Question

A wave travelling along the x-axis with y representing its displacement is described by (𝑣 is the speed of the wave)

• A. \(\frac{∂y}{∂x}+\frac{1}{v}\frac{∂y}{∂t}=0\)
• B. \(\frac{∂y}{∂x}-\frac{1}{v}\frac{∂y}{∂t}=0\)
• C. \(\frac{∂^2y}{∂x^2}-\frac{1}{v^2}\frac{∂^2y}{∂t^2}=0\)
• D. \(\frac{∂^2y}{∂x^2}+\frac{1}{v^2}\frac{∂^2y}{∂t^2}=0\)

Answer 1

The Correct Option is A, B, D

The question involves understanding the equation of a wave travelling along the x-axis with displacement \( y \), characterized by partial derivatives. Let's analyze the options in the context of wave equations.

Wave equations generally involve second-order partial derivatives and are derived from considering wave motion in terms of space and time. For a one-dimensional wave travelling along the x-axis, the standard wave equation is:

\(\frac{∂^2y}{∂x^2}-\frac{1}{v^2}\frac{∂^2y}{∂t^2}=0\)

This represents the relationship between the spatial and temporal variations of the wave. The wave speed is represented by \( v \), and its square affects the time derivative to ensure dimensional consistency.

Now, let's assess the given options:

1. \(\frac{∂y}{∂x}+\frac{1}{v}\frac{∂y}{∂t}=0\): This option implies a first-order relationship between space and time derivatives. While it may represent a condition for specific wave scenarios, it is not the general wave equation.
2. \(\frac{∂y}{∂x}-\frac{1}{v}\frac{∂y}{∂t}=0\): Similar to the first option, this represents a linear relationship and is likely used for specific methodologies, not for general wave description.
3. \(\frac{∂^2y}{∂x^2}-\frac{1}{v^2}\frac{∂^2y}{∂t^2}=0\): As derived earlier, this is the standard wave equation, correctly describing the wave behaviour in terms of second derivatives with respect to space and time.
4. \(\frac{∂^2y}{∂x^2}+\frac{1}{v^2}\frac{∂^2y}{∂t^2}=0\): This form would imply a different type of wave motion, potentially representing decaying solutions or non-standard wave behaviour.

Therefore, the correct answer, representing the well-known wave equation, is:

• \(\frac{∂^2y}{∂x^2}-\frac{1}{v^2}\frac{∂^2y}{∂t^2}=0\)

This option correctly describes a wave travelling along the x-axis with the displacement \( y \), and \( v \) as the speed of the wave.