Experiment 123 Determination of the sound wave velocity with the method of Lissajous figures

Transcription

1 perimen 3 Deerminaion of he sound wave veloci wih he mehod of Lissajous figures The aim of he eercise To sud acousic wave propagaion in he air To deermine of he sound wave veloci in he air Mehodolog of he measuremen The veloci of he longiudinal wave in he solid saes can be obained as follows: ssume ha he rod of he densi is eended along he -ais Consider a small clinder of a lengh d and a cross-secion area The relaive eension of he bar equals: and ( ) d d d in posiion and d, respecivel or ssems ha obe Hooke's law, he relaive eension equals:, where is he modulus of elasici and he resoring force eered b he maerial The resulan force acing on he mass dm of he clinder reads: d d ccording o he second Newon s law he resulan force is given b he epression: d dm Comparing boh equaions one can derive he following differenial equaion: which is a classical propagaing wave equaion: ( ), c where c refers o he propagaion speed of he longiudinal perurbaion and equals: c or he acousic wave propagaing in he air one has o find he equivalen of he elasici modulus for a gas Hence we can consider a clinder wih a gas and a moving pison wih a cross-secion area The saring pressure of he gas is given b p and he pison eers force

2 on he gas, which produces addiional pressure p / s a resul he saring volume of he gas will be reduced The iniial and final lenghs of he clinder fulfilled wih he gas equal: l V / and l l V / V/, respecivel The Hooke s law for he gas reads: l l V p l l V p V dp V K V dv where, and K is he modulus of gas compressibili nder he assumpion ha flucuaions of he gas are fas enough o disable he hea ransfer κ he gas undergoes he adiabaic process: p V cons, where κ c p /c V, c p being he specific hea for consan pressure and c V being he specific hea for consan volume Differeniaing κ κ V dp κ p V dv and dividing he resul b V κ- he above-menioned epression ( ) dp p ield: κ, dv V Combining his equaion wih he speed of sound equaion one obains he formula c κ p ha describes he speed of sound in a gas The main componens of he air are diaomic gases (N, O ) hence κ 4 Tpical amospheric pressure equals 5 N/m or such condiions he densi of air equals 3 kg/m 3 and, consequenl, he speed of sound in he air equals 38 m/s The measuremen process

3 The measuremen of he speed of sound in he air can be performed in he eperimenal seup presened above The signal emied b he acousic generaor is ransmied o he speaker and o he horizonal plaes of he oscilloscope The acousic wave propagaes oward he microphone and is ransformed ino he elecric signal ransmied o he verical plaes of he oscilloscope The microphone can be moved on he bench and he posiion of he microphone can be read from he ape-measure The signal from microphone is shifed wih respec o he signal from he speaker according o he relaion: Φ Φ Φ( r), where Φ sands for he phase shif relaed o all phenomena no relaed o he sound waves, Φ( r) is a phase shif deermined b he posiions of he speaker and he microphone If one applies he harmonic signal o he horizonal and verical plaes of he oscilloscope, namel () sin( ω ), and () sin( ω Φ), he resulan figure observed on he screen will be described b he following epression cos ( Φ) sin ( Φ) The above equaion corresponds o he ellipse or he phase shif Φ k π, where k is he ineger, he equaion below describes he line ± Therefore, for Φ k π one obains

4 , and for Φ (k ) π ( ) The illusraive eample is given below Φ kπ Φ (k-)π The ransiion from one figure o anoher can be realized b moving he microphone wih respec o he speaker, which resuls in he phase shif Φ π The minimal disance beween he posiions corresponding o boh figures is D, which is relaed o he wave period T, he wavelengh λ and he frequenc f b means of he formula: λ c T c D, f Performing a number of he measuremens of D for a given frequenc f one can deermine he veloci of sound in he air: c f D Procedure Turn on he oscilloscope and he generaor e he sinusoidal signal wih he maimal frequenc of f 5 khz on he generaor 3 Measure he frequenc of he signal using he oscilloscope simae he error f Inroduce he resuls o he able 4 Move he microphone o he ereme far posiion and observe he figures on he screen There should be an ellipse inscribed wihin cm cm recangular on he screen 5 Move microphone back ill he line appears on he screen Inroduce he posiion of he microphone ino he able 6 Move he microphone furher and record he posiions of he microphone corresponding o he line appearing on he screen 7 Reverse he direcion of moving he microphone and record he characerisic posiions

5 8 Change he frequenc of he acousic wave following he advice of he insrucor and repea poins ner all he daa in he able simae he error of he posiion of he microphone m Repor preparaion The repor should conain: shor descripion of he eperimenal mehod The eperimenal daa colleced in he able ignal from generaor Number of he measuremen Posiion of he microphone Disance c c f [Hz] f [Hz] I r i [ m ] D r i r i [m] [ m/s ] Calculaions and he error analsis or each se of D measuremens and for each frequenc f calculae: a) rihmeic mean of D b) Mean squared error according o he formula: D n [ D Di] i n ( n ), c) The measuremen error D D α α using uden isher error wih α 95 d) The speed of he acousic wave in he air according o he epression: c fd e) The error of he sound speed according o he formula: c 4( D f f D α ) f) The final speed for each frequenc should be given in he form: c c ± c 4 Conclusions