Waves revision

This page contains a summary of the full waves topic for physics! It’s useful as a revision guide.

Key terms in waves

Wave speed equation

\text{wave speed} = \text{frequency} \times \text{wavelength}

Law of reflection

• The angle of incidence is equal to the angle of reflection
  • The angle of incidence is the angle to the normal that the wave comes in at.
  • The angle of reflection is the angle to the normal that the wave goes out at.
• We can write this as i=r:
  • i is the angle of incidence
  • r is the angle of reflection

Polarisation

A polarised wave is a wave that is ‘filtered’ to only be let through if it’s rotated the correct way.

We can use a polariser to do this. They have a tiny slit(s) in one rotation, which blocks all waves which oscillate in the ‘wrong’ direction.

Only transverse waves can be polarised.

Phase difference

The phase difference of two waves is an angle which we usually measure in radians.

It tells us how ‘offset’ the waves’ peaks and troughs are from each other.

• If waves have a phase difference of 0\pi radians, 2\pi radians, or any multiple of 2\pi radians, they will be in phase as their peaks and troughs perfectly match up.
• If they have a phase difference of 1\pi radians, 3\pi rads, or any odd multiple of \pi radians, then they are in antiphase as the peak of one directly lines up to the trough of the other.
• Otherwise, they’re out of phase in no meaningful way.

We can also compare the phase difference of two points on the same wave:

• Just find the phase angle of each point, then subtract them to find the phase difference.

Principle of superposition

When two waves interfere, the resultant amplitude at any point is equal to the sum of the amplitudes of the individual waves, at that point.

Constructive interference

When the waves interfere (superpose), if the peaks perfectly align with the other peaks and the troughs are perfectly aligned to the other troughs, that’s constructive interference.

In other words, the phase difference is 0 or a multiple of 2\pi.

The amplitudes at any point will double if the waves have the same amplitude, or add together if they’re somewhat different.

The waves are in phase in constructive interference

Destructive interference

When the waves interfere (superpose), if the peaks perfectly align with the other troughs and the troughs are perfectly aligned to the other peaks, that’s destructive interference.

In other words, the phase difference is 1\pi or another odd multiple of \pi.

The amplitudes at any point will cancel out if the waves have the same amplitude, or subtract from each other to make a much smaller wave otherwise.

The waves are in antiphase in destructive interference

Refraction

Refraction is when the speed of light or another wave changes when it passes through a barrier into a new medium.

The light will bend, as its speed changes so therefore so does its direction.

Refractive index

The refractive index tells us how the speed of a wave will change when passing between two mediums.

n = \frac{c}{c_s}

Snell’s law

n_1\sin\theta_1=n_2\sin\theta_2

Total internal reflection

For total internal reflection to occur, the angle of incidence must be greater than the critical angle.

\theta_c=\frac{n_2}{n_1}

The key thing is that, for TIR to occur, the refractive index of the material the wave is currently in must be greater than the refractive index of the material it’s moving into/towards.

Single-slit diffraction pattern

• The width of the central fringe is double the width of the other fringes.
• If the slit width is increased, the fringes get narrower and closer together.
• Larger wavelength means wider fringes, as the waves spread out more.
• The central fringe is the brightest, because it has the most constructive interference.
• The intensity of the fringes decreases as you move away from the central fringe, as the waves are less in phase and therefore less constructive.

Coherence

If light is coherent, it means that the waves have a constant phase difference and the same frequency.

We need coherent light to get a stable interference pattern, e.g. when investigating single or double slit diffraction patterns.

We can make light coherent by using a laser, or by using a single slit to ‘filter’ the light from a non-coherent source, like. a light bulb.

Double-slit interference pattern

• The central fringe is the brightest, because it has the most constructive interference.
• The intensity of the fringes decreases as you move away from the central fringe, as the waves are less in phase and therefore less constructive.
• The fringes are equally spaced, as the path difference between the two slits is constant.
• The fringe separation increases if the wavelength increases, as the waves spread out more.
• The fringe separation decreases if the distance between the slits increases, as the path difference between the two slits is smaller.

Double slit equation

Ws=\lambda D

Where:

• n is the fringe number (e.g. 1 for the first bright fringe, 2 for the second, etc.)
• \lambda is the wavelength of the light
• d is the distance between the slits
• \theta is the angle between the central fringe and the nth fringe

Path difference

The difference in the distance travelled by two waves is called the path difference.

White light in double-slit interference

• The central fringe is white as all the wavelengths interfere constructively
• Side fringes are spectra (basically rainbow) as the different wavelengths are projected on the screen at different angles, so show up at different positions.
• Blue light is closest to the central fringe, as it has the smallest wavelength so is diffracted the least.

Diffraction grating

• Instead of having just two slits, a diffraction grating has loads of slits.

Diffraction grating pattern

• Bright spot at the centre, called order 0.
• On either side of the central spot, there are more bright spots called orders.

Diffraction grating equation

n\lambda = d\sin\theta

Calculating distance between slits given lines per millimetre

• First, convert lines per millimetre to lines per metre by multiplying by 1000.
• Then do \text{metres per line} = \frac{1}{\text{lines per metre}}

Calculating the maximum number of orders

• Use our equation n\lambda = d\sin\theta.
• Substitute \theta for 90\degree (or \sin\theta for 1) to find the maximum possible value of n.
• Round down to the nearest whole number, as n must be an integer (because we can’t have a fractional number of orders).

How are stationary waves made?

• When two waves in the opposite direction have the same wavelength speed and amplitude, they can interfere to create a stationary wave.
• The points where the waves always interfere destructively are called nodes.
  • Nodes don’t move at all - they have zero amplitude and displacement. They don’t move because the two waves are always in antiphase at the nodes, so they always completely cancel out.
• The points where the waves always interfere constructively are called antinodes.
  • They have the maximum amplitude and displacement, because the two waves are in phase and interfere constructively to add together to make a bigger wave.
  • They oscillate between the maximum positive and negative amplitude each phase.

Harmonics on a stationary wave

• The fundamental frequency (or first harmonic) is the lowest frequency that can be produced on a stationary wave. It has one antinode and two nodes.
• Each extra harmonic adds an extra node and antinode, and increases the frequency by a multiple of the fundamental frequency.

Equation for stationary waves

f=\frac1{2l}\sqrt{\frac{T}{\mu}}

Density in stationary waves

\mu = \frac{m}{l}

where:

• \mu is the mass per unit length of the string
• m is the mass of the string
• l is the length of the string