As the Mathematical treatment yields, the pressure wave leads the displacement wave by $\frac{\pi}{2}$. But I want to ask why is it so? According to me, supposing if the displacement wave is a sine curve (hence pressure change is a cosine curve), then from phase 0 to $\frac{\pi}{2}$ displacement increases and $∆P$ is still postive. How? Shouldn't $∆P$ be negative where the displacement is increasing because the air column gets stretched and rarefaction takes place?
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