Let $U ( x , t ) = e ^ { - ( x - c t ) ^ { 2 } }$ .Simplify $U _ { t t } - C ^ { 2 } U _ { x x }$ .
A) $U _ { t t } - c ^ { 2 } U _ { x x } = 0$ .
B) $U _ { t t } - C ^ { 2 } U _ { x x } = U$ .
C) $U _ { t t } - c ^ { 2 } U _ { x x } = - U$ .
D) $U _ { t t } - c ^ { 2 } U _ { x x } = c ^ { 2 } U$ .

Question

Quizplus · Accepted Answer

To show that $U _ { t t } - c ^ { 2 } U _ { x x } = 0$, we differentiate $U(x,t) = e^{- (x - ct)^2}$ with respect to $t$ and $x$, and then substitute into the expression. Both $U_{tt}$ and $c^2 U_{xx}$ will yield terms that cancel each other out, leading to the result $0$. This is characteristic of the wave equation solution.

Let U(x,t)=e−(x−ct)2U ( x , t ) = e ^ { - ( x - c t ) ^ { 2 } }U(x,t)=e−(x−ct)2

Let $U ( x , t ) = e ^ { - ( x - c t ) ^ { 2 } }$