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.

Key concepts

Wave equation

The wave equation is an essential mathematical formula used to describe the behavior of waves. It generally takes the form \[ \frac{\partial^2 \psi}{\partial t^2} = v^2 \frac{\partial^2 \psi}{\partial x^2} \] where

• \( \psi \) is the wave function, often representing the displacement of the wave.
• \( v \) is the speed of the wave.
• \( x \) and \( t \) represent space and time coordinates, respectively.

Understanding the wave equation allows us to predict how waves behave over time and space.
The solutions to this equation can represent many wave types, including sound waves, electromagnetic waves, and water waves.

Wave propagation

Wave propagation refers to how waves travel through different media. The process involves the transmission of energy and momentum from one point to another.

• Waves can propagate through solids, liquids, and gases.
• The energy and momentum carried are distributed without the permanent displacement of the medium as a whole.

Understanding wave propagation is crucial for numerous applications, such as telecommunications, acoustics, and optics.
The wave's ability to reorganize itself after encountering obstacles demonstrates fascinating aspects of wave propagation, such as reflection, refraction, and diffraction.

Directional wave motion

Directional wave motion describes the path that a wave takes as it travels. Waves can move in different directions depending on the source and medium.

• Positive directional waves are expressed as \( \psi(x - vt) \), indicating movement along the x-axis direction.
• Negative directional waves are expressed as \( \psi(x + vt) \), indicating movement in the opposite direction.

Directional wave motion helps understand how waves, such as light and sound, move and interact with their environment, including phenomena like Doppler Effect and wave interference.

Gaussian wave packet

A Gaussian wave packet is a specific wave form characterized by a Gaussian function. It represents a wave that is well localized in space.

• Mathematically, it is expressed as \( e^{-k(x + vt)^2} \), where \( k \) indicates the spread of the packet.
• These packets are fundamental in quantum mechanics for describing particles with wave-like properties.

Gaussian wave packets demonstrate aspects of both wave and particle behavior, showing how quantum objects can be described mathematically.
They provide insights into the principle of uncertainty, where the exact location and momentum cannot be precisely determined simultaneously.

Physics problem-solving

Physics problem-solving involves using mathematical tools and logical reasoning to address questions about the physical world.

• Identify known quantities and variables.
• Apply relevant physical laws and equations.
• Break down complex problems into simpler components.

It is essential to understand the principles underlying the models and assumptions used.
Effective physics problem-solving often involves iterative methods, where hypotheses and approaches are continually refined based on observations and results.
This approach helps in developing new technologies and solving real-world challenges such as energy production, environmental issues, and medical advancements.