Question

[1pt] Which of the following six functions represent a travelling wave pulse whose shape does not change with time?

E.g., if the first one can represent such a pulse, but the rest cannot, answer A.

2e-5(x-300t)^2

(2t-5x)^3

9x^2-42xt+49t^2

x2-49t^2

sqrt[((3t-5x)^2)+5]

(5+2x-3t)/(5+2x+3t)

Answer 1

For a function to represent a wave, it must comply with the following equality of partial derivatives:

$\frac{\partial^{2} \Psi }{\partial x^{2}}=\frac{1}{v^{2}}\frac{\partial^{2}\Psi }{\partial t^{2}}$

to know if some of the functions shown above fulfill a wave function, we will derive the function twice with respect to x and then twice with respect to t. If the iguladad is fulfilled, then if they are wave functions.

Funtion 1

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

in the previous equation we solved the remarkable product

First step: We look for the second partial derivative of x

$\frac{\partial \Psi }{\partial x}=2e^{-5}(2x-300t)$

$\frac{\partial^{2} \Psi }{\partial x^{2}}=2e^{-5}(2)=4e^{-5}$

Second step: We look for the second partial derivative of t

$\frac{\partial \Psi }{\partial t}=2e^{-5}(-300x+2t)$

$\frac{\partial^{2} \Psi }{\partial t^{2}}=2e^{-5}(2)=4e^{-5}$

in this case $\frac{1}{v^{2}}=1$ and equality is fulfilled. The function if it represents a wave function and its form does not change with time because the function after being derived twice does not depend on t.

funtion 2

$\Psi =9x^{2}-42xt+49t^{2}x$

$\frac{\partial \Psi }{\partial x}=18x-42t$

$\frac{\partial^{2} \Psi }{\partial x^{2}}=18$

now we derive with respect to t

$\frac{\partial \Psi }{\partial t}=-42x+98t$

$\frac{\partial^{2} \Psi }{\partial t^{2}}=98$

in the case of function 2, equality is not met and consequently it is not a wave function.

if the following functions are derived, the equality of the partial derivatives is not fulfilled and they are not wave functions. The only function that meets equality is function 1.