Question

Among the following, the equation representing a progressive wave is:
\[ \text{(A)}\quad y=2\cos3x\sin10t \] \[ \text{(B)}\quad y=2\sqrt{x-vt} \] \[ \text{(C)}\quad y=3\sin(5x-0.5t)+4\cos(x-0.5t) \] \[ \text{(D)}\quad y=\cos x\sin t+\cos2x\sin2t \]

• A. A and D
• B. C
• C. A, C, D
• D. B

Hint:

A progressive wave always depends on the combined variable: \[ (x\pm vt). \] If space and time parts appear separately, the equation generally represents a standing wave.

Answer 1

The Correct Option is B

Step 1: Recall the standard form of a progressive wave.
A progressive wave is generally represented as \[ y=f(x\pm vt) \] or in sinusoidal form, \[ y=A\sin(kx-\omega t) \] or \[ y=A\cos(kx-\omega t) \] The displacement must depend on the combination \[ (x\pm vt) \] which indicates propagation of the wave.

Step 2: Analyze option (A).
\[ y=2\cos3x\sin10t \] This expression is a product of separate space and time terms. It represents a standing wave, not a progressive wave.
Hence, option (A) is not correct.

Step 3: Analyze option (B).
\[ y=2\sqrt{x-vt} \] Although it contains \[ (x-vt), \] it is not a valid wave equation because wave displacement must remain finite and continuous for all positions and times. This form does not represent a standard physical progressive wave.
Hence, option (B) is not correct.

Step 4: Analyze option (C).
\[ y=3\sin(5x-0.5t)+4\cos(x-0.5t) \] Both terms contain expressions of the form \[ (kx-\omega t), \] which clearly represent travelling waves. Their superposition also represents a progressive wave.
Hence, option (C) is correct.

Step 5: Analyze option (D).
\[ y=\cos x\sin t+\cos2x\sin2t \] Again, space and time variables are separated. Such forms represent standing waves rather than progressive waves.
Hence, option (D) is not correct.

Step 6: Final conclusion.
Therefore, the equation representing a progressive wave is \[ \boxed{\text{C}} \]