Answer to Question #286382 in Field Theory for Pranali

Question #286382

Which of the following six functions represent a travelling wave pulse whose shape does not change with time?

  1. 2e-5(x-300t)2
  2. (2t-5x)3
  3. 9x2-42xt+49t2
  4. x2-49t2
  5. sqrt[(3t-5x)2+5]
  6. (5+2x-3t)/(5+2x+3t)
1
Expert's answer
2022-01-12T08:31:20-0500

Solution

For checking six functions represent a travelling wave pulse whose shape does not change with time

2ψx2=1v22ψt2\frac{ ∂^2\psi}{ ∂x^2}=\frac{1}{v^2}\frac{ ∂^2\psi}{ ∂t^2} ............ (1)


Now function

ψ=2e5(x300t)2\psi=2e^{{-5}}{(x-300t) ^2}

ψ=2e5(x300tx+t2)\psi=2e^{{-5}}{(x-300tx+t^2)}

Then it's partial derivatives

2ψx2=4e5\frac{ ∂^2\psi}{ ∂x^2}=4e^{-5}

And also

2ψt2=4e5\frac{ ∂^2\psi}{ ∂t^2}=4e^{-5}

Then putting in equation (1)

Then velocity is found as fullfilled condition.


Now other

ψ=9x242xt+49t2\psi=9x^2-42xt+49t^2

Then partial derivatives

2ψx2=18\frac{ ∂^2\psi}{ ∂x^2}=18

And

2ψt2=98\frac{ ∂^2\psi}{ ∂t^2}=98

Whicha also satisfy equation.


So this is answer.


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