What is the relationship between wavelength and frequency for a wave?

The relationship between wavelength and frequency for a wave is governed by the wave equation, which states that the speed of a wave is equal to the product of its wavelength and frequency. In mathematical terms, this can be expressed as:

[ \text{Speed} = \text{Wavelength} \times \text{Frequency} ]

Given that the speed of light in a vacuum is a constant, if the wavelength of a wave increases (becomes longer), the frequency must decrease for the product (speed) to remain constant. Thus, longer wavelengths correspond to lower frequencies. This is why the correct statement indicates that longer wavelengths equal lower frequencies.

In contrast, shorter wavelengths would indicate a higher frequency, as they would need to compensate for the relatively shorter length to maintain the same overall speed. The independence of wavelength from frequency is not accurate in the context of wave behavior since they are directly related through the equation mentioned above.