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13.1Mechanical pulses and waves

If we take the free end of a taut, horizontal rope and we give it a sudden up and down jerk, a pulse will be produced that will travel down the length of the rope at a certain speed (Figure 13.1).

Figure 13.1: A pulse on a taut rope. The rope itself moves up and down. What moves to the right is the pulse.

The upward force due to the hand forces a section of the rope to move up. Because of the tension in the rope, this section pulls the section in front of it upwards. In this way the pulse moves forward. In the meantime the hand has moved down, forcing sections of the rope to return to their horizontal equilibrium position. Neighbouring sections again do the same because of the tension in the rope. If the motion of the hand holding the free end is continuous then a mechanical wave a disturbance that transfers energy and momentum through oscillations of the particles of a medium is established on the rope. Figure 13.2 shows two full waves travelling down the length of the rope.

Figure 13.2: A continuous wave travelling along the rope.

Now if the right end of the rope is attached to a body that is free to move, the body will move when the wave gets to it. This means that the wave transfers energy and momentum.

KEY POINT

A mechanical wave is a disturbance that travels in a medium through oscillations of the particles of the medium. A wave transfers energy and momentum. The direction of propagation of the wave is the direction of energy transfer.

How does a wave transfer energy? As we will learn in the next chapter, when a wave travels in a medium the particles of the medium perform simple harmonic oscillations about their equilibrium positions. In Chapter 12.1 we learned that in simple harmonic motion we have kinetic and elastic potential energy. This energy has to be supplied by the source that creates the wave. Suppose that the source provides an amount of energy E in order to produce one full wave. This happens in a time equal to one period. In the next period the source has to provide an additional amount E and so on. In this way, as the wave progresses, energy is carried along by the wave, flowing from the source through the medium in the direction of propagation of the wave. It follows that the speed at which energy is being transferred is the speed of the wave.

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Figure 13.3 shows one full wave.

Figure 13.3: One full wave.

The length of one full wave is known as the wavelength the length of one full wave, λ.

It is also the distance from crest to crest or trough to trough (Figure 13.4). (A crest is the highest point on the wave and a trough the lowest.)

Figure 13.4: Three distances that all give the wavelength.

The time to create one full wave is known as the period the time it takes for an orbiting object to make one complete revolution; the time for one full oscillation or one full wave of the wave, T.

frequency the number of complete revolutions or full waves made in one second; f=1 T, f, is the number of full waves created in 1 s. f=1 T.

If the period of wave is T = 0.25 s, for example, then the number of full waves produced in one second is 4.

Suppose we have a wave on a rope, of wavelength λ, period T and frequency f. Figure 13.5 shows the rope at time zero when we have not yet produced any waves, and at time T where we have produced one full wave. The wave has moved a distance equal to one wavelength in a time equal to one period and so the speed of the wave is:

v= distance for one full wave time for one full wave =λ T

Since f=1 T we also have that:

v=f λ

Figure 13.5: In a time of one period, the wave has moved forward a distance of one wavelength. The speed of the wave depends only on the properties of the medium and not on how it is produced.

KEY POINT

The speed of a wave is determined only by the properties of the medium and not on how the wave is produced.

CHECK YOURSELF 1

A wave travels in a medium with speed v and wavelength λ. The frequency of the wave is doubled. What happens to the speed of the wave and its wavelength?

Worked Example 13.1

A sound wave of frequency 450 Hz is emitted from A and travels towards B, a distance of 150 m away. How many full waves fit in the distance from A to B? (Take the speed of sound to be 340 m s−1.)

Answer

The wavelength is:

λ=340 450=0.756 m

Thus the number of full waves that fit in the distance 150 m is:

N=150 0.756

N=198 full wavelengths.

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Worked Example 13.2

The noise of thunder is heard 3 s after the flash of lightning. How far away is the place where lightning struck? (Take the speed of sound to be 340 m s−1.)

Answer

Light travels so fast that we can assume that lightning struck exactly when we see the flash of light.

If thunder is heard 3 s later, it means that it took 3 s for sound to cover the unknown distance, d. Thus:

d=v t

d=340×3

d=1020 m.

Worked Example 13.3

Water wave crests in a lake are 5.0 m apart and pass by an anchored boat every 2.0 s. What is the speed of the water waves?

Answer

Use v = f λ. The wavelength is 5.0 m and the period is 2.0 s. So:

v=5.0 2.0

v=2.5 m s−1.

TEST YOUR UNDERSTANDING

1In football stadiums fans often create a ‘wave’. Suggest factors that determine the speed of the ‘wave’.

2A number of dominoes are stood next to each other along a straight line. A small push is given to the first domino, and one by one the dominoes fall over.

aOutline how this is an example of wave motion.

bSuggest how the speed of the wave pulse could be increased.

cDesign an experiment in which this problem can be investigated.

3Draw a labelled diagram to explain the terms:

●wavelength

●period

●amplitude

●crest

●trough.

4Calculate the frequency of light (speed 3.0 × 108 m s–1) of wavelength 656 nm.

5Calculate the wavelength that corresponds to a sound frequency of:

a256 Hz

b25 kHz.

(Take the speed of sound to be 330 m s–1.)

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6A boy shouts towards a vertical cliff a distance of 250 m away and 1.5 s later hears an echo. What is an estimate of the speed of sound from these data?

7A sound wave (speed 330 m s–1) and a light wave (speed 3.0 × 108 m s–1) have the same wavelength. What is the ratio of the light to the sound frequency?