Question

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)

Answer 1

Solution

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

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

Now function

$\psi=2e^{{-5}}{(x-300t) ^2}$

$\psi=2e^{{-5}}{(x-300tx+t^2)}$

Then it's partial derivatives

$\frac{ ∂^2\psi}{ ∂x^2}=4e^{-5}$

And also

$\frac{ ∂^2\psi}{ ∂t^2}=4e^{-5}$

Then putting in equation (1)

Then velocity is found as fullfilled condition.

Now other

$\psi=9x^2-42xt+49t^2$

Then partial derivatives

$\frac{ ∂^2\psi}{ ∂x^2}=18$

And

$\frac{ ∂^2\psi}{ ∂t^2}=98$

Whicha also satisfy equation.

So this is answer.