Standing Wave in a Organ Pipe
Category : JEE Main & Advanced
Organ pipes are the musical instrument which are used for producing musical sound by blowing air into the pipe. Longitudinal stationary waves are formed on account of superimposition of incident and reflected longitudinal waves.
Equation of standing wave \[y=2a\cos \frac{2\pi vt}{\lambda }\,\sin \frac{2\pi x}{\lambda }\]
Frequency of vibration \[n=\frac{v}{\lambda }\]
Different mode of vibration in organ pipe
| Closed organ pipe | ||
| Fundamental mode | Third harmonic First over tone | Fifth harmonic Second over tone |
|
\[{{n}_{1}}\,=\,\frac{v}{4l}\] |
\[{{n}_{2}}\,=\,\frac{v}{{{\lambda }_{2}}}\,\,=\,\frac{3v}{4l}\,=3\,{{n}_{1}}\] |
\[{{n}_{3}}\,=\,\frac{5v}{4l}=\,5{{n}_{1}}\] |
| Open organ pipe | ||
| Fundamental mode | Second harmonic | Third harmonic |
|
\[{{n}_{1}}=\frac{v}{{{\lambda }_{1}}}=\,\frac{v}{2l}\] |
\[{{n}_{2}}=\frac{v}{{{\lambda }_{2}}}=\frac{v}{L}=\,2{{n}_{1}}\] |
\[{{n}_{3}}=\frac{v}{{{\lambda }_{3}}}=\,\frac{3v}{2l}=3{{n}_{1}}\] |
(1) Closed organ pipe
(i) In closed organ pipe only odd harmonic are present. Ratio of harmonic is \[{{n}_{1}}:{{n}_{3}}:{{n}_{5}}~....=~1\text{ }:\text{ }3\text{ }:\text{ }5\text{ }....\]
(ii) pth overtone \[={{(2p+1)}^{th}}\] harmonics
(iii) Ratio of overtones = 3 : 5 : 7 ....
(iv) The maximum possible wavelength is 4l
(v) General formula for wavelength is \[\lambda \,=\,\frac{4l}{\left( 2N-1 \right)}\]; where N = 1, 2, 3, ... corresponds to order of mode of vibration.
(vi) General formula for frequency \[n=\frac{(2N-1)v}{4l}\]
(vii) Position of nodes from closed end \[x=0,\,\,\frac{\lambda }{2},\lambda ,\frac{3\lambda }{2}.....\]
(viii) Position of antinodes from closed end \[x=\frac{\lambda }{4},\frac{3\lambda }{4},\frac{5\lambda }{4}...\]
(2) Open organ pipe
(i) In open organ pipe all (even and odd) harmonic are present. Ratio of harmonic is \[{{n}_{1}}:{{n}_{3}}:{{n}_{5}}~....=~1\text{ }:\text{ }2\text{ }:\text{ }3~....\]
(ii) pth overtone \[={{(p+1)}^{th}}\] harmonics
(iii) Ratio of overtones = 2 : 3 : 5 ....
(iv) The maximum possible wavelength is 2l
(v) General formula for wavelength is \[\lambda \,=\,\frac{2l}{N}\]; where N = 1, 2, 3, ... corresponds to order of mode of vibration.
(vi) General formula for frequency \[n=\frac{Nv}{2l}\]
(vii) Position of nodes from one end \[x=\frac{\lambda }{4},\frac{3\lambda }{4},\,\frac{5\lambda }{4}.....\]
(viii) Position of antinodes from one end \[x=0,\,\frac{\lambda }{2},\lambda ,\frac{3\lambda }{2}.....\]
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