Question

The transverse displacement $y(x,t)$ of a wave on a string is given by $y(x,t)=e^{-(a{x}^2+b{t}^2+2\sqrt{ab}xt\}}$. This represents a :

• A. wave moving in $+x$ direction with speed $\sqrt{\frac{a}{b}}$
• B. wave moving in $-x$ direction with speed $\sqrt{\frac{b}{a}}$
• C. standing wave of frequency $\sqrt{b}$
• D. standing wave of frequency $\frac{1}{\sqrt{b}}$

Answer 1

Answer: (B)

wave moving in $-x$ direction with speed $\sqrt{\frac{b}{a}}$

Given that $y=e^{-(a{x}^2+2\sqrt{ab}xt+b{t}^2)}=e^{-(\sqrt{a}x+\sqrt{b}t{)}^2}=e^{-\sqrt{a}(x+\sqrt{\frac{b}{a}}t{)}^2}$

The given displacement equation is of the form $y=f(x+vt)$

Thus, the wave moves in $-\text{x}$ with a speed of $v=\sqrt{\frac{b}{a}}$