While learning Fourier transforms, I have this little doubt about wavelength.
Why is that wavelength for a sine wave, or any other wave, is measured with a unit of distance?
If it travels w.r.t. time, why is it not measured as a unit of time?
I think, in the below drawing, the green line is the wavelength. So, why measure it with a unit of distance?
Best Answer
The plot you provide gives the amplitude versus time so there is no length, other than "length" of time, i.e., period) to speak of. Without further context, what you have plotted there is a simple sinusoidal function of time, not a wave.
A wave is a function of both time and space. For example:
$$f(x,t) = \cos(\frac{2\pi}{\lambda}x - \frac{2\pi}{T} t) $$
where the wavelength \$\lambda \$ (measured in units of length) and period T (measured in units of time) are explicit.
Often, this is written as:
$$f(x,t) = \cos(kx - \omega t) $$
where k, the wavenumber is:
$$ k = \frac{2\pi}{\lambda} $$
and \$\omega\$, the angular frequency is:
$$\omega = \frac{2\pi}{T}$$
The wave propagates with phase velocity
$$v_p = \dfrac{\lambda}{T} = \dfrac{\omega}{k} $$
Then we can write:
$$f(x,t) = \cos\frac{2\pi}{T}(\frac{x}{v_p} - t) = \cos\omega(\frac{x}{v_p} - t)$$
or
$$f(x,t) = \cos\frac{2\pi}{\lambda}(x - tv_p) = \cos k(x - tv_p)$$