Can the wavelengths (sound, light) be infinite?

Question

Can the wavelengths (sound, light) be infinite or are they limited by something?

Answer 1

By the relation $c=\lambda\nu,$ for an infinite wavelength $\lambda$ you would have zero frequency $\nu$.

Consider a plane wave with the usual phase factor $$\cos\left(2\pi\left(\frac{x}{\lambda}-\nu t\right)+\phi_0\right).$$ For $\lambda=\infty$ and $\nu=0$ this simply reduces to $\cos(\phi_0)$, i.e. a constant independent of position and time.

For light this just means a homogenous static electric field $\vec{E}$ and a homogenous static magnetic field $\vec{B}$.
For sound this means a homogenous static pressure $p$ and a homogenous steady wind with velocity $\vec{v}$.

Physically and mathematically there is no problem with that.

Answer 2

By the Planck-Einstein relation, $$ E = hc/ \lambda$$ For an infinite wavelength, you would have zero energy, which is problematic. Also, you would need to measure this "infinite wavelength". What kind of mathematical shape would this look like? A constant? I guess if you consider the constant state to be "infinite wavelength", then yes; but noone would call an homogeneous electromagnetic field "infinite wavelength".