Question

If $y=A\sin (\omega t-kx)$, then the value of $\frac{d^{2}y}{d{t}^2}/\frac{d^{2}y}{d{x}^2}$ is

• A. $\frac{k^2}{{\omega }^2}$
• B. $-\frac{k^2}{{\omega }^2}$
• C. $\frac{{\omega }^2}{k^2}$
• D. $-\frac{{\omega }^2}{k^2}$

Answer 1

The correct option is C $\frac{{\omega }^2}{k^2}$

Given,

$y=A\sin (\omega t-kx)\to \left(1\right)$

Differentiate for two times on both sides w.r.t. t

$\frac{dy}{dt}=A\omega \cos (\omega t-kx)$

$\frac{d^{2}y}{d{t}^2}=-A{\omega }^2\sin (\omega t-kx)$

Differentiate $\left(1\right)$ for twice on both sides w.r.t. x

$\frac{dy}{dx}=-Ak\cos (\omega t-kx)$

$\frac{d^{2}y}{d{x}^2}=-A{k}^2\sin (\omega t-kx)$

$\frac{\frac{d^{2}y}{d{t}^2}}{\frac{d^{2}y}{d{x}^2}}=\frac{-A{\omega }^2\sin (\omega t-kx)}{-A{k}^2\sin (\omega t-kx)}=\frac{{\omega }^2}{k^2}$