse is transverse. Each motion will be discussed.

Consider the situation shown in Figure 6. The dot represents one particle of the medium. We see that as the pulse moves to the right the particle only moves up and down.

Motion of a Particle of the Medium

First we consider the motion of a particle of the medium when a pulse moves through the medium. For the explanation we will zoom into the medium so that we are looking at the atoms of the medium. These atoms are connected to each other as shown in Figure 5.

Figure 5: Particles in a medium.

When a pulse moves through the medium, the particles in the medium only move up and down. We can see this in Figure 6 which shows the motion of a single particle as a pulse moves through the medium.

Figure 6: Positions of a pulse in a rope at different times. The pulse moves to the right as shown by the arrow. You can also see the motion of a point in the medium through which the pulse is travelling. Each block is 1 cm.

Tip:

A particle in the medium only moves up and down when a transverse pulse moves through the medium. The pulse moves from left to right (or right to left). The motion of the particle is perpendicular to the motion of a transverse pulse.

If you consider the motion of the particle as a function of time, you can draw a graph of position vs. time and velocity vs. time.

Investigation : Drawing a position-time graph

1. Study Figure 6 and complete the following table:

   Table 2

   time (s)	0	1	2	3	4	5	6	7	8	9
   position (cm)

2. Use your table to draw a graph of position vs. time for a particle in a medium.

The position vs. time graph for a particle in a medium when a pulse passes through the medium is shown in Figure 7

Figure 7: Position against Time graph of a particle in the medium through which a transverse pulse is travelling.

Investigation : Drawing a velocity-time graph

1. Study Figure 7 and complete the following table:

   Table 3

   time (s)	0	1	2	3	4	5	6	7	8	9
   velocity (cm.s-1-1)

2. Use your table to draw a graph of velocity vs time for a particle in a medium.

The velocity vs. time graph far a particle in a medium when a pulse passes through the medium is shown in Figure 8.

Figure 8: Velocity against Time graph of a particle in the medium through which a transverse pulse is travelling.

Motion of the Pulse

The motion of the pulse is much simpler than the motion of a particle in the medium.

Tip:

A point on a transverse pulse, eg. the peak, only moves in the direction of the motion of the pulse.

Exercise 2: Transverse pulse through a medium

Figure 9: Position of the peak of a pulse at different times (since we know the shape of the pulse does not change we can look at only one point on the pulse to keep track of its position, the peak for example). The pulse moves to the right as shown by the arrow. Each square is 0,5cm0,5cm.

Given the series of snapshots of a transverse pulse moving through a medium, depicted in Figure 9, do the following:

• draw up a table of time, position and velocity,
• plot a position vs. time graph,
• plot a velocity vs. time graph.

Solution

1. Step 1. Interpreting the figure :

   Figure 9 shows the motion of a pulse through a medium and a dot to indicate the same position on the pulse. If we follow the dot, we can draw a graph of position vs time for a pulse. At t= 0st=0s the dot is at 0cm0cm. At t= 1st=1s the dot is 1cm1cm away from its original postion. At t= 2st=2s the dot is 2cm2cm away from its original postion, and so on.

2. Step 2. We can draw the following table: :

   Table 4

   time (s)	0	1	2	3	4	5	6	7	8	9
   position (cm)	0	1	2	3	4	5	6	7	8	9
   velocity (cm.s-1-1)	1	1	1	1	1	1	1	1	1	1

3. Step 3. A graph of position vs time is drawn as is shown in the figure. :

   Figure 10

4. Step 4. Similarly, a graph of velocity vs time is drawn and is shown in the figure below. :

   Figure 11

[ Show Solution ] [ Hide Solution ]

Travelling Pulse

1. A pulse is passed through a rope and the following pictures were obtained for each time interval:

   Figure 12

   1. Complete the following table for a particle in the medium:

      Table 5

      time (s)	0,00	0,25	0,50	0,75	1,00	1,25	1,50	1,75	2,00
      position (mm)
      velocity (mm.s-1-1)

   2. Draw a position vs. time graph for the motion of the particle at 3cm3cm.
   3. Draw a velocity vs. time graph for the motion of the particle at 3cm3cm.
   4. Draw a position vs. time graph for the motion of the pulse through the rope.
   5. Draw a velocity vs. time graph for the motion of the pulse through the rope.
   
   Click here for the solution.

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

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 13 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.

Figure 13: Reflection and transmission of a pulse at the boundary between two media.

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 14.

Figure 14: Reflection and transmission of a pulse at the boundary between two media.

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 15

Figure 15: Reflection and transmission of a pulse at the boundary between two media.

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.

Figure 16: Reflection of a pulse from a fixed end.

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 16.

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 17. 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.

Figure 17: Reflection of a pulse from a free end.

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.
   Click here for the solution.
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.
   Click here for the solution.

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.

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 19.

Definition 5: Constructive interference is when two pulses meet, resulting in a bigger pulse.

Figure 19: Superposition of two pulses: constructive interference.

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 20. In general, amplitudes of individual pulses add together to give the amplitude of the resultant pulse.

Definition 6: Destructive interference is when two pulses meet, resulting in a smaller pulse.

Figure 20: 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.

Exercise 3: 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.

Figure 21

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.

   Figure 22

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.

   Figure 23

3. Step 3. After 5s5s :

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

   Figure 24

[ Show Solution ] [ Hide Solution ]