In the context of wave functions, how does increasing the frequency $f$ of a wave affect its wavelength $\lambda$, assuming the wave speed $v$ remains constant?

• A. The wavelength $\lambda$ first increases, then decreases as frequency $f$ increases.
• B. The wavelength $\lambda$ increases as frequency $f$ increases.
• C. The wavelength $\lambda$ remains unchanged as frequency $f$ increases.
• D. The wavelength $\lambda$ decreases as frequency $f$ increases.

Answer: (D)

Verified step by step guidance

1. Recall the fundamental relationship between wave speed (v), frequency (f), and wavelength (\lambda), which is given by the equation: \[v = f \times \lambda\]
2. Since the wave speed \(v\) is constant, we can rearrange the equation to express wavelength in terms of frequency: \[\lambda = \frac{v}{f}\]
3. Analyze how wavelength \(\lambda\) changes when frequency \(f\) increases: because \(v\) is constant, increasing \(f\) means the denominator of the fraction increases.
4. As the denominator increases, the overall value of \(\lambda = \frac{v}{f}\) decreases, indicating an inverse relationship between wavelength and frequency.
5. Therefore, when frequency increases while wave speed remains constant, the wavelength decreases.