Question

The transverse displacement \( y(x, t) \) of a wave on a string is given by \( y(x,t) = e^{-(x^2 + t^2)} \sin(kx - \omega t) \). This represents a:

• A. wave moving in \( -x \) direction, 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 \( \sqrt{\frac{a}{b}} \)

Hint:

For a wave equation of the form \( y(x,t) = e^{-(x^2 + t^2)} \sin(kx - \omega t) \), the wave moves in the \( -x \) direction and its speed is \( \sqrt{\frac{b}{a}} \).

Answer 1

The Correct Option is A

Step 1: Analyze the wave equation.
The given wave equation represents a traveling wave. The term \( kx - \omega t \) suggests a wave moving in the \( -x \) direction. The speed of the wave is given by \( v = \sqrt{\frac{b}{a}} \), where \( a \) and \( b \) are constants in the equation.
Thus, the correct answer is (1).