I am struggling with the following question.
Consider $$U(x,t)=2\cos\left(kx-\sqrt{\frac{a}{b}}k^2+(\Delta k)^2)t\right)\cos\left((\Delta k)(x-2\sqrt{\frac{a}{b}}kt\right)$$ and suppose that $\Delta k$ is much smaller than $k$ . Determine the spatial periodicity of each cosine factor.
I am having trouble finding any reference regarding how to find spacial periodicity. All I find is that spacial periodicity equals $\dfrac{2\pi}{k}$, but I am sure that for this wave equation the spacial periodicity should be a bit more complex then this. For example, for the first cosine factor I would expect it looks more like $$\dfrac{2\pi}{k}-\dfrac{2\pi}{k^2}$$ but I have no justification for this.
Any help with this would be greatly appreciated. Thanks.
Not sure why you would want to find spatial periodicity separately for each cos factor. However, a general expression for a traveling wave is $U(x,t) = a\cos(2\pi)(kx-nt)$. The expression $U(x,t) = a\cos(2\pi(kx-nt)+g)$ represents another wave with phase difference $g$ relative to the first expression (Waves, C.A. Coulson). If you recast your first cos term in this fashion, you should be able to find out the $k$ for it quite simply.