Question

A wave is represented by the equation \(y=0.2\,\sin 4\pi \left[\frac{t}{0.08}+\frac{x}{0.8}+\frac{1}{12}\right]\) in SI units.
The expression represents a wave traveling in the

• A. positive x-direction with velocity 10 m/s and with wavelength 0.4 m.
• B. negative x-direction with velocity 10 m/s and with wavelength 0.4 m.
• C. positive x-direction with velocity 20 m/s and with wavelength 0.6 m.
• D. negative x-direction with velocity 20 m/s and with wavelength 0.6 m.

Hint:

Rewrite the equation in the standard form with angular frequency and wave number multiplying t and x, then read off the direction, speed and wavelength.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
Expand the argument: \(y=0.2\sin\left(50\pi t+5\pi x+\dfrac{\pi}{3}\right)\). So \(\omega=50\pi\) and \(k=5\pi\).

Step 2: Direction
The \(t\) and \(x\) terms have the same sign, so the wave moves in the negative x-direction.

Step 3: Speed and wavelength
\(v=\dfrac{\omega}{k}=\dfrac{50\pi}{5\pi}=10\ \text{m/s}\)
\(\lambda=\dfrac{2\pi}{k}=\dfrac{2\pi}{5\pi}=0.4\ \text{m}\)

Final Answer:
The wave travels in the negative x-direction at 10 m/s with wavelength 0.4 m. \[ \boxed{\text{(B) }\text{negative x-direction, 10 m/s, wavelength 0.4 m}} \]