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A-pipe-closed-at-one-end-emits-sound-wave-of-frequency-440-Hz-i-Determine-the-length-of-the-air-column-ii-Frequencies-of-the-second-harmonic-and-the-third-harmonic-Velocity-of-sound-is-330m-s-Re




Question Number 10728 by Saham last updated on 23/Feb/17
A pipe closed at one end emits sound wave of frequency 440 Hz.   (i) Determine the length of the air column  (ii) Frequencies of the second harmonic and the third harmonic  Velocity of sound is 330m/s  Refractive index of glass is 1.50, that of  air is 1.00,   and that of water is 1.33
$$\mathrm{A}\:\mathrm{pipe}\:\mathrm{closed}\:\mathrm{at}\:\mathrm{one}\:\mathrm{end}\:\mathrm{emits}\:\mathrm{sound}\:\mathrm{wave}\:\mathrm{of}\:\mathrm{frequency}\:\mathrm{440}\:\mathrm{Hz}.\: \\ $$$$\left(\mathrm{i}\right)\:\mathrm{Determine}\:\mathrm{the}\:\mathrm{length}\:\mathrm{of}\:\mathrm{the}\:\mathrm{air}\:\mathrm{column} \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{Frequencies}\:\mathrm{of}\:\mathrm{the}\:\mathrm{second}\:\mathrm{harmonic}\:\mathrm{and}\:\mathrm{the}\:\mathrm{third}\:\mathrm{harmonic} \\ $$$$\mathrm{Velocity}\:\mathrm{of}\:\mathrm{sound}\:\mathrm{is}\:\mathrm{330m}/\mathrm{s} \\ $$$$\mathrm{Refractive}\:\mathrm{index}\:\mathrm{of}\:\mathrm{glass}\:\mathrm{is}\:\mathrm{1}.\mathrm{50},\:\mathrm{that}\:\mathrm{of}\:\:\mathrm{air}\:\mathrm{is}\:\mathrm{1}.\mathrm{00},\: \\ $$$$\mathrm{and}\:\mathrm{that}\:\mathrm{of}\:\mathrm{water}\:\mathrm{is}\:\mathrm{1}.\mathrm{33} \\ $$
Answered by sandy_suhendra last updated on 23/Feb/17
(i) f_1 =440 Hz  v=f_1 .λ_1  ⇒ λ_1 =(v/f_1 )=((330)/(440))=0.75 m     the air column=h_1 =(1/4)λ_1 =(1/4)×0.75=0.1875 m     (ii) f_1  : f_2  : f_3  : ... = 1 : 3 : 5 : ...  second harmonic=f_2 =(3/1)×440=1,320 Hz  third harmonic=f_3 =(5/1)×440=2,200 Hz
$$\left(\mathrm{i}\right)\:\mathrm{f}_{\mathrm{1}} =\mathrm{440}\:\mathrm{Hz} \\ $$$$\mathrm{v}=\mathrm{f}_{\mathrm{1}} .\lambda_{\mathrm{1}} \:\Rightarrow\:\lambda_{\mathrm{1}} =\frac{\mathrm{v}}{\mathrm{f}_{\mathrm{1}} }=\frac{\mathrm{330}}{\mathrm{440}}=\mathrm{0}.\mathrm{75}\:\mathrm{m}\:\:\: \\ $$$$\mathrm{the}\:\mathrm{air}\:\mathrm{column}=\mathrm{h}_{\mathrm{1}} =\frac{\mathrm{1}}{\mathrm{4}}\lambda_{\mathrm{1}} =\frac{\mathrm{1}}{\mathrm{4}}×\mathrm{0}.\mathrm{75}=\mathrm{0}.\mathrm{1875}\:\mathrm{m}\:\:\: \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{f}_{\mathrm{1}} \::\:\mathrm{f}_{\mathrm{2}} \::\:\mathrm{f}_{\mathrm{3}} \::\:…\:=\:\mathrm{1}\::\:\mathrm{3}\::\:\mathrm{5}\::\:… \\ $$$$\mathrm{second}\:\mathrm{harmonic}=\mathrm{f}_{\mathrm{2}} =\frac{\mathrm{3}}{\mathrm{1}}×\mathrm{440}=\mathrm{1},\mathrm{320}\:\mathrm{Hz} \\ $$$$\mathrm{third}\:\mathrm{harmonic}=\mathrm{f}_{\mathrm{3}} =\frac{\mathrm{5}}{\mathrm{1}}×\mathrm{440}=\mathrm{2},\mathrm{200}\:\mathrm{Hz} \\ $$
Commented by Saham last updated on 23/Feb/17
Thanks for your help. God bless you sir.
$$\mathrm{Thanks}\:\mathrm{for}\:\mathrm{your}\:\mathrm{help}.\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}. \\ $$

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