Waves

A wave is a transfer of energy or information using oscillations of a medium, without moving the particles of the medium themselves.

Waves can be modelled mathematically and have a number of different properties.

Key Concepts

Wave phenomena

All waves have the following chracteristics:

Reflection: When a wave hits a barrier it comes back. The angle of reflection is equal to the angle of incidence.

Refraction: When a wave passes into another medium, it changes direction due to a change in speed.

Interference: When two waves of the same type meet, they combine to construct and destruct one another.

Superposition: The total displacement of interfering waves is the vector sum of the individual displacements.

Diffraction: When a wave passes through a narrow opening it spreads out. A wave will also pass around the edge of a barrier.

Wave quantities

You will need to memorise the following definitions:

Amplitude (A): The maximum displacement from the equilibrium position.

Wave speed (v): The distance moved by the wave per second.

Wavelength (λ): The distance between two equivalent points on consecutive waves (e.g. peak to peak)

Frequency (f): The number of complete cycles passing a point per second or the number of waves produced every second.

Wave types

In a transverse wave, the direction of displacement is perpendicular to the direction of propagation. Examples include:

• water waves
• waves on a string
• electromagnetic waves
• S-earthquake waves

Transverse waves have peaks and troughs.

A wave is said to be polarised if the displacement is restricted to one plane. This plane could be vertical, horizontal or anything in between. If a wave can be polarised, it must be transverse.

In a longitudinal wave, the direction of displacement is parallel to the direction of propagation. Examples include:

• sound waves
• compression waves on a slinky spring
• P-earthquake waves

NB: Where possible, avoid using 'slinky spring' as an example of a transverse or longitudinal wave, as you will not be giving an unambigous response.

There are no peaks and troughs in a longitudinal wave, instead there are compressions and rarefactions.

Graphical representation of progressive waves

Displacement vs position

Displacement vs time

The displacement time graph for a longitudinal wave is plotted in the same way as for a transverse wave, but the displacement position is a bit more tricky since the displacement is parallel to the wave direction.

Wave equations

We recall from Kinematics that speed is distance divided by time. Given that all individual waves travel at the same speed in a given medium, it is often more convenient to consider calculating the speed of just one wave:

$$v={\lambda \over T}$$

Since $$T={1\over f}$$$$v=f\lambda$$

NB: A stretched string has a wave speed related to the tension in the string and the mass per unit length of the string:

T (N) = tension
μ (kg m-1) = mass per unit length

It is possible to find an equation for how the y-displacement of a wave varies with the distance travelled (x) through the medium: $$y=A\sin(2\pi ft-{2\pi \over \lambda}x)$$

Standing waves

Standing waves (or stationary waves) form when the following occurs:

• Two waves
• With the same freqency and amplitude
• Travelling in opposite directions
• Interfere with each other

The most regular occurence of this is a single wave reflecting back onto itself.

Standing waves are represented by drawing the two extremes in position.

Node: Point that has zero amplitude

Antinode: Point with maximum amplitude

All sections between two nodes oscillate perfectly in phase.

Essentials

Sound

When air is disturbed it causes a change in pressure. This in turn disturbs the surrounding air, resulting in propagation of changing pressure throughout the medium.

Sound waves can be simulated using Algodoo.

Sound is not just something to be measured; it can also be experienced with our senses. The loudness and pitch of a sound are related to the physical quantities of amplitude and frequency, respectively.

High pitch = high frequency
Loud sound = large amplitude
Speed = 340 ms-1

Sound changes speed when it enters a new material (the denser the material, the more particles present, and the faster the propagation of pressure. In fact, sound will not travel through a vacuum.

Sound is also refracted as it passes through air of a different temperature.

Light

The properties of electromagnetic waves depend on their wavelength. Radio waves have the longest wavelength and gamma waves the shortest. Conversely (due to the constant speed of light, $$c=3\times10^8$$ms-1), radio waves have the lowest frequency and gamma waves the highest.

We can display all the different wavelengths on a chart; this is called a spectrum.

We can produce a spectrum of visible white light by separating the wavelengths using a prism. Red light maintains its speed best in glass and so refracts least, whereas violet light reduces in speed the most and refracts most.

As with sound waves, humans are able to sense visible light:

Wavelength → colour
Amplitude → brightness

Brightness: directly related to the intensity (power per unit area), brightness is proportional to the square of the amplitude.

Reflection: the reason we see objects is because they reflect light into our eyes. When light reflects, the angle of reflection equals the angle of incidence. However, this is only noticeable when the surface is smooth.

Refraction: When light passes from one medium to another its velocity changes resulting in a change of direction.

A mathematical relationship can be used to calculate the angle of refraction in the new material. It depends on the refractive indices (plural: index) of the materials:

$$n_1 \sin i_1=n_2 \sin i_2$$

Refractive index is defined of the ratio of the speed of light in a vacuum to the speed of light in the material:

$$n_A={c\over v_A}$$

NB: We take air to have a refractive index = 1 (as light is not refracted between a vacuum and air).

Critical angle: When light travels from glass to air, it refracts away from the normal. If the angle of incidence is large enough then the angle of refraction will be 90°. The angle at which this happens is called the critical angle.

Total internal reflection: If the critical angle is exceeded, no ray is refracted and all of the light is reflected inside the more optically dense material.

Single slit diffraction: When light passes through a narrow slit (<0.1mm), it can be observed to spread out. Diffraction is optimised when the slit width approximately equals the wavelength of the light ($$\approx 10^{-7}$$m).

Two slit interference: If light passes through two narrow slits, interference takes place where the light overlaps.

Polarisation: Light can be polarised by passing it through a special plastic called Polaroid.

Malus' law gives the relationship between the intensity and angle ($$\theta$$) between the polarisers:

$$I=I_0\cos^2 \theta$$

I (Wm-2) = transmitted intensity

I0 (Wm-2) is the incident intensity

Summary

There are some key differences between progressive and standing waves:

 Progressive Standing Amplitude of all points is equal Amplitude of all points between a node and antinode different All points within one wavelength out of phase All points between 2 nodes in phase Energy transfer No energy transfer Wave profile progresses Wave profile stationary

Test Yourself

Use flashcards to practise your recall.

Just for Fun

Check out this πg physics summary.