The given wave function is: $y(x,t)=e^{-(a^2{x}^2+b^2{t}^2+2\sqrt{ab}xt)}$

Let's analyze this step by step to understand the nature of the wave.

Step 1: Rewrite the Argument

First, look at the exponent: $-(a^2{x}^2+b^2{t}^2+2\sqrt{ab}xt)$

Notice that this looks like the expansion of a squared term. In fact, it is a perfect square:

${(ax+bt)}^2=a^2{x}^2+b^2{t}^2+2abxt$

Our exponent has $2\sqrt{ab}xt$ instead of $2abxt$. But note that $\sqrt{ab}=\sqrt{a}\sqrt{b}$. Therefore, we can rewrite the exponent as:

$-{(\sqrt{a}x+\sqrt{b}t)}^2$

So the wave function simplifies to: $y(x,t)=e^{-{(\sqrt{a}x+\sqrt{b}t)}^2}$

Step 2: Identify the Wave Type

A general form for a traveling wave is , where $v$ is the wave speed.

Our function is $y(x,t)=f(\sqrt{a}x+\sqrt{b}t)$, where $f\left(z\right)=e^{-z^2}$.

This matches the form $f(x+vt)$. The plus sign indicates the wave is moving in the negative x-direction.

Step 3: Find the Wave Speed

Compare the argument $\sqrt{a}x+\sqrt{b}t$ to the standard form $x+vt$.

We can factor this argument to make the coefficient of x equal to 1: $\sqrt{a}\left(x+\frac{\sqrt{b}}{\sqrt{a}}t\right)=\sqrt{a}\left(x+\sqrt{\frac{b}{a}}t\right)$

From this, we can see the wave speed $v$ is $\sqrt{\frac{b}{a}}$.

Therefore, this is a wave moving in the -x direction with speed $\sqrt{\frac{b}{a}}$.

Final Answer: The wave is moving in the -x direction with speed $\sqrt{\frac{b}{a}}$.

Related Topics & Formulae

Traveling Waves: A disturbance that moves through a medium, transferring energy without permanently displacing the medium itself. The general mathematical form is $y(x,t)=f(x\pm vt)$.

• $y(x,t)=f(x-vt)$ → Wave moving in the +x direction.
• $y(x,t)=f(x+vt)$ → Wave moving in the -x direction.

Wave Speed (v): The speed at which the wave profile moves. It is a property of the medium and is constant for a given wave on a given string.