Sure, let's explore the relationships between the speed, wavelength, and frequency of a wave.
The speed [tex]\( v \)[/tex] of a wave is related to its wavelength [tex]\( \lambda \)[/tex] (the distance between successive crests or troughs) and its frequency [tex]\( f \)[/tex] (the number of oscillations per unit time).
The fundamental relationship between these quantities is:
[tex]\[ \text{Speed} = \text{Wavelength} \times \text{Frequency} \][/tex]
Given this understanding, let's review the options provided:
1. Frequency = speed + wavelength
This option suggests that frequency is the sum of speed and wavelength, which doesn't make sense dimensionally. Frequency (measured in Hz or [tex]\( \text{s}^{-1} \)[/tex]) cannot simply be added to a speed (measured in meters/second) and a wavelength (measured in meters).
2. Speed = wavelength × frequency
This option correctly matches the fundamental relationship. The speed of a wave is the product of its wavelength and its frequency. Dimensionally, this makes sense as well:
[tex]\[
\text{Speed (meters/second)} = \text{Wavelength (meters)} \times \text{Frequency (1/second)}
\][/tex]
Thus, this relationship is accurate.
3. Speed = wavelength - frequency
This option is incorrect because speed is not the difference between wavelength and frequency. Similar to option 1, subtracting a frequency (1/second) and wavelength (meters) does not create a meaningful or dimensionally consistent result for speed.
4. Wavelength = speed × frequency
This option is incorrect as well according to our wave relationship. The correct relationship is speed = wavelength × frequency, not wavelength = speed × frequency. The units also don't match properly using this formula.
Given our analysis, the correct relationship is:
[tex]\[ \text{Speed} = \text{Wavelength} \times \text{Frequency} \][/tex]
Thus, the correct option is:
[tex]\[ \boxed{2} \][/tex]
