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# Transverse waves

## Introduction

Waves occur frequently in nature. The most obvious examples are waves in water, on a dam, in the ocean, or in a bucket. We are most interested in the properties that waves have. All waves have the same properties, so if we study waves in water, then we can transfer our knowledge to predict how other examples of waves will behave.

## What is a transverse wave?

We have studied pulses in Transverse Pulses, and know that a pulse is a single disturbance that travels through a medium. A wave is a periodic, continuous disturbance that consists of a train or succession of pulses.

Definition 1: Wave

A wave is a periodic, continuous disturbance that consists of a train of pulses.

Definition 2: Transverse wave

A transverse wave is a wave where the movement of the particles of the medium is perpendicular (at a right angle) to the direction of propagation of the wave.

### Investigation : Transverse Waves

Take a rope or slinky spring. Have two people hold the rope or spring stretched out horizontally. Flick the one end of the rope up and down continuously to create a train of pulses.

1. Describe what happens to the rope.
2. Draw a diagram of what the rope looks like while the pulses travel along it.
3. In which direction do the pulses travel?
4. Tie a ribbon to the middle of the rope. This indicates a particle in the rope.
5. Flick the rope continuously. Watch the ribbon carefully as the pulses travel through the rope. What happens to the ribbon?
6. Draw a picture to show the motion of the ribbon. Draw the ribbon as a dot and use arrows to indicate how it moves.

In the Activity, you have created waves. The medium through which these waves propagated was the rope, which is obviously made up of a very large number of particles (atoms). From the activity, you would have noticed that the wave travelled from left to right, but the particles (the ribbon) moved only up and down.

When the particles of a medium move at right angles to the direction of propagation of a wave, the wave is called transverse. For waves, there is no net displacement of the particles (they return to their equilibrium position), but there is a net displacement of the wave. There are thus two different motions: the motion of the particles of the medium and the motion of the wave.

The following simulation will help you understand more about waves. Select the oscillate option and then observe what happens.

Figure 4
Phet simulation for Transverse Waves

### Peaks and Troughs

Waves have moving peaks (or crests) and troughs. A peak is the highest point the medium rises to and a trough is the lowest point the medium sinks to.

Peaks and troughs on a transverse wave are shown in Figure 5.

Definition 3: Peaks and troughs

A peak is a point on the wave where the displacement of the medium is at a maximum. A point on the wave is a trough if the displacement of the medium at that point is at a minimum.

### Amplitude and Wavelength

There are a few properties that we saw with pulses that also apply to waves. These are amplitude and wavelength (we called this pulse length).

Definition 4: Amplitude

The amplitude is the maximum displacement of a particle from its equilibrium position.

#### Investigation : Amplitude

Fill in the table below by measuring the distance between the equilibrium and each peak and troughs in the wave above. Use your ruler to measure the distances.

 Peak/Trough Measurement (cm) a b c d e f
2. Are the distances between the equilibrium position and each peak equal?
3. Are the distances between the equilibrium position and each trough equal?
4. Is the distance between the equilibrium position and peak equal to the distance between equilibrium and trough?

As we have seen in the activity on amplitude, the distance between the peak and the equilibrium position is equal to the distance between the trough and the equilibrium position. This distance is known as the amplitude of the wave, and is the characteristic height of wave, above or below the equilibrium position. Normally the symbol AA is used to represent the amplitude of a wave. The SI unit of amplitude is the metre (m).

#### Exercise 1: Amplitude of Sea Waves

If the peak of a wave measures 2m2m above the still water mark in the harbour, what is the amplitude of the wave?

##### Solution
1. Step 1. Think about what amplitude means :

The definition of the amplitude is the height of a peak above the equilibrium position. The still water mark is the height of the water at equilibrium and the peak is 2m2m above this, so the amplitude is 2m2m.

#### Investigation : Wavelength

Fill in the table below by measuring the distance between peaks and troughs in the wave above.

 Distance(cm) a b c d
2. Are the distances between peaks equal?
3. Are the distances between troughs equal?
4. Is the distance between peaks equal to the distance between troughs?

As we have seen in the activity on wavelength, the distance between two adjacent peaks is the same no matter which two adjacent peaks you choose. There is a fixed distance between the peaks. Similarly, we have seen that there is a fixed distance between the troughs, no matter which two troughs you look at. More importantly, the distance between two adjacent peaks is the same as the distance between two adjacent troughs. This distance is called the wavelength of the wave.

The symbol for the wavelength is λλ (the Greek letter lambda) and wavelength is measured in metres (mm).

#### Exercise 2: Wavelength

The total distance between 44 consecutive peaks of a transverse wave is 6m6m. What is the wavelength of the wave?

##### Solution
1. Step 1. Draw a rough sketch of the situation :

2. Step 2. Determine how to approach the problem :

From the sketch we see that 4 consecutive peaks is equivalent to 3 wavelengths.

3. Step 3. Solve the problem :

Therefore, the wavelength of the wave is:

3 λ = 6 m λ = 6 m 3 = 2 m 3 λ = 6 m λ = 6 m 3 = 2 m
(1)

### Points in Phase

#### Investigation : Points in Phase

Fill in the table by measuring the distance between the indicated points.

 Points Distance (cm) A to F B to G C to H D to I E to J

What do you find?

In the activity the distance between the indicated points was the same. These points are then said to be in phase. Two points in phase are separate by an integer (0,1,2,3,...) number of complete wave cycles. They do not have to be peaks or troughs, but they must be separated by a complete number of wavelengths.

We then have an alternate definition of the wavelength as the distance between any two adjacent points which are in phase.

Definition 5: Wavelength of wave

The wavelength of a wave is the distance between any two adjacent points that are in phase.

Points that are not in phase, those that are not separated by a complete number of wavelengths, are called out of phase. Examples of points like these would be AA and CC, or DD and EE, or BB and HH in the Activity.

### Period and Frequency

Imagine you are sitting next to a pond and you watch the waves going past you. First one peak arrives, then a trough, and then another peak. Suppose you measure the time taken between one peak arriving and then the next. This time will be the same for any two successive peaks passing you. We call this time the period, and it is a characteristic of the wave.

The symbol TT is used to represent the period. The period is measured in seconds (ss).

Definition 6: Period (TT)
The period (TT) is the time taken for two successive peaks (or troughs) to pass a fixed point.

Imagine the pond again. Just as a peak passes you, you start your stopwatch and count each peak going past. After 1 second you stop the clock and stop counting. The number of peaks that you have counted in the 1 second is the frequency of the wave.

Definition 7: Frequency
The frequency is the number of successive peaks (or troughs) passing a given point in 1 second.

The frequency and the period are related to each other. As the period is the time taken for 1 peak to pass, then the number of peaks passing the point in 1 second is 1T1T. But this is the frequency. So

f = 1 T f = 1 T
(2)

or alternatively,

T = 1 f T = 1 f
(3)

For example, if the time between two consecutive peaks passing a fixed point is 12s12s, then the period of the wave is 12s12s. Therefore, the frequency of the wave is:

f = 1 T = 1 1 2 s = 2 s - 1 f = 1 T = 1 1 2 s = 2 s - 1
(4)

The unit of frequency is the Hertz (HzHz) or s-1s-1.

#### Exercise 3: Period and Frequency

What is the period of a wave of frequency 10Hz10Hz?

##### Solution
1. Step 1. Determine what is given and what is required :

We are required to calculate the period of a 10Hz10Hz wave.

2. Step 2. Determine how to approach the problem :

We know that:

T = 1 f T = 1 f
(5)
3. Step 3. Solve the problem :
T = 1 f = 1 10 Hz = 0 , 1 s T = 1 f = 1 10 Hz = 0 , 1 s
(6)
4. Step 4. Write the answer :

The period of a 10Hz10Hz wave is 0,1s0,1s.

### Speed of a Transverse Wave

In Motion in One Dimension, we saw that speed was defined as

speed = distance traveled time taken speed = distance traveled time taken
(7)

The distance between two successive peaks is 1 wavelength, λλ. Thus in a time of 1 period, the wave will travel 1 wavelength in distance. Thus the speed of the wave, vv, is:

v = distance traveled time taken = λ T v = distance traveled time taken = λ T
(8)

However, f=1Tf=1T. Therefore, we can also write:

v = λ T = λ · 1 T = λ · f v = λ T = λ · 1 T = λ · f
(9)

We call this equation the wave equation. To summarise, we have that v=λ·fv=λ·f where

• v=v= speed in m·s-1m·s-1
• λ=λ= wavelength in mm
• f=f= frequency in HzHz

#### Exercise 4: Speed of a Transverse Wave 1

When a particular string is vibrated at a frequency of 10Hz10Hz, a transverse wave of wavelength 0,25m0,25m is produced. Determine the speed of the wave as it travels along the string.

##### Solution
1. Step 1. Determine what is given and what is required :
• frequency of wave: f=10Hzf=10Hz
• wavelength of wave: λ=0,25mλ=0,25m

We are required to calculate the speed of the wave as it travels along the string. All quantities are in SI units.

2. Step 2. Determine how to approach the problem :

We know that the speed of a wave is:

v = f · λ v = f · λ
(10)

and we are given all the necessary quantities.

3. Step 3. Substituting in the values :
v = f · λ = ( 10 Hz ) ( 0 , 25 m ) = 2 , 5 m · s - 1 v = f · λ = ( 10 Hz ) ( 0 , 25 m ) = 2 , 5 m · s - 1
(11)
4. Step 4. Write the final answer :

The wave travels at 2,5 m·s-12,5m·s-1 along the string.

#### Exercise 5: Speed of a Transverse Wave 2

A cork on the surface of a swimming pool bobs up and down once every second on some ripples. The ripples have a wavelength of 20cm20cm. If the cork is 2m2m from the edge of the pool, how long does it take a ripple passing the cork to reach the edge?

##### Solution
1. Step 1. Determine what is given and what is required :

We are given:

• frequency of wave: f=1Hzf=1Hz
• wavelength of wave: λ=20cmλ=20cm
• distance of cork from edge of pool: D=2mD=2m

We are required to determine the time it takes for a ripple to travel between the cork and the edge of the pool.

The wavelength is not in SI units and should be converted.

2. Step 2. Determine how to approach the problem :

The time taken for the ripple to reach the edge of the pool is obtained from:

t = D v ( from v = D t ) t = D v ( from v = D t )
(12)

We know that

v = f · λ v = f · λ
(13)

Therefore,

t = D f · λ t = D f · λ
(14)
3. Step 3. Convert wavelength to SI units :
20 cm = 0 , 2 m 20 cm = 0 , 2 m
(15)
4. Step 4. Solve the problem :
t = D f · λ = 2 m ( 1 Hz ) ( 0 , 2 m ) = 10 s t = D f · λ = 2 m ( 1 Hz ) ( 0 , 2 m ) = 10 s
(16)
5. Step 5. Write the final answer :

A ripple passing the leaf will take 10s10s to reach the edge of the pool.

The following video provides a summary of the concepts covered so far.

Figure 13
Khan academy video on waves - 1

#### Waves

1. When the particles of a medium move perpendicular to the direction of the wave motion, the wave is called a .................. wave.
2. A transverse wave is moving downwards. In what direction do the particles in the medium move?
3. Consider the diagram below and answer the questions that follow:
1. the wavelength of the wave is shown by letter

.
2. the amplitude of the wave is shown by letter

.
4. Draw 2 wavelengths of the following transverse waves on the same graph paper. Label all important values.
1. Wave 1: Amplitude = 1 cm, wavelength = 3 cm
2. Wave 2: Peak to trough distance (vertical) = 3 cm, peak to peak distance (horizontal) = 5 cm
5. You are given the transverse wave below. Draw the following:
1. A wave with twice the amplitude of the given wave.
2. A wave with half the amplitude of the given wave.
3. A wave travelling at the same speed with twice the frequency of the given wave.
4. A wave travelling at the same speed with half the frequency of the given wave.
5. A wave with twice the wavelength of the given wave.
6. A wave with half the wavelength of the given wave.
7. A wave travelling at the same speed with twice the period of the given wave.
8. A wave travelling at the same speed with half the period of the given wave.
6. A transverse wave travelling at the same speed with an amplitude of 5 cm has a frequency of 15 Hz. The horizontal distance from a crest to the nearest trough is measured to be 2,5 cm. Find the
1. period of the wave.
2. speed of the wave.
7. A fly flaps its wings back and forth 200 times each second. Calculate the period of a wing flap.
8. As the period of a wave increases, the frequency increases/decreases/does not change.
9. Calculate the frequency of rotation of the second hand on a clock.
10. Microwave ovens produce radiation with a frequency of 2 450 MHz (1 MHz = 106106 Hz) and a wavelength of 0,122 m. What is the wave speed of the radiation?
11. Study the following diagram and answer the questions:
1. Identify two sets of points that are in phase.
2. Identify two sets of points that are out of phase.
3. Identify any two points that would indicate a wavelength.
12. Tom is fishing from a pier and notices that four wave crests pass by in 8 s and estimates the distance between two successive crests is 4 m. The timing starts with the first crest and ends with the fourth. Calculate the speed of the wave.

## Summary

1. A wave is formed when a continuous number of pulses are transmitted through a medium.
2. A peak is the highest point a particle in the medium rises to.
3. A trough is the lowest point a particle in the medium sinks to.
4. In a transverse wave, the particles move perpendicular to the motion of the wave.
5. The amplitude is the maximum distance from equilibrium position to a peak (or trough), or the maximum displacement of a particle in a wave from its position of rest.
6. The wavelength (λλ) is the distance between any two adjacent points on a wave that are in phase. It is measured in metres.
7. The period (TT) of a wave is the time it takes a wavelength to pass a fixed point. It is measured in seconds (s).
8. The frequency (ff) of a wave is how many waves pass a point in a second. It is measured in hertz (Hz) or s-1s-1.
9. Frequency: f=1Tf=1T
10. Period: T=1fT=1f
11. Speed: v=fλv=fλ or v=λTv=λT.
12. When a wave is reflected from a fixed end, the resulting wave will move back through the medium, but will be inverted. When a wave is reflected from a free end, the waves are reflected, but not inverted.

## Exercises

1. A standing wave is formed when:
1. a wave refracts due to changes in the properties of the medium
2. a wave reflects off a canyon wall and is heard shortly after it is formed
3. a wave refracts and reflects due to changes in the medium
4. two identical waves moving different directions along the same medium interfere
2. How many nodes and anti-nodes are shown in the diagram? Click here for the solution.
3. Draw a transverse wave that is reflected from a fixed end.
4. Draw a transverse wave that is reflected from a free end.
5. A wave travels along a string at a speed of 1,5m·s-11,5m·s-1. If the frequency of the source of the wave is 7,5 Hz, calculate:
1. the wavelength of the wave
2. the period of the wave
6. Water waves crash against a seawall around the harbour. Eight waves hit the seawall in 5 s. The distance between successive troughs is 9 m. The height of the waveform trough to crest is 1,5 m.
1. How many complete waves are indicated in the sketch?
2. Write down the letters that indicate any TWO points that are:
1. in phase
2. out of phase
3. Represent ONE wavelength.
3. Calculate the amplitude of the wave.
4. Show that the period of the wave is 0,67 s.
5. Calculate the frequency of the waves.
6. Calculate the velocity of the waves.

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