A water wave travels at a constant speed. If the frequency increases from to while the wave speed stays the same, what happens to the wavelength ?
- The wavelength doubles
- The wavelength is cut in half (correct answer)
- The wavelength stays the same
- The wavelength increases by a factor of 4
Explanation: This question tests understanding of the relationship between wavelength, frequency, and wave speed, described by the equation v = fλ. The wave equation v = fλ states that wave speed (v) equals the product of frequency (f, measured in Hz or cycles per second) and wavelength (λ, the distance between successive wave crests), and this relationship can be rearranged to solve for any of the three quantities: f = v/λ or λ = v/f. For waves traveling at constant speed v, the equation v = fλ shows that frequency and wavelength are inversely proportional: if frequency increases by a factor of 2 (from 0.40 Hz to 0.80 Hz), wavelength must decrease by the same factor to keep their product (wave speed) constant. Mathematically, fλ = constant, so f₁λ₁ = f₂λ₂, which means λ₂ = λ₁ × (f₁/f₂) = λ₁ × (0.40/0.80) = λ₁/2. Choice B is correct because it accurately describes the inverse relationship between f and λ, recognizing that doubling the frequency halves the wavelength when wave speed remains constant. Choice A incorrectly claims frequency and wavelength are directly proportional (both increase together), when actually they're inversely proportional: as frequency increases, wavelength must decrease to maintain constant wave speed. The key insight is that at constant wave speed, frequency and wavelength are inversely related—double the frequency means half the wavelength, which explains why higher frequency waves always have shorter wavelengths when traveling at the same speed.