Wave equation Notes

How Do Wave Quantities Describe Both The Shape And Timing Of Oscillations?

To model a wave, we describe both its spatial pattern (what it looks like along a line) and its time behavior (how fast it oscillates).

1. Equilibrium Position and Displacement

1. The equilibrium position is the undisturbed position of the medium (for example, the flat surface of still water).
2. The wave causes a displacement from equilibrium.

2. Wave Speed (v)

1. Wave speed describes how fast a wave travels through a medium.
2. It tells us how much distance the wave covers in one second.
3. Wave speed depends on:
   1. The medium the wave is travelling through
   2. The properties of the medium, such as tension or depth (depending on the wave)
4. It is measures in Metres per second (m s⁻¹)

3. Frequency (f)

1. Frequency describes how often a wave repeats.
2. It counts how many complete waves pass a fixed point every second.
3. A higher frequency means the wave oscillates more rapidly.
4. It is measured in Hertz (Hz), where 1 Hz = 1 wave per second

4. Wavelength (λ)

1. Wavelength is the distance between repeating points on a wave.
2. It can be measured between:
   1. Crest to crest (transverse waves)
   2. Trough to trough (transverse waves)
   3. Compression to compression (longitudinal waves)
3. Wavelength shows how “spread out” a wave is.
4. It is measured in Metres (m).

5. Amplitude

1. The amplitude $A$ tells you how "large" the oscillation is.
2. Larger amplitude generally means more energy carried by the wave.

6. Time Period

1. The time period (T) is the time taken for one complete wave cycle.
2. The time period $T$ is measured in seconds (s).
3. Frequency and time period are inversely related.
4. $$f = \frac{1}{T}.$$

The Wave Equation Links Speed, Frequency, And Wavelength

1. The wave equation shows that wave speed depends on how frequently waves pass a point and how long each wave is.
2. When a wave pattern moves through a medium, it has a wave speed $v$ (sometimes called wave velocity).
3. This equation can be understood by thinking in one second:
   1. if $f$ waves pass a point each second, then $f$ wavelengths pass that point each second,
   2. so the wave travels a distance $f\lambda$ each second,
   3. therefore speed is $v = f\lambda$.

The Wave Equation Helps Explain "Seeing" At Different Scales

1. Waves are not only for water and sound. In modern physics, particles such as electrons can show wave-like behavior.
2. A powerful idea is that waves can only resolve (distinguish) objects that are larger than their wavelength.
3. Visible light has a wavelength roughly a thousand times larger than an atom, so an optical microscope cannot show individual atoms.
4. To probe much smaller structures, you need waves with much smaller wavelengths (for example, electron waves).