NZ NCEA Level 3 Physics

Describing a wave

A periodic wave repeats in both space and time. Its wavelength $\lambda$ (metres) is the distance between two neighbouring points that are in step, its frequency $f$ (hertz) is the number of complete oscillations each point makes per second, and its period is $T = \frac{1}{f}$. The speed at which the pattern moves is

where $v$ is in $\mathrm{m\,s^{-1}}$. For a given medium the speed is fixed by the medium, so a wave of higher frequency has a shorter wavelength. Sound in air at room temperature travels at about $343\,\mathrm{m\,s^{-1}}$, and light in air at very nearly $3.00\times10^{8}\,\mathrm{m\,s^{-1}}$.

Two points on a wave are in phase when they are at the same stage of their oscillation, which happens when they are a whole number of wavelengths apart. They are in antiphase when one is at a crest while the other is at a trough, which happens when they are an odd number of half-wavelengths apart. The phase difference is measured as an angle, where one whole wavelength corresponds to $2\pi$ radians, so a separation of $\Delta$ along the wave gives

The principle of superposition

When two waves pass through the same point, the displacement of the medium there is the sum of the displacements each wave would produce on its own. The waves do not change each other, and once they have passed through one another each carries on as before. This is the principle of superposition, and it is the basis of interference, diffraction, standing waves and beats.

Adding two displacements is ordinary addition with signs. If one wave is pushing a point up by $3\,\mathrm{mm}$ and the other is pushing it down by $1\,\mathrm{mm}$, the point is $2\,\mathrm{mm}$ up. If two equal waves are exactly in step the displacements add to double, which is constructive interference. If they are exactly in antiphase the displacements are equal and opposite and the sum is zero, which is destructive interference. Anything in between gives an intermediate amplitude.

The energy carried by a wave is proportional to the square of its amplitude, so the intensity at a point where two equal waves interfere constructively is four times that of one wave alone, not twice. At a point of destructive interference the intensity is zero. The energy is not lost: it is redistributed, so the extra at the bright places is exactly made up by the shortfall at the dark places when the whole pattern is averaged.

Coherent sources

For a steady pattern of maxima and minima to be seen, the two sources must be coherent. Coherent sources have the same frequency and a constant phase difference between them. If the phase difference wandered at random, the positions of the maxima would shift too quickly to be noticed and only an even average would be seen.

Two separate lamps are not coherent, because each atom emits light in short bursts with no relationship to its neighbours. Light is made coherent by taking a single beam and splitting it, for instance by passing it through two narrow slits, so that any change in the source affects both halves together. Two loudspeakers driven by the same signal generator are coherent for the same reason, and so are two dippers on a ripple tank that are driven by the same motor. A laser gives light with a single wavelength and a fixed phase relationship over a wide beam, which makes it convenient.

Path difference

Waves leave coherent sources in phase and travel different distances to reach a point. The path difference is the difference between those two distances, and it decides the phase difference on arrival. If the two sources are in phase with each other:

• constructive interference occurs where the path difference is a whole number of wavelengths, $\Delta = n\lambda$ with $n = 0, 1, 2, \dots$
• destructive interference occurs where the path difference is an odd number of half-wavelengths, $\Delta = \left(n + \tfrac{1}{2}\right)\lambda$

If the sources are in antiphase, the conditions swap: a whole number of wavelengths then gives destructive interference and a half-odd number gives constructive interference. A point midway between two sources has zero path difference, so it is a maximum for in-phase sources and a minimum for sources in antiphase.

Worked example

Two loudspeakers A and B are driven in phase by one signal generator at $686\,\mathrm{Hz}$. The speed of sound is $343\,\mathrm{m\,s^{-1}}$. A microphone is $4.00\,\mathrm{m}$ from A and $5.25\,\mathrm{m}$ from B.

The path difference is two and a half wavelengths, which is an odd number of half-wavelengths. The two waves arrive in antiphase and the microphone sits at a minimum, where the signal is close to zero. If the microphone were moved until it was $5.50\,\mathrm{m}$ from B, the path difference would be $1.50\,\mathrm{m}$, which is $3\lambda$, and the signal would be at a maximum.

A minimum is seldom exactly zero in practice. The speakers give slightly different amplitudes at the microphone because of the different distances, and sound reflected from walls adds further waves, so the cancellation is only partial.

Changing the frequency

Raising the frequency shortens the wavelength, so the same path difference is a larger number of wavelengths. In the example above, doubling the frequency to $1372\,\mathrm{Hz}$ gives $\lambda = 0.250\,\mathrm{m}$ and $\frac{\Delta}{\lambda} = 5$, which turns the minimum into a maximum. This is why the positions of interference patterns depend on wavelength, and why white light, which is a mixture of wavelengths, spreads into colours where a single colour would give a clean set of fringes.