Name: Longitudinal Standing Waves
Uploaded: 2019-07-19
Duration: 13 min 37 s
Description: We have already studied standing waves, but do you know standing waves can be formed in longitudinal waves also, watch this video to know more.

Question

Show that all harmonics are present in case of an air column vibrating in a pipe open at both ends.

Accepted Answer

In fundamental mode, the vibration of air column open at both ends, there are two anti-nodes and one node as shown in the figure.

Let $λ$ be the wavelength of the wave, $v$ be the velocity and $L$ be the length.

Now, as we can see $L = \frac{λ}{2}$ or $λ = 2 L$

and $n = \frac{v}{λ} = \frac{v}{2 L}$ , this is the frequency of fundamental mode in a pipe open at both the ends.

Now for $1^{s t}$ harmonic let $λ_{1}$ be the wavelength, from the figure, $L = λ_{1}$ and $n_{1} = \frac{v}{λ_{1}} = \frac{v}{L}$

$∴ n_{1} = 2 n$

Now for $2^{n d}$ harmonic let $λ_{2}$ be the wavelength, from the figure, $L = \frac{3 λ_{2}}{2}$ and $n_{2} = \frac{v}{λ_{2}} = \frac{3 v}{2 L}$

$∴ n_{2} = 3 n$

therefore, $n, 2 n, 3 n$ ..... are the harmonics which are both even as well as odd.

Show that all harmonics are present in case of an air column vibrating in a pipe open at both ends.

Important Questions on Stationary Waves

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