Waves

7.1 Progressive waves

A progressive wave transfers energy from one place to another without transferring matter. You can see this on ropes, on springs and on water in a ripple tank.

Wave terms

Using a CRO

A cathode-ray oscilloscope displays voltage against time.

• The time-base setting (such as 2 ms/div) is the time per horizontal division. Measure the divisions for one cycle, multiply, and that is the period. Then $f = 1/T$.
• The y-gain setting (such as 5 V/div) is the voltage per vertical division. Amplitude is the divisions from the centre to a peak, multiplied by the y-gain.

The wave equation

In one period $T$, a wave travels one wavelength. So $v = \lambda/T$, and since $f = 1/T$:

$v = f\lambda$

Intensity

Intensity is the power per unit area, at right angles to the direction of travel, $I = P/A$, measured in W m⁻². For a progressive wave:

$I \propto A^2$

Doubling the amplitude makes the intensity four times as large. From a point source spreading out equally, $I = P/(4\pi r^2)$.

7.2 Transverse and longitudinal waves

Reading wave graphs

Two graphs look alike, and you must check the axis:

• Displacement against distance is a snapshot of the wave. It gives the wavelength.
• Displacement against time shows one point as time passes. It gives the period.

For a longitudinal wave, a graph of displacement against distance is still a sine curve, even though the particles move along the wave's direction. Compressions are where neighbouring particles move towards a point.

7.3 Doppler effect for sound waves

When a source of sound moves relative to a stationary observer, the observed frequency differs from the frequency the source emits. Coming towards you, the wavefronts are bunched up, so you hear a higher frequency. Going away, they are spread out, so you hear a lower one.

$f_o = \frac{f_s\, v}{v \pm v_s}$

where $v$ is the speed of sound and $v_s$ is the speed of the source. Use minus when the source approaches (the observed frequency goes up) and plus when it recedes.

7.4 Electromagnetic spectrum

All electromagnetic waves are transverse and travel at the same speed in free space, $c = 3.00 \times 10^{8}$ m/s.

The X-ray and gamma-ray ranges overlap because they are named by how they are made, not just their wavelength.

7.5 Polarisation

Polarisation can only happen to transverse waves. In a plane-polarised wave, the oscillations are all in one plane that contains the direction of travel. Unpolarised light has oscillations in many planes.

A polarising filter only transmits the component of the oscillations along its transmission axis. So the amplitude after the filter is $A_0\cos\theta$. Since intensity depends on amplitude squared:

$I = I_0\cos^2\theta$

This is Malus's law, where $\theta$ is the angle between the wave's plane of polarisation and the transmission axis.