Welcome to the World of Waves!

Have you ever wondered how music reaches your ears from a speaker, or how your phone receives a signal from miles away? The answer is waves! In this chapter, we are going to explore how energy travels through space and materials. Don't worry if it seems like a lot of definitions at first—once you see the patterns, it all clicks together!

1. Progressive Waves

A progressive wave is a disturbance that travels through a medium (like air or water) or a vacuum, transferring energy from one place to another without transferring matter.

Think of a "Mexican Wave" in a sports stadium: the people move up and down (the disturbance), but no one actually moves to a different seat. The "wave" travels around the stadium, but the people stay put!

Key Terms to Master

  • Displacement: The distance of a point on the wave from its equilibrium (rest) position.
  • Amplitude (\(A\)): The maximum displacement. It’s the "height" of the wave from the middle.
  • Wavelength (\(\lambda\)): The distance between two identical points on a wave (e.g., from one peak to the next).
  • Period (\(T\)): The time taken for one complete wave to pass a point.
  • Frequency (\(f\)): The number of waves passing a point per second. Measured in Hertz (Hz).
  • Phase Difference: A measure of how "out of sync" two points on a wave are, usually measured in degrees or radians.

The Wave Equation

There is a very important relationship between speed (\(v\)), frequency (\(f\)), and wavelength (\(\lambda\)).
Formula: \(v = f\lambda\)

How to derive it:
1. Speed = Distance / Time.
2. For one wave, the distance is \(\lambda\) and the time is \(T\).
3. So, \(v = \lambda / T\).
4. Since \(f = 1/T\), we substitute to get \(v = f\lambda\).

Intensity and Amplitude

The intensity (\(I\)) of a wave is the power per unit area.
Important Fact: Intensity is proportional to the square of the amplitude.
\(I \propto A^2\)

If you double the amplitude of a wave, the intensity (and the energy it carries) actually increases by four times (\(2^2 = 4\))!

Quick Review: Progressive Waves

Key Takeaway: Waves transfer energy, not matter. Use \(v = f\lambda\) for calculations. Remember that if amplitude doubles, intensity quadruples!

2. Transverse and Longitudinal Waves

Waves come in two main "flavors" based on how they vibrate.

Transverse Waves

The vibrations are perpendicular (at 90 degrees) to the direction of energy travel.
Example: Light waves, waves on a string, or S-waves in earthquakes.

Longitudinal Waves

The vibrations are parallel to the direction of energy travel. These waves consist of compressions (squashed together) and rarefactions (stretched apart).
Example: Sound waves or P-waves in earthquakes.

Common Mistake to Avoid: Students often think all waves move up and down. Remember, sound waves move back and forth like a slinky being pushed and pulled!

3. Using the Cathode-Ray Oscilloscope (CRO)

A CRO is a device that lets us "see" sound waves by turning them into an electrical graph.

  • Y-gain: Controls the vertical axis (Voltage). Use this to find the Amplitude.
  • Time-base: Controls the horizontal axis (Time). Use this to find the Period (\(T\)).

Step-by-step to find Frequency:
1. Count how many boxes (divisions) one full wave takes on the screen.
2. Multiply by the "Time-base" setting (e.g., \(5\,ms/div\)) to get the Period \(T\).
3. Use \(f = 1/T\) to calculate the frequency.

4. The Doppler Effect

Have you noticed how an ambulance siren sounds higher-pitched as it rushes toward you and lower-pitched as it moves away? This is the Doppler Effect.

When a source of sound moves:
- Towards you: The waves get "bunched up," the wavelength decreases, and the frequency (pitch) sounds higher.
- Away from you: The waves get "stretched out," the wavelength increases, and the frequency sounds lower.

The Formula: \(f_o = \frac{f_s v}{v \pm v_s}\)
Where \(f_o\) is the observed frequency, \(f_s\) is the source frequency, \(v\) is the speed of sound, and \(v_s\) is the speed of the source.

Trick for the \(\pm\) sign:
- Use minus (\(-\)) when the source is moving Towards you (this makes the denominator smaller and the frequency bigger).
- Use plus (\(+\)) when moving Away.

5. The Electromagnetic Spectrum

All electromagnetic (EM) waves are transverse and travel at the same speed in a vacuum: the speed of light (\(c = 3.0 \times 10^8\,m/s\)).

The Order (from longest wavelength to shortest):
Radio waves \(\rightarrow\) Micro waves \(\rightarrow\) Infrared \(\rightarrow\) Visible Light \(\rightarrow\) Ultraviolet \(\rightarrow\) X-rays \(\rightarrow\) Gamma rays.

Did you know? Visible light only ranges from about 400 nm (violet) to 700 nm (red). That is a tiny slice of the whole spectrum!

6. Polarisation

Polarisation is a phenomenon that only happens to transverse waves. It restricts the wave's vibrations to a single plane.

Analogy: Imagine waving a rope through a picket fence. If the gaps in the fence are vertical, you can only make the rope vibrate up and down. If you try to vibrate it sideways, the fence blocks the wave!

Malus’s Law

When polarised light passes through a second filter (an analyser), its intensity changes depending on the angle (\(\theta\)) between the filters.
Formula: \(I = I_0 \cos^2\theta\)

Quick Tip:
- If \(\theta = 0^{\circ}\) (parallel), \(I = I_0\) (maximum light passes).
- If \(\theta = 90^{\circ}\) (crossed), \(I = 0\) (no light passes).

Final Summary Checklist
  • Can you use \(v = f\lambda\)?
  • Do you know the difference between transverse and longitudinal?
  • Can you calculate frequency from a CRO screen?
  • Do you remember the visible light range (400–700 nm)?
  • Can you apply Malus's Law for polarisation?

Great job! Waves can be a wavy ride, but keep practicing these formulas and analogies, and you'll be an expert in no time.