Can the equation $v=\lambda f$ be made true even for non sinusoidal waves?

The known relation between the speed of a propagating wave, the wave length of the wave, and its frequency is
$$v=\lambda f$$ which is always true for any periodic sinusoidal waves.

Now consider:

1-a case were the wave is not sinusoidal but still periodic, like a sawtooth pattern say, that propagates with speed $v$, and of frequency $f$. It is obvious in this case we will not have a single $\lambda$ but infinite number of them, $\lambda_i$ (from Fourier analyzing that pattern). Can one define/calculate a new wave length $\lambda$ using $\lambda_i$'s such that $\lambda=v/f$?

2-another case where the propagating wave is not periodic, just a gaussian pulse say that propagates with speed $v$. Can one define something like $v=\lambda f$ in this case? how to calculate $f$ and $\lambda$ such that $v=f\lambda$

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For any spatially periodic wave, sinusoidal or not, the wavelength is just the spatial period, by definition. It is true that if you compute the Fourier decomposition of a non-sinusoidal wave, you'll get a potentially infinite number of frequency components, but all of these will have wavelengths that are integer factors of the wavelength of the overall wave. In other words, suppose your wave is described by a function $F(x,t)$ satisfying $F(x,t) = F(x + \lambda,t)$. Then its spectral decomposition takes the form

$$F(x,t) = \sum_{n=0}^{\infty} a_n g_n(x,t)$$

where $g_n(x,t) = g_n(x + \lambda/n, t)$.

For the Fourier decomposition specifically, you'll have something like $g_n(x,t) = \sin(\lambda x/n - vt + \phi_n)$. Every component $g_n$ propagates with the same speed $v$, because of the wave equation. You can compute the frequencies of the various Fourier components using $f_n = v/(\lambda/n) = nv/\lambda = nf_1$, showing that all the frequencies are multiples of one fundamental frequency $f \equiv f_1$. This is the basic idea behind harmonics.

For a waveform that is not periodic, it doesn't make sense to define a wavelength or frequency for the wave as a whole. However, you can get some useful results by treating an aperiodic signal as a periodic signal that has an infinitely large wavelength, i.e. use the argument above in the limit $\lambda\to\infty,f\to 0$.

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Think through what these three quantities represent

• $\lambda$ is the wave length--the spacial period on which the wave repeats
• $f$ is the frequency--the temporal period on which the wave repeats
• $v$ is velocity--a ratio of distance traveled over time take to travel it

so...yeah, the relation holds for any wave that will maintain it's shape in space and time.

Note that if there is a frequency dispersion in the propagation of the wave (which there is for most physical systems), then the "holds it shape" condition is not more than approximately true, and this relation may not be useful over large time or distance scales.

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