Chapter 12: Waves (Numerical Problems)

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Chapter Overview & SLOs

How is the wavelength determined from the physical length of a medium? The wavelength (λ) is calculated by dividing the total length (L) of the string or tank by the number of waves (n) produced: λ = L/n.

What is the fundamental relationship between wave speed, frequency, and wavelength? The speed of a wave (v) is the product of its frequency (f) and wavelength (λ), expressed by the universal wave equation v = fλ.

How are frequency and time period related? Frequency and time period are inversely related: T = 1/f. As frequency increases, time period decreases.

What is the speed of electromagnetic waves in a vacuum? For electromagnetic waves (light, radio waves) in a vacuum, the speed is constant: c ≈ 3.0 × 10⁸ m/s.

Example 1 - Stretched string: If 10 waves are produced on a 20 m string, the wavelength is 20/10 = 2 m. If the wave speed is 12 m/s, the frequency is f = v/λ = 12/2 = 6 Hz.

Example 2 - Ripple tank: For a 1.5 m tank where waves take 2 s to cross, the speed is v = distance/time = 1.5/2 = 0.75 m/s. If the distance between a crest and trough is 10 cm, the wavelength is 20 cm (0.2 m) because crest to trough = λ/2. Resulting frequency f = v/λ = 0.75/0.2 = 3.75 Hz, and time period T = 1/f = 1/3.75 = 0.267 s.

Example 3 - Red light frequency: For red light with a wavelength of 700 nm (700 × 10⁻⁹ m) in a vacuum, the frequency is f = c/λ = (3.0 × 10⁸)/(700 × 10⁻⁹) ≈ 4.29 × 10¹⁴ Hz.

Example 4 - FM radio wavelength: For an FM station transmitting at 90 MHz (90 × 10⁶ Hz), the wavelength is λ = c/f = (3.0 × 10⁸)/(90 × 10⁶) = 3.33 m.

Important notes:

• The distance between a successive crest and trough is half a wavelength (λ/2).
• Unit conversions: 1 nm = 10⁻⁹ m, 1 MHz = 10⁶ Hz.

These notes are strictly aligned with the Student Learning Outcomes (SLOs) for the FBISE 2026 annual examination.

• How do we calculate wavelength and frequency on a stretched string? If 10 waves are produced on a 20 m string, the wavelength is 20/10 = 2 m; if the speed is 12 m/s, the frequency is $f = 12/2 = 6\text{ Hz}$.
• How are ripple tank dimensions used to find wave properties? For a 1.5 m tank where waves take 2 s to cross, the speed is 0.75 m/s; if the distance between a crest and trough is 10 cm, the wavelength is 20 cm (0.2 m), resulting in a frequency of 3.75 Hz and a time period of 0.267 s.
• How do we find the frequency of light from its wavelength? For red light with a wavelength of 700 nm ($700 \times 10^{-9}\text{ m}$) in a vacuum, the frequency is calculated using $f = c/\lambda$, resulting in approximately $4.29 \times 10^{14}\text{ Hz}$.
• How is the wavelength of radio signals determined? For an FM station transmitting at 90 MHz ($90 \times 10^6\text{ Hz}$), the wavelength is found by dividing the speed of light by the frequency ($\lambda = c/f$), which equals 3.33 m.

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