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Question-113243




Question Number 113243 by ShakaLaka last updated on 11/Sep/20
Commented by ShakaLaka last updated on 11/Sep/20
Can anybody solve this  question?
$$\mathrm{Can}\:\mathrm{anybody}\:\mathrm{solve}\:\mathrm{this} \\ $$$$\mathrm{question}? \\ $$
Commented by mr W last updated on 11/Sep/20
sin ωt ∙ sin ωat  has no x as variable!  please check the question!
$$\mathrm{sin}\:\omega{t}\:\centerdot\:\mathrm{sin}\:\omega{at} \\ $$$${has}\:{no}\:{x}\:{as}\:{variable}! \\ $$$${please}\:{check}\:{the}\:{question}! \\ $$
Commented by ShakaLaka last updated on 11/Sep/20
I will have to ask my teacher  that what is the correct question.  because this question I wrote is  totally same as it is.  Actually, this is the question of  the assignmemt, which was  written directly from university,  not by my teacher.  By the way, there is also  another mistake in the question  that there is written (∂^2 v/∂c^2 ) instead  of (∂^2 v/∂t^2 )
$$\mathrm{I}\:\mathrm{will}\:\mathrm{have}\:\mathrm{to}\:\mathrm{ask}\:\mathrm{my}\:\mathrm{teacher} \\ $$$${that}\:{what}\:{is}\:{the}\:{correct}\:{question}. \\ $$$$\mathrm{because}\:\mathrm{this}\:\mathrm{question}\:\mathrm{I}\:\mathrm{wrote}\:\mathrm{is} \\ $$$$\mathrm{totally}\:\mathrm{same}\:\mathrm{as}\:\mathrm{it}\:\mathrm{is}. \\ $$$$\mathrm{Actually},\:\mathrm{this}\:\mathrm{is}\:\mathrm{the}\:\mathrm{question}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{assignmemt},\:{which}\:{was} \\ $$$${written}\:{directly}\:{from}\:{university}, \\ $$$${not}\:{by}\:{my}\:{teacher}. \\ $$$${By}\:{the}\:{way},\:{there}\:{is}\:{also} \\ $$$${another}\:{mistake}\:{in}\:{the}\:{question} \\ $$$${that}\:{there}\:{is}\:{written}\:\frac{\partial^{\mathrm{2}} {v}}{\partial{c}^{\mathrm{2}} }\:{instead} \\ $$$${of}\:\frac{\partial^{\mathrm{2}} {v}}{\partial{t}^{\mathrm{2}} } \\ $$
Answered by mathdave last updated on 12/Sep/20
if only u(x,t)=sinwx.sinat  then is a solution that satisfy the wave  equation (∂^2 u/∂t^2 )=a^2 .(∂^2 u/∂x^2 )  by letting  ((∂^2 (sinwx.sinwat))/∂t^2 )=−w^2 a^2 sinwx.sinwat  and  a^2 .((∂^2 (sinwx.sinwat))/∂x^2 )=−w^2 a^2 sinwx.sinwat  ∵u(x,t)=sinwx.sinwat is a solution to  wave equation   (∂^2 u/∂t^2 )=a^2 .(∂^2 u/∂x^2 )         Q.E.D
$${if}\:{only}\:{u}\left({x},{t}\right)=\mathrm{sin}{wx}.\mathrm{sin}{at} \\ $$$${then}\:{is}\:{a}\:{solution}\:{that}\:{satisfy}\:{the}\:{wave} \\ $$$${equation}\:\frac{\partial^{\mathrm{2}} {u}}{\partial{t}^{\mathrm{2}} }={a}^{\mathrm{2}} .\frac{\partial^{\mathrm{2}} {u}}{\partial{x}^{\mathrm{2}} } \\ $$$${by}\:{letting} \\ $$$$\frac{\partial^{\mathrm{2}} \left(\mathrm{sin}{wx}.\mathrm{sin}{wat}\right)}{\partial{t}^{\mathrm{2}} }=−{w}^{\mathrm{2}} {a}^{\mathrm{2}} \mathrm{sin}{wx}.\mathrm{sin}{wat} \\ $$$${and} \\ $$$${a}^{\mathrm{2}} .\frac{\partial^{\mathrm{2}} \left(\mathrm{sin}{wx}.\mathrm{sin}{wat}\right)}{\partial{x}^{\mathrm{2}} }=−{w}^{\mathrm{2}} {a}^{\mathrm{2}} \mathrm{sin}{wx}.\mathrm{sin}{wat} \\ $$$$\because{u}\left({x},{t}\right)=\mathrm{sin}{wx}.\mathrm{sin}{wat}\:{is}\:{a}\:{solution}\:{to} \\ $$$${wave}\:{equation}\: \\ $$$$\frac{\partial^{\mathrm{2}} {u}}{\partial{t}^{\mathrm{2}} }={a}^{\mathrm{2}} .\frac{\partial^{\mathrm{2}} {u}}{\partial{x}^{\mathrm{2}} }\:\:\:\:\:\:\:\:\:{Q}.{E}.{D} \\ $$
Commented by ShakaLaka last updated on 13/Sep/20
yeah mathdave, it can be  possible
$${yeah}\:{mathdave},\:{it}\:{can}\:{be} \\ $$$${possible} \\ $$
Commented by ShakaLaka last updated on 15/Sep/20
mathdave you were right. my  teacher is also saying that the  function “sin ωt ∙ sin ωat” is  incorrect, the correct function  is “sin ωx ∙ sin ωat”; as you said.
$${mathdave}\:{you}\:{were}\:{right}.\:{my} \\ $$$${teacher}\:{is}\:{also}\:{saying}\:{that}\:{the} \\ $$$${function}\:“\mathrm{sin}\:\omega{t}\:\centerdot\:\mathrm{sin}\:\omega{a}\mathrm{t}''\:{is} \\ $$$${incorrect},\:{the}\:{correct}\:{function} \\ $$$${is}\:“\mathrm{sin}\:\omega{x}\:\centerdot\:\mathrm{sin}\:\omega{a}\mathrm{t}'';\:{as}\:{you}\:{said}. \\ $$

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