It is much more difficult to picture a longitudinal wave than a transverse wave. If the spring with the wave travelling down it (Fig. 2.2) is compared with an unstretched spring, some regions can be seen where the coils are closer together, and other regions where the coils are further apart. The part of the spring where the coils are closely spaced is called a region of compression, and the region where they are separated more widely, is the rarefaction region.

Waves on the sea are generally described in terms of peaks and troughs. The movement up to a wave crest, down to a trough, and back up to the crest again is known as a cycle of oscillation. A cork floating in the sea bobs up and down as the waves go past. The difference in height of the cork between a crest and a trough is twice the amplitude. Perhaps a simpler way of visualising the amplitude is as the difference in water height above the seabed between a flat, calm sea and the crest of the wave. The number of wave crests passing the cork in a second is the wave frequency (f). Frequency is measured in hertz (Hz), where 1Hz is 1 cycle/second. The time that elapses between two adjacent wave crests passing the cork is the period (τ) of the oscillation. This has units of time: if each cycle takes t seconds, there must be 1/t cycles in each second. The number of cycles that occur in a second has already been defined as the frequency, and so can be written as follows:

The distance between two adjacent wave crests is the wavelength (λ).

Figure 2.3A and B show a wave frozen at two moments, a short time apart. It can be seen that the different points on the wave have changed position relative to the central line but have not moved in space. In fact, if you tracked the motion of point A over several periods, the movement up and down would look like the picture shown in Figure 2.3C. The speed at which the wave crests move is known as the wave speed. As the wave moves a wavelength (λ) in one cycle, and as one cycle takes a time equal to the period t, then the wave speed (c) is given by the equation:

Figure 2.3 A and B, the position of two points A and C in the path of a wave as it passes through. The displacements shown are frozen at two different times, between which the wave has moved on a fraction of a wavelength. C, the displacement of the point during two cycles.

It is known that 1/t is the same as the frequency f, and so:

In Figure 2.4, points A and B on the wave (or, equally, A¹ and B¹) are moving in the same way and will reach the crest (or trough) together. These points are said to be in phase with each other. The movement from A to B (or A¹ to B¹) represents one cycle of the wave motion. A and C, however, are not in phase: C is a quarter of a cycle ahead of A and they are said to have a phase difference (ϕ) of a quarter cycle. Phase is usually expressed as an angle, where a complete cycle is 2π radians (or 360°). A quarter cycle therefore represents a phase difference of π/2 radians (90°). This is illustrated in Figure 2.5.

Figure 2.4 Points A and B, and also A¹ and B¹, are always in the same relative position in the wave. They are in phase. Points A and C are out of phase.

Figure 2.5 The phase angle can be likened to the turning of a water wheel. Imagine two wheels, A and B, both with a mark on their rim. A does not move but B turns and, as it does, the rim mark executes the circles, each complete turn representing one cycle. The angle through which the mark turns in one cycle is 360° (2π radians). Thus, for example, compared with A, when the mark on B’s rim has moved around a quarter of a turn (cycle), the angle between the two marks is a quarter of 360° (90° or π/2 radians); after half a turn, the angle between the two marks is 180° or π radians. This angle between the two marks is analogous to the phase difference. As B rotates, the height of the mark above the wheel’s hub varies. If the wheel turns at a constant speed, then the mark’s height traces out a sine wave when plotted against time.

REFLECTION AND REFRACTION OF WAVES

When waves travelling through a medium arrive at the surface between two media, some of the energy is reflected back into the first medium and some is transmitted through into the second medium. The proportion of the total energy that is reflected is determined by the properties of the two media involved. Figure 2.6 shows what happens when waves are reflected by a flat (plane) surface. An imaginary line that is perpendicular to the surface is called the normal. The law of reflection states that the angle between the incident (incoming) wave and the normal is always equal to the angle between the reflected wave and the normal. If the incident wave is at normal incidence (perpendicular to the surface), the wave is reflected back along its path.

Figure 2.6 The law of reflection states that the angle of incidence equals the angle of reflection.

The waves that are transmitted into the second medium may also undergo refraction. This is the bending of light towards the normal when it travels from one medium into one in which the wave speed is lower, or away from the normal when the wave speed in the second medium is higher (Fig. 2.7). For example, light bends towards the normal as it enters water from air since it travels more slowly in water than in air, and so a swimming pool may appear shallower than it really is.

Figure 2.7 When a beam passes from one medium to another it can be refracted (i.e. it changes its direction).

As has been discussed earlier, waves carry energy. There are conditions, however, in which the transport of energy can be stopped, and the energy can be localised. This happens in a standing (stationary) wave. A standing wave is produced when an incident wave meets a returning reflected wave with the same amplitude. When the two waves meet, the total amplitude is the sum of the two individual amplitudes. Thus, as can be seen in Figure 2.8A, if the trough of one wave coincides with the crest of the other, the two waves cancel each other out. If, however, the crest of one meets the crest of the other, the wave motion is reinforced (Fig. 2.8B) and the total amplitude doubles. In the reinforced standing wave there are points that always have zero amplitude; these are called nodes. Similarly, there are points that always have the greatest amplitude, and these are called antinodes. Nodes and antinodes are shown in Figure 2.8B. The distance between adjacent nodes, or adjacent antinodes, is one-half of the wavelength (λ/2).

Figure 2.8 A standing wave is formed when two waves of equal amplitude travelling in opposite directions meet. A, the two waves cancel each other out; B, the two waves add to reinforce each other.

POLARISATION

When flicking the Slinky spring up and down to produce a transverse wave, one has an infinite number of choices as to the direction in which to move it, so long as the motion is at right angles to the line of the spring. If the spring is always moved in a fixed direction, the wave is said to be polarised – the waves are in that plane only. However, if the waves (or directions in which the spring is moved) are in a number of different directions, the waves are unpolarised. It is possible to polarise the waves by passing them through a filter that allows only waves that are in one plane through. This can be visualised by envisaging a piece of card with a long narrow slit in it. This will allow the waves formed in the plane of the slit to go through, but no others – the card therefore acts as a polarising filter.

ELECTRICITY AND MAGNETISM

Everyone is familiar with effects of electrical charges, even if they are not aware of their causes. The ‘static’ experienced when brushing newly washed hair, or undressing, and the electrical discharge obvious in lightning are examples of the effects of charges.

ELECTRICITY

Matter is made up of atoms, an atom being the smallest particle of an element that can be identified as being from that element. The atom consists of a positively charged central nucleus (made up of positively charged protons and uncharged neutrons), with negatively charged particles (electrons) orbiting around it, resembling a miniature solar system. An atom contains as many protons as there are electrons, and so has no net charge. If this balance is destroyed, the atom has a non-zero net charge and is called an ion. If an electron is removed from the atom it becomes a positive ion, and if an electron is added the atom becomes a negative ion.

Two particles of opposite charge attract each other, and two particles of the same charge repel each other (push each other away). Hence, an electron and a proton are attracted to each other, whereas two electrons repel each other.

The unit of charge is a coulomb (C). An electron has a charge of 1.6 × 10⁻¹⁹ C, so it takes a very large number (6.2 × 10¹⁸) of electrons to make up one coulomb.

The force between two particles of charge q₁ and q₂ is proportional to the product of q₁ and q₂ (q₁ × q₂), and inversely proportional to the distance between them (d) squared (Fig. 2.9). Thus, the force is proportional to q₁q₂/d². The constant of proportionality (i.e. the invariant number) necessary to allow one to calculate the force between two charges is 1/4πɛ, where ɛ is the permittivity of the medium containing the two charges:

Figure 2.9 Two particles of charge q₁ and q₂ a distance of d apart experience a force between them that is proportional to q₁q₂/

If one of the charges is negative, then the force is attractive. If the particles are in a vacuum, the permittivity used is ɛ₀; this is known as the permittivity of free space. For a medium other than a vacuum, the permittivity is often quoted as a multiple of ɛ₀, where the multiplying factor, κ, is known as the relative permittivity or dielectric constant. So:

Electric fields

An electric field exists around any charged particle. If a smaller charge that is free to move is placed in the field, the paths it will move along are called lines of force (or field lines). Examples of fields and their patterns are shown in Figure 2.10.

Figure 2.10 Examples of electric fields near charged particles and plates. A, field between two particles of equal and opposite charges; B, field between two positively charged particles; C, field between a charged particle and an oppositely charged plate; D, field between two oppositely charged plates.

The electric field strength, E, is defined as the force per unit charge on a particle placed in the field. A little thought shows that E = F/q, where F is the force and q is the particle’s charge. The units used to describe E are newtons/coulomb (N/C).

If E is the same throughout a field, it is said to be uniform. In this case, the field lines are parallel to each other as shown in Figure 2.10D. If a charged particle is moved in this field, work is done on it, unless it moves perpendicular to the field lines. This is somewhat analogous to moving a ball around on Earth. If the ball is always kept at the same height, and moved horizontally, its potential energy remains constant. If the ball is raised or lowered, its potential energy is changed. The ball has no potential energy when it lies on the ground. In a non-uniform field where the lines are not parallel, moving a charged particle always results in a change of potential energy. The electric potential, V, is defined as the potential energy per unit charge of a positively charged particle placed at that point. Electric potential is measured in units of volts. As the position at which the electric potential energy is zero is taken as infinity, another way of thinking of the electrical potential at a point is as the work done in moving the charge to that point from infinity. In practice it is easier to compare the electrical potential at two points in the field than to consider infinity. The difference in the work required to move a charge from infinity to a point, A, and that required to move it to another point, B, is called the potential difference (p.d.) between the two points; this is also measured in volts. The p.d. is best thought of as a kind of pressure difference. Between the two points there will be a gradient in potential (just as there is a pressure gradient between the top and bottom of a waterfall). This gradient is described in units of volts/metre. In a uniform field between parallel plates with potential difference V, and separation d, the potential gradient is given by V/d. If a particle of charge q is moved from one plate to another, the work done is qV. Work is force × distance, and so the force, F, is given by: