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Which of the following functions represent wave ? Plzz Explain each option by differential equation of wave... →d²y/dx²=1/v²(d²y/dt²)
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Asked by jhajuhi19 | 05 Mar, 2020, 13:28: PM
answered-by-expert Expert Answer
if a function y(x,t) is solution to the differential equation of wave motion ,  begin mathsize 14px style fraction numerator d squared y over denominator d x squared end fraction space equals space 1 over v squared fraction numerator d squared y over denominator d t squared end fraction end style  ..........................(1)
where v is speed of wave, then y(x,t) represents wave .
 
(a) y(x,t) = sin(kx) cos(ωt )
 
we find,  begin mathsize 14px style fraction numerator d squared y over denominator d x squared end fraction space equals space minus space k to the power of 2 to the power of space end exponent space sin left parenthesis k x right parenthesis space cos left parenthesis omega t right parenthesis space equals space minus k squared space y end style ...........................(2)
we find begin mathsize 14px style fraction numerator d squared y over denominator d t squared end fraction space equals space minus space omega to the power of 2 to the power of space end exponent space sin left parenthesis k x right parenthesis space cos left parenthesis omega t right parenthesis space equals space minus omega squared space y end style.......................(3)
In above equations ω is angular frequency and k is wave vector .
 
Speed of the wave v is related to angular frequency ω and wave vector k is ,  v = ω/k  ..................(4)
 
from eqns.(2) , (3) and (4), we verify the function y(x,t) = sin(kx) cos(ωt) is solution to the differential equation
of wave motion as given by eqn.(1).
 
hence the function y(x,t) represents wave
---------------------------------------------------------------------------
 
(b) y(x,t) = k2 x2 - ω2 t2
 
we find , begin mathsize 14px style fraction numerator d squared y over denominator d x squared end fraction space equals space 2 space k to the power of 2 to the power of space end exponent space space end style  and    begin mathsize 14px style fraction numerator d squared y over denominator d x squared end fraction space equals space minus 2 space omega to the power of 2 to the power of space end exponent space space end style
from the above relations we find y(x,t) = k2 x2 - ω2 t2 is not the solution of differential equation (1)
 
Hence y(x,t) = k2 x2 - ω2 t2 do not represent wave
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similarly we can verify for other functions
 



Answered by Thiyagarajan K | 05 Mar, 2020, 19:24: PM

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