Describe the features of a wave in terms of wavefront, wavelength, frequency, crest (peak), trough, amplitude and wave speed

3.1 General Properties of Waves

Understanding the terminology and the quantitative relationships between wave quantities is essential for solving any Cambridge IGCSE 0625 physics problem involving wave motion.

1. Core Wave Terminology

• Wavefront – an imaginary line (in 2‑D) or surface (in 3‑D) joining points that are in the same phase of vibration. For a transverse wave the wavefront is perpendicular to the direction of propagation.
• Wavelength (λ) – distance between two consecutive points that are in phase (e.g. crest‑to‑crest, trough‑to‑trough, or any two points with the same displacement).
• Frequency (f) – number of complete cycles that pass a given point each second. Unit: hertz (Hz).
• Period (T) – time taken for one complete cycle.

  \$T=\frac{1}{f}\$

  Unit: seconds (s).

• Periodic motion – a motion that repeats itself at regular intervals; for waves the interval is the period T.
• Crest (Peak) – highest point of a transverse wave.
• Trough – lowest point of a transverse wave.
• Amplitude (A) – maximum displacement of the medium from its equilibrium position. For a transverse wave it is the distance from the equilibrium line to a crest (or trough).
• Wave speed (v) – rate at which a wave propagates through the medium.

2. Fundamental Relationships

The three core quantities are linked by:

Using T = 1/f this can also be written as:

These equations are required for AO2 (application) questions – changing one quantity forces a corresponding change in another while the speed v remains fixed for a given medium.

3. Transverse vs. Longitudinal Waves

• Transverse waves: particle motion is perpendicular to the direction of travel.

  Examples: water‑surface waves, waves on a stretched string, electromagnetic waves.

  Features: crests, troughs, amplitude measured perpendicular to the direction of propagation.

• Longitudinal waves: particle motion is parallel to the direction of travel.

  Examples: sound waves in air, compression waves in a spring.

  Features: compressions and rarefactions instead of crests/troughs.

4. Wave Speed and the Medium

Wave speed is a property of the medium, not of the wave itself. Changing the medium changes v even if the frequency stays the same.

• For a given medium, v is constant. Changing the frequency therefore changes the wavelength according to v = fλ.
• String waves: \(v = \sqrt{\dfrac{T}{\mu}}\) where T is the tension and μ the mass per unit length.
• Sound waves (gases, liquids, solids): \(v = \sqrt{\dfrac{B}{\rho}}\) where B is the bulk modulus and ρ the density.
• Typical values: sound in air ≈ 340 m s⁻¹, in water ≈ 1500 m s⁻¹; light in vacuum ≈ 3.0 × 10⁸ m s⁻¹.

5. Wave Phenomena

5.1 Reflection

• The wave bounces back into the original medium.
• Angle of incidence = angle of reflection.
• Relevant for both light and sound (e.g., mirrors, echoes).

5.2 Refraction

• When a wave passes into a medium where its speed changes, the direction bends.
• Derivation of Snell’s law (required for the syllabus):
  

  1. Consider a wavefront striking the boundary at two points A and B.
  2. If the wave travels a distance v₁Δt in medium 1 and v₂Δt in medium 2 during the same time Δt, the angles satisfy

     \(\displaystyle \frac{\sin\theta1}{\sin\theta2}= \frac{v1}{v2}\).

  3. Since in a medium v = fλ and the frequency is unchanged, we obtain the familiar form

     \(\displaystyle n1\sin\theta1 = n2\sin\theta2\) where \(n = c/v\) is the refractive index.

• Critical angle (total internal reflection): when light travels from a higher‑index medium to a lower‑index medium, the angle of incidence for which \(\sin\thetac = \dfrac{v2}{v1} = \dfrac{n1}{n2}\) produces a refracted angle of 90°. For \(\theta > \thetac\) the wave is totally reflected.

5.3 Diffraction

• The wave spreads out after passing through a narrow opening or around an obstacle.
• Quantitative condition (syllabus requirement): diffraction is most noticeable when the slit width a is comparable to the wavelength, i.e. \(a \approx λ\).
• Examples: ripples from a slit in a ripple tank, sound through a doorway.

6. Practical Link – Ripple‑Tank Experiments

The ripple‑tank set‑up demonstrates all three phenomena in one experiment and provides a quantitative way to determine wave speed.

1. Generate a plane wave with a vibrator at one end of the tank.
2. Reflection: place a smooth barrier at an angle, measure incident and reflected angles with a protractor (they should be equal).
3. Refraction: create a step in water depth (shallow → deep). Measure the wavelength before (λ₁) and after (λ₂) the step while keeping the vibrator frequency constant.

   
   Numerical example:

   Frequency set to 40 Hz. In shallow water (depth = 2 cm) the measured wavelength is 0.10 m, giving

   \(v1 = fλ1 = 40 × 0.10 = 4.0 \text{m s}^{-1}\).

   In deeper water (depth = 5 cm) the wavelength increases to 0.14 m, so

   \(v_2 = 40 × 0.14 = 5.6 \text{m s}^{-1}\).

   The frequency is unchanged, confirming that the change in speed is due to the change in medium (depth).

4. Diffraction: place a narrow slit (width ≈ λ) or a small obstacle and observe the spreading of the wavefront beyond the opening.

7. Connections to Other Syllabus Topics

• Sound (3.2.1) – a longitudinal wave; the same relationship \(v = fλ\) holds, with speed given by \(v = \sqrt{B/ρ}\).
• Light (3.2.2) – an electromagnetic transverse wave; in vacuum \(c = 3.0 × 10⁸ \text{m s}^{-1}\). Refraction is described by Snell’s law derived above.
• Electromagnetic spectrum (3.2.3) – all EM waves travel at the same speed in vacuum, differing only in λ and f (still linked by \(c = fλ\)).
• Energy transfer – a wave transports energy from one place to another without permanent displacement of the medium (a one‑sentence reminder required by the syllabus).

8. Table of Wave Features

9. Example Calculation (Core Formula)

Given f = 50 Hz and λ = 0.20 m:

10. Suggested Diagram

11. Common Misconceptions

1. Wavelength vs. amplitude – wavelength is a horizontal distance between identical points; amplitude is the vertical height of the wave.
2. Wave speed depends on frequency – in a single medium v is fixed; changing f changes λ so that v = fλ remains true.
3. Wavefront is a physical object – it is an imaginary construct used to describe points of equal phase.
4. All waves behave the same – transverse waves have crests/troughs; longitudinal waves have compressions/rarefactions.
5. Diffraction only occurs with “big” openings – it is strongest when the slit width is comparable to the wavelength (a ≈ λ).

12. Summary

A wave is characterised by its wavefront, wavelength, frequency (or period), amplitude, and speed. The fundamental relationship v = fλ (or v = λ/T) links the three core quantities. Wave speed is determined by the medium, so changing the medium changes the wavelength while the frequency stays constant. Transverse and longitudinal waves have distinct features, and the phenomena of reflection, refraction (including Snell’s law, critical angle and total internal reflection) and diffraction (most evident when slit width ≈ λ) are central to the IGCSE syllabus. A ripple‑tank experiment provides a hands‑on method to observe these behaviours and to calculate wave speed quantitatively. Remember that waves transport energy without permanently moving the medium – a point often tested in exam questions.