Question

Among the following, the equation representing a progressive wave is:

• A. \( y = 2 \cos 3x \sin 10t \)
• B. \( y = 2 \sqrt{x - vt} \)
• C. \( y = 3 \sin (5x - 0.5t) + 4 \cos (5x - 0.5t) \)
• D. \( y = \cos x \sin t + \cos 2x \cdot \sin 2t \)

Hint:

Progressive waves vary as \( f(x \pm vt) \). If the argument is the same in both sine and cosine (like C), it's a valid progressive wave.

Answer 1

The Correct Option is B

A progressive wave has the general form: \[ y = f(x \pm vt) \quad \text{(or)} \quad y = A \sin(kx - \omega t + \phi) \] Let’s analyze each: - (A) \( y = 2 \cos 3x \sin 10t \) is a product of functions in \(x\) and \(t\) — represents a stationary wave, not progressive.
- (B) \( y = 2 \sqrt{x - vt} \) is not a harmonic wave (not sinusoidal).
- (C) Combine terms:
\[ y = 3 \sin (5x - 0.5t) + 4 \cos (5x - 0.5t) = R \sin(5x - 0.5t + \phi) \] This represents a progressive wave.
- (D) Sum of two standing wave expressions — not a single progressive wave.