Waves
Wave Properties
5th Year · 6th Year (Leaving Cert)
- ✓By the end of this lesson students will be able to define and apply the wave equation v = fλ.
- ✓By the end of this lesson students will be able to describe the Doppler effect and solve related problems.
- ✓By the end of this lesson students will be able to explain diffraction and its dependence on wavelength and aperture size.
- ✓By the end of this lesson students will be able to describe interference and state the conditions for constructive and destructive interference.
- ✓By the end of this lesson students will be able to solve numerical problems involving wave properties.
Key concepts
The wave equation relates the speed (v) of a wave to its frequency (f) and wavelength (λ). It applies to all types of waves, including sound waves, light waves, and water waves. Speed is measured in metres per second (m/s), frequency in hertz (Hz), and wavelength in metres (m).
The Doppler effect is the apparent change in the frequency of a wave due to the relative motion between the source of the wave and the observer. When the source and observer are moving towards each other, the observed frequency increases. When they are moving away from each other, the observed frequency decreases. This effect is observed with both sound and light waves, and has applications in areas such as radar, medical imaging (ultrasound), and astronomy.
Diffraction is the spreading out of waves as they pass through an opening (aperture) or around an obstacle. The extent of diffraction depends on the ratio of the wavelength (λ) of the wave to the size of the opening or obstacle (d). Significant diffraction occurs when the wavelength is comparable to or larger than the size of the opening (λ ≈ d). If the wavelength is much smaller than the opening (λ << d), diffraction is minimal.
Interference is the superposition of two or more waves to form a resultant wave of greater, lower, or the same amplitude. For sustained interference patterns to be observed, the sources must be coherent (i.e., they must have a constant phase relationship and the same frequency). Interference can be constructive or destructive: Constructive Interference: Occurs when waves meet in phase, resulting in a larger amplitude (e.g., a louder sound or a brighter light). This happens when the path difference between the waves is an integer multiple of the wavelength. Destructive Interference: Occurs when waves meet out of phase (180° out of phase), resulting in a smaller or zero amplitude (e.g., a quieter sound or a darker region). This happens when the path difference between the waves is an odd multiple of half a wavelength.
Key facts to remember
- 1The wave equation is v = fλ, where v is wave speed, f is frequency, and λ is wavelength.
- 2The Doppler effect is the apparent change in frequency due to relative motion between source and observer.
- 3Diffraction is the spreading of waves as they pass through an opening or around an obstacle.
- 4Significant diffraction occurs when the wavelength is comparable to the size of the opening (λ ≈ d).
- 5Interference is the superposition of two or more waves.
- 6For sustained interference, sources must be coherent (constant phase relationship and same frequency).
- 7Constructive interference occurs when path difference = nλ (n = 0, 1, 2, ...).
- 8Destructive interference occurs when path difference = (n + 1/2)λ (n = 0, 1, 2, ...).
Worked examples
Example 1
A sound wave has a frequency of 250 Hz and travels through air at a speed of 340 m/s. Calculate its wavelength.
Answer
λ = 1.36 m
Remember that Hz is equivalent to s⁻¹.
Example 2
A car horn emits a sound with a frequency of 400 Hz. The car is approaching a stationary observer at a speed of 20 m/s. Calculate the frequency of the sound heard by the observer. (Speed of sound in air = 340 m/s).
Answer
f' = 425 Hz
Always consider the relative motion to determine whether the frequency should increase or decrease, and choose the signs accordingly.
Example 3
Two coherent sources of sound waves, S₁ and S₂, are 3.0 m apart. They emit waves of wavelength 0.5 m. A listener is at a point P such that the path difference (S₂P - S₁P) is 1.5 m. What type of interference occurs at point P?
Answer
Constructive interference occurs at point P.
For interference problems, drawing a diagram can often help visualise the path difference.
Common mistakes
- ✗Incorrectly applying the signs in the Doppler effect formula, leading to an incorrect increase or decrease in frequency.
- ✗Confusing diffraction with interference; diffraction is the spreading of a single wave, while interference is the interaction of two or more waves.
- ✗Not stating the conditions for sustained interference (coherence, same frequency).
- ✗Using incorrect units or forgetting to include units in the final answer.
- ✗Assuming all waves diffract or interfere equally; the extent depends on wavelength and conditions.
Exam tips
- ★Always draw a diagram for Doppler effect and interference problems to help visualise the scenario and determine the correct signs or path differences.
- ★Clearly state all formulae used and show all steps in your calculations, even for simple algebraic manipulations.
- ★Pay close attention to units and ensure consistency throughout your calculations. Convert all quantities to SI units where necessary.
- ★Learn the definitions of key terms (e.g., coherent sources, diffraction, interference) as these are frequently examined.
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