# Why pulse waves results in wave packets?

I was doing experiments of measuring sonic velocity and I generate pulse waves from sensor 1, but when they are received by sensor 2, I saw wave packets on the oscilloscope, can you explain why? I was told that it might be related to Fourier transform but I don't understand.

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Yes this is related to Fourier transforms. If you have a signal $s(t)$ that you measure over a time window $\Delta t$, then you will have a resulting dispersion in the frequencies i.e. a $\tilde{s}(\omega)$ that differs from a delta function for a purely harmonic signal. The "physical" reason for this is that for your sensor the signal received is zero before and after a time window $\Delta t$ and you need to sum many harmonic waves together to get vanishing amplitudes after a given time.

More mathematically, if you have a harmonic signal $s_h(t)$, then you need to convolute it with some window function $f_w(t)$ to generate the pulse. The total signal in the output of your sensor 1 is therefore $s(t)=s_h \star f_w$, where $\star$ denotes the convolution operation.

Now, the Fourier transform of a convolution is nothing but the simple product of the Fourier transform of the functions convoluted. It gives then:

$\mathrm{FT}[s](\omega)= \mathrm{FT}[s_h](\omega)\cdot \mathrm{FT}[f_w](\omega)$

Hence, even if $\mathrm{FT}s_h](\omega)= A\delta(\omega-\omega_0)+B\delta(\omega+\omega_0)$, $\mathrm{FT}[f_w](\omega)$ has no reason a priori to be a pure delta function, except if the corresponding time window is in fact infinite...which is never the case in the real world and evn less the case for pulses.

That is why you have a wave packet.

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