Wave speed is directly proportional to frequency and inversely proportional to wavelength.

To elaborate, the speed of a wave is intricately linked to its frequency and wavelength, as expressed in the wave equation:

In this equation, $v$ represents the wave speed, $f$ is the frequency, and $\lambda$ is the wavelength. This formula indicates that the speed of a wave is equal to the product of its frequency and wavelength.

When the speed of a wave increases while the frequency remains constant, the wavelength must also increase. This occurs because the wave has to travel a greater distance in the same amount of time. Conversely, if the speed decreases while the frequency remains unchanged, the wavelength will decrease, as the wave covers less distance within the same time frame.

Similarly, if the speed of a wave increases while the wavelength remains constant, the frequency must also increase. This is due to the wave needing to complete more cycles in the same amount of time. Conversely, if the speed decreases while the wavelength remains constant, the frequency will decrease, as the wave completes fewer cycles in that duration.

It is important to note, however, that in many physical systems, the speed of a wave is determined by the medium through which it propagates and cannot be easily altered. For instance, the speed of light is constant in a vacuum, and the speed of sound is influenced by the properties of the air or any other medium. In these scenarios, altering the frequency of the wave will lead to a corresponding adjustment in the wavelength, and vice versa, in order to satisfy the wave equation.

In summary, wave speed is interconnected with both frequency and wavelength. When the speed of a wave changes, the frequency and wavelength adjust to maintain equilibrium. Think of it as a balancing act: if one parameter increases, the others must adjust accordingly to ensure the wave continues to propagate effectively. Remember, the medium plays a crucial role in determining wave speed, which subsequently affects the relationships between frequency and wavelength.