Question

there's a equation for wavespeed=wavelength*f

the wavespeed doesn't depend on the wavelength. Why not?

if you double the wavelength, won't the speed increase as well. In my book, it says that v is unchanged. I don't understand

Answer 1

Velocity = wavelength * frequency

The velocity changes with the wavelength. Your book probably refers to a situation in which both the wavelength and frequency change.

Answer 2

Generally, in physical mediums, the speed of propagation is a constant of the material, independent of the incoming wave. As a first approximation, treating velocity as a constant is good enough. But some material exhibit a very slightly changing wave speed that is related to the incoming wavelength, you see this as changing index of refraction as a function of wavelength. Remember Newtons rainbow in the prism? That works because the index of refraction is different (very slightly) for each color, and of course, the index of refraction is a measure of light velocity in the medium.