Given wave equation

$y(x,t)=e^{-(ax^2+bt^2+2\sqrt{ab}\,xt)}$

Rewrite the exponent

Observe that $ax^2+bt^2+2\sqrt{ab}\,xt=(\sqrt a\,x+\sqrt b\,t)^2$

Hence, $y(x,t)=e^{-(\sqrt a\,x+\sqrt b\,t)^2}$

Compare with standard travelling wave form

A travelling wave moving without changing shape has the form $y(x,t)=f(x-vt) \quad \text{or} \quad y(x,t)=f(x+vt)$

where:

• $f(x-vt)$ represents motion in the $+x$ direction,
• $f(x+vt)$ represents motion in the $-x$ direction.

Now, $y(x,t)=e^{-(\sqrt a\,x+\sqrt b\,t)^2}$

Factor out $\sqrt a$ inside the argument: $\sqrt a\,x+\sqrt b\,t=\sqrt a\left(x+\sqrt{\frac{b}{a}}\,t\right)$

Therefore, $y(x,t)=e^{-a\left(x+\sqrt{\frac{b}{a}}\,t\right)^2}$

This is of the form $f\left(x+vt\right)$ with $v=\sqrt{\frac{b}{a}}$

Direction of motion

Since the form is $f(x+vt)$, the wave moves in the negative $x$-direction.

Check options

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

Final answer

The given expression represents a travelling wave moving in the $-x$ direction with speed $\boxed{\sqrt{\frac{b}{a}}}$