Question

Which of the following functions represents a travelling wave? (A) \((\mathrm{x}-\mathrm{vt})^{2}\) (B) in \((\mathrm{x}+\mathrm{vt})\) (C) \(\mathrm{e}^{-(\mathrm{x}+\mathrm{vt}) 2}\) (D) \(\\{1 /(\mathrm{x}+\mathrm{vt})\\}\)

Short answer

Options A, C, and D represent traveling waves, while option B is invalid due to a misprint.

Step-by-step solution

Option A

\((\mathrm{x}-\mathrm{vt})^{2}\): This function has the traveling wave form \((\mathrm{x}-\mathrm{vt})\) inside the square. This could represent a wave moving in the positive direction along the x-axis. So, option A seems to be a possible solution.

Option B

"in \((\mathrm{x}+\mathrm{vt})\)": This expression doesn't make sense symbolically as a function, and it seems to be a misprint. Therefore, it cannot represent a traveling wave.

Option C

\(\mathrm{e}^{-(\mathrm{x}+\mathrm{vt}) 2}\): This function has the form \(\mathrm{e}^{-\mathrm{k}(\mathrm{x}+\mathrm{vt})^2}\) (where k=1), which shows a Gaussian modulated wave traveling in the negative direction along the x-axis (due to the minus sign before the exponent). Option C represents a traveling wave.

Option D

\(\\{1 /(\mathrm{x}+\mathrm{vt})\\}\): This function has the form \(\frac{1}{\mathrm{x}+\mathrm{vt}}\), which represents a wave moving in the negative direction along the x-axis. Therefore, option D also represents a traveling wave. In conclusion, options A, C, and D represent traveling waves, and the only invalid option is option B.