Waves

A progressive wave is a disturbance that transfers energy from one place to another without any net transfer of matter. We can represent waves using graphs. A displacement-distance graph shows the displacement of all particles along the wave at a single instant in time. A displacement-time graph shows the oscillation of a single particle over a period of time. From these, we can define key properties. Amplitude (A) is the maximum displacement from the equilibrium position. Wavelength (λ) is the distance between two consecutive points in phase, like two crests. Period (T) is the time for one complete oscillation of a particle. Frequency(f) is the number of complete oscillations per second, measured in Hertz (Hz). Frequency and period are reciprocals: f = 1/T.

Waves can be classified based on the direction of oscillation relative to the direction of energy transfer. In a transverse wave, the particles of the medium oscillate perpendicular (at 90°) to the direction of energy transfer. Examples include all electromagnetic waves (like light), waves on a guitar string, and ripples on water. In a longitudinal wave, the particles of the medium oscillate parallel to the direction of energy transfer. These waves travel as a series of compressions (regions of high pressure) and rarefactions (regions of low pressure). The most common example is a sound wave.

The speed of a wave(v) is related to its frequency(f) and wavelength (λ) by the wave equation: v = fλ. This is one of the most fundamental equations in wave physics. It tells us that for a given speed, a higher frequency means a shorter wavelength, and vice versa. We can measure wave properties using a Cathode-Ray Oscilloscope (CRO). A CRO displays a voltage-time graph. The 'Y-gain' or 'volts/div' setting controls the vertical scale, allowing us to determine the amplitude (peak voltage). The 'time-base' or 's/div' setting controls the horizontal scale, allowing us to measure the period (T) of the wave. From the period, we can calculate the frequency using f = 1/T.

The intensity of a wave is a measure of the energy it carries per unit time, per unit area. Formally, intensity (I) is defined as the power (P) transmitted per unit area (A) perpendicular to the wave's velocity. This gives the equation: Intensity = Power / Area. The unit of intensity is watts per square metre (W m⁻²). For any wave, the intensity is directly proportional to the square of its amplitude (A). This is written as I ∝ A². This is a crucial relationship. It means that if you double the amplitude of a wave, you multiply its intensity by 2², which is 4. If you triple the amplitude, the intensity increases by a factor of 3², which is 9. This is why a slightly louder sound (small increase in pressure amplitude) can carry significantly more energy.

The Doppler effect is the change in the observed frequency of a wave when there is relative motion between the source of the wave and the observer. For sound, this results in a change in pitch. As a sound source (like an ambulance siren) moves towards you, the wave crests arrive more frequently, so you hear a higher frequency (higher pitch). As it moves away, the wave crests are more spread out and arrive less frequently, so you hear a lower frequency (lower pitch). The observed frequency (f_o) can be calculated using the formula f_o = f_s * v / (v ± v_s), where f_s is the source frequency, v is the speed of sound, and v_s is the speed of the source. Use the minus sign (-) in the denominator when the source is moving towards the observer, and the plus sign (+) when it is moving away.

Electromagnetic (EM) waves are transverse waves made of oscillating electric and magnetic fields. They are unique because they do not require a medium and can travel through a vacuum. In a vacuum, all EM waves travel at the same speed, the speed of light, c = 3.00 x 10⁸ m/s. The EM spectrum is the continuous range of all EM waves, organised by frequency and wavelength. The wave equation c = fλ applies to all of them. In order of increasing wavelength (and decreasing frequency), the spectrum is: Gamma rays, X-rays, Ultraviolet (UV), Visible light, Infrared (IR), Microwaves, Radio waves. The small part of this spectrum that our eyes can detect is visible light, with wavelengths from approximately 400 nm (violet) to 700 nm (red).

In a normal, unpolarised transverse wave (like light from the sun or a bulb), the oscillations occur in all possible planes perpendicular to the direction of energy transfer. Polarisation is the process of restricting these oscillations to a single plane. A polarising filter (or 'polariser') achieves this by acting like a set of microscopic parallel slits, only allowing oscillations aligned with its 'transmission axis' to pass through. The resulting wave is plane-polarised. A key fact is that only transverse waves can be polarised. Longitudinal waves, like sound, cannot be polarised because their oscillations are already fixed in one direction: parallel to the direction of energy transfer. This property is considered definitive proof that light is a transverse wave. Polarised sunglasses work by blocking horizontally-polarised light, which is the primary component of glare reflected from surfaces like water or roads.

When unpolarised light passes through a polarising filter, its intensity is halved (I = I₀/2). If this now plane-polarised light then passes through a second filter (called an 'analyser'), the transmitted intensity depends on the angle between the two filters. Malus's Law describes this relationship: I = I_max cos²θ. Here, I_max is the intensity of the light incident on the analyser, I is the intensity transmitted through the analyser, and θ is the angle between the transmission axis of the polariser and the analyser. If the analyser is aligned with the polariser (θ=0°), cos²(0°)=1, and all the light gets through (I = I_max). If the analyser is 'crossed' with the polariser (θ=90°), cos²(90°)=0, and no light gets through (I = 0).