Waves

Wave motion, and the words you need to describe it

A wave is a way of transferring energy from one place to another without transferring matter. Shake the end of a rope and a pulse runs along it, but the rope itself does not travel: each small section of rope oscillates about its own fixed position and passes the energy on to the next section. The same is true of a compression travelling along a spring and of water waves in a ripple tank, where a motor-driven dipper makes a ripple pattern that moves outwards while the water itself only bobs up and down. Every wave needs something to oscillate, and the energy travels through that material, or through the electric and magnetic fields in the case of light.

The seven quantities in the syllabus describe the same wave from different angles. Displacement is the distance of an oscillating point from its undisturbed position, and it is positive on one side of that position and negative on the other. Amplitude is the maximum displacement, so it is measured from the undisturbed position to a crest, never from trough to crest. Wavelength is the distance between two successive points that are in phase, such as two successive crests or two successive troughs, and it is also the distance the wave travels in one period.

The remaining three terms describe the timing. Period is the time taken for one complete oscillation of a point, or for one wavelength to pass a fixed point. Frequency is the number of complete oscillations passing a fixed point each second, measured in hertz (Hz), so frequency and period are related by f = 1 / T. Phase difference is how far one oscillation leads or lags behind another at the same instant, measured in degrees or radians: one whole wavelength apart is 360 degrees (or 2 pi radians), half a wavelength apart is 180 degrees, and a quarter of a wavelength is 90 degrees.

Two points do not have to be a whole number of wavelengths apart to be in phase. The crest of one wave and the crest of the next are in phase because they are one wavelength apart, and so are points 270 degrees apart in the other sense, because phase difference is really the fraction of a cycle by which one oscillation is ahead of the other.

• Displacement: how far a point has moved from its undisturbed position, in metres.
• Amplitude: the maximum displacement, in metres.
• Wavelength: the distance between two successive points in phase, in metres.
• Period: the time for one complete oscillation, in seconds.
• Frequency: the number of oscillations per second, f = 1 / T, in hertz (Hz).
• Phase difference: the fraction of a cycle by which one oscillation leads another, in degrees or radians.

Deriving and using v = f lambda

The wave equation follows from the definitions you already have, and an exam question may ask you to derive it rather than recall it. Speed is distance travelled divided by time taken. During one period T, a wavefront moves forward by exactly one wavelength, lambda, so the speed of the wave is v = lambda / T. Frequency is the number of oscillations per second, f = 1 / T, which rearranges to 1 / T = f. Substituting this into v = lambda / T gives v = f x lambda, where v is in metres per second, f is in hertz and lambda is in metres.

A worked example. A ripple tank dipper vibrates at 8.0 Hz and the waves it makes are 4.5 cm apart. Convert the wavelength to metres first: 4.5 cm = 0.045 m. Then v = f lambda = 8.0 x 0.045 = 0.36 m s-1. For a sound wave of frequency 440 Hz travelling through air at 340 m s-1, the wavelength is lambda = v / f = 340 / 440 = 0.77 m. Notice that the two rearrangements are used in opposite directions: in the first the wavelength is known and the speed is wanted, in the second the speed is known and the wavelength is wanted.

The frequency of a wave is fixed by the source and does not change as the wave travels: a 440 Hz tuning fork sends out 440 oscillations every second whether the sound is in air or in water. The speed, however, depends on the medium, because the medium decides how quickly one oscillating point can pass the disturbance to the next. Since f is fixed and v changes, the wavelength must change too, by lambda = v / f, so sound travelling from air into water keeps its frequency but has a longer wavelength because it travels faster.

Reading frequency and amplitude from a cathode-ray oscilloscope

A cathode-ray oscilloscope (CRO) draws the signal as a graph of voltage against time, and its two controls set the scales of that graph. The time-base sets the horizontal scale, in milliseconds per division or microseconds per division, so it tells you how much time each square on the screen represents. The y-gain sets the vertical scale, in volts per division, so it tells you how much voltage each square represents. With a periodic signal the trace is a steady repeating pattern, and measuring that pattern gives the period and the amplitude of the wave.

To find the frequency, measure the horizontal distance for one complete cycle in divisions and multiply by the time-base setting to get the period; then use f = 1 / T. To find the amplitude, measure the vertical peak-to-peak height in divisions, multiply by the y-gain setting to get the peak-to-peak voltage, and halve it. A trace spanning 4.0 divisions with a time-base of 5.0 ms per division has a period of 4.0 x 5.0 = 20 ms = 0.020 s, so f = 1 / 0.020 = 50 Hz. If the same trace has a peak-to-peak height of 6.0 divisions with a y-gain of 2.0 V per division, the peak-to-peak voltage is 6.0 x 2.0 = 12 V, so the amplitude is 12 / 2 = 6.0 V.

• Time-base: the number of milliseconds (or microseconds) each division represents, read from the horizontal scale.
• Y-gain: the number of volts each division represents, read from the vertical scale.
• Period = (number of divisions for one cycle) x (time-base setting).
• Frequency = 1 / period.
• Amplitude = (peak-to-peak height in divisions) x (y-gain setting) / 2.

Energy, intensity and amplitude

A progressive wave carries energy away from the source, and intensity is the rate at which that energy arrives at each square metre of surface held at right angles to the direction the wave is travelling: intensity = power / area, measured in W m-2. Intensity is not the same as power. A large loudspeaker emits more power than a small one, but the intensity at your ear also depends on how far away it is, because the same power is spread over a larger area.

For a progressive wave the intensity is proportional to the square of the amplitude, I is proportional to A^2. Doubling the amplitude therefore multiplies the intensity by four, and multiplying the amplitude by three multiplies the intensity by nine. The reason is that the energy carried by a wave depends on how far each oscillating particle is displaced and how fast it is moving, and both of those quantities increase in proportion to the amplitude, so the energy depends on the amplitude twice over.

Worked example. A wave of amplitude 2.0 mm has an intensity of 0.50 W m-2. A wave of the same frequency but amplitude 4.0 mm has an amplitude twice as large, so its intensity is 2^2 = 4 times greater: 4 x 0.50 = 2.0 W m-2. If instead you wanted to double the intensity, you would have to increase the amplitude by a factor of 2^0.5, that is 1.41, because (1.41)^2 is approximately 2.

When a wave spreads out from a small source in three dimensions, the same power passes through a spherical surface of ever greater area, so the intensity falls with the square of the distance from the source: doubling the distance quarters the intensity. That is why a siren is much quieter at 200 m than at 100 m, and it is the same geometrical spreading that makes amplitude fall with distance.

Transverse and longitudinal waves

Waves are classified by the direction in which the oscillations point compared with the direction in which the energy travels. In a transverse wave the oscillations are perpendicular to the direction of travel of the wave: shake a rope up and down and the crests travel away from your hand while the rope moves across the line of travel. In a longitudinal wave the oscillations are parallel to the direction of travel: push and pull the end of a spring and compressions travel along it, with the coils moving backwards and forwards along the same line that the energy follows.

Sound in air is longitudinal. The vibrating source creates regions where the air is squeezed and its pressure and density are slightly above the surrounding value, called compressions, and regions where the air is stretched and the pressure is slightly below it, called rarefactions, and these travel away from the source. A microphone records the pressure changes, and the signal is usually displayed as a graph, but that graph is a representation of the wave, not a picture of the air molecules. A longitudinal wave can also be drawn as a row of dots that are closer together at a compression and further apart at a rarefaction.

Electromagnetic waves, waves on a rope or a string and water surface waves are all transverse. Sound in air or water, compression waves in a spring and the ultrasound used in medical scanners are longitudinal. Only transverse waves can be polarised, because polarisation means choosing a single plane for oscillations that are already perpendicular to the direction of travel; there is no equivalent choice for oscillations that lie along the direction of travel, which is why a beam of sound cannot be polarised.

The Doppler effect for sound

When a source of sound moves relative to a stationary observer, the observer hears a frequency that is different from the frequency the source is emitting. A source approaching the observer is heard at a higher frequency than the source frequency, and a source moving away is heard at a lower frequency. The source itself has not changed: only 500 compressions leave the siren each second, but they arrive at the observer more often when the siren is closing in and less often when it is drawing away.

The reason is the crowding of the wavefronts. As the source moves towards the observer, each new wavefront is emitted from a point closer to the observer than the one before, so the wavefronts are squeezed together in front of the source and the distance between them, which is the observed wavelength, is shorter than the wavelength you would measure with the source at rest. The speed of sound in the air is unchanged, so f = v / lambda with a smaller lambda means a higher observed frequency. Behind a receding source the wavefronts are stretched further apart, and the observed frequency is lower.

The observed frequency is given by f_o = f_s v / (v +/- v_s), where f_s is the source frequency, f_o is the frequency heard by the stationary observer, v is the speed of sound in the air and v_s is the speed of the source. Use the minus sign when the source is approaching, because the source is moving into its own wavefronts, and the plus sign when the source is receding. The equation only applies to a source moving directly towards or directly away from the observer, and to a stationary observer: the syllabus requires the source to move, not the observer and the source together.

Worked example. A siren emits 500 Hz and the speed of sound is 340 m s-1. Approaching at 30 m s-1, f_o = 500 x 340 / (340 - 30) = 500 x 340 / 310 = 548 Hz. After passing, f_o = 500 x 340 / (340 + 30) = 500 x 340 / 370 = 459 Hz. The observer hears a sudden drop of about 90 Hz as the vehicle goes past, which is why an ambulance seems to change note at the moment it passes you.

The electromagnetic spectrum

All electromagnetic waves are transverse waves, in which oscillating electric and magnetic fields travel together, and in free space they all travel at the same speed, c = 3.00 x 10^8 m s-1, whatever their frequency. In a material the speed is lower and depends on the material, but the frequency is set by the source and does not change. Because v = f lambda in free space becomes c = f lambda, a high-frequency region of the spectrum always has a short wavelength and a low-frequency region always has a long wavelength.

The spectrum is usually split into seven regions, from the longest wavelength to the shortest: radio waves, microwaves, infrared, visible light, ultraviolet, X-rays and gamma rays. The boundaries between neighbouring regions are conventions rather than sharp walls, and the two shortest regions overlap: the real difference between X-rays and gamma rays is where they come from, since X-rays are produced by fast electrons hitting a metal target in an X-ray tube while gamma rays are produced by changes inside a nucleus.

Visible light occupies only a tiny window: wavelengths from about 700 nm (red) to about 400 nm (violet), which is 7 x 10-7 m to 4 x 10-7 m, and it is the only region the human eye detects. Red light has the longest wavelength of the visible colours and violet the shortest.

Polarisation and Malus's law

Polarisation is the restriction of the oscillations of a transverse wave to a single plane. Light from a lamp or from the Sun is unpolarised: its electric field oscillates in every plane containing the direction of travel. A polarising filter only transmits the component of the oscillation parallel to its transmission axis, so light leaving a filter is plane-polarised and cannot pass through a second filter whose axis is at right angles to the first. Because polarisation requires oscillations that are perpendicular to the direction of travel, it is direct evidence that light is a transverse wave, and it is impossible for a longitudinal wave such as sound.

Unpolarised light passing through a single filter always comes out at half the original intensity, whatever the orientation of the filter, because on average half of the oscillations lie closer to the transmission axis than to the perpendicular direction. When unpolarised light meets a filter, the angle between the axis of the filter and the direction of polarisation is therefore meaningless; the quantity that matters is the angle between the plane of polarisation of the incoming light and the transmission axis of the filter.

For light that is already plane-polarised, the transmitted intensity is given by Malus's law, I = I0 cos^2 theta, where I0 is the intensity of the light arriving at the filter, I is the intensity leaving it and theta is the angle between the plane of polarisation of the incoming light and the transmission axis of the filter. Worked example: polarised light of intensity 400 W m-2 falls on a filter with its plane of polarisation at 60 degrees to the transmission axis. Then I = 400 x cos^2 60 = 400 x (0.50)^2 = 400 x 0.25 = 100 W m-2.

A series of filters is handled one at a time, using the light that leaves each filter as the input to the next. If a second filter is placed after the first with its axis 30 degrees away, the light reaching it is the 100 W m-2 polarised in the direction of the first axis, so I = 100 x cos^2 30 = 100 x 0.75 = 75 W m-2. The angle that matters is always between the polarisation direction of the light and the axis of the next filter, never between the first and last filter alone.

• Polarisation can only happen to transverse waves, so it is evidence that light is transverse.
• A polarising filter transmits only oscillations parallel to its transmission axis.
• Unpolarised light loses half its intensity in one filter, at any orientation.
• For plane-polarised light, I = I0 cos^2 theta.
• For several filters, apply Malus's law once for each filter in turn.
• Malus's law uses cos^2, not cos: at 60 degrees the transmitted intensity is 25 percent, not 50 percent.