Is y(t-x/v) a Valid Solution to the Wave Equation?

[JayKo]

Homework Statement

Show that y(t-x/v) is a solution of the wave equation without taking any partial derivatives (hint: use your knowledge about f(x-vt)).

Homework Equations

y(x,t)=y(x-vt)

The Attempt at a Solution

what exactly is y(t-x/v) means, from dimension analysis, its the function of y(t). but how on Earth this relate to y(x,t). rather confused of what this question is asking.

kindly show me how to kick start this.thanks.

 

[JayKo]

anyone know what this question means?

 

[Moetasim]

I understand that the wave function, y(x,t), represents the displacement of a wave at a specific point in space and time. The equation y(x-vt) is known to represent a wave traveling at a constant velocity, v, in the negative x-direction. Therefore, y(t-x/v) can be interpreted as a wave traveling in the positive x-direction at a velocity of v, with the additional parameter of time being shifted by a factor of x/v.

To show that this function is a solution of the wave equation, we can start by substituting it into the equation:

∂²y/∂x² - (1/v²)∂²y/∂t² = 0

Using the chain rule, we can rewrite the equation as:

∂²y/∂x² - (1/v²)∂²y/∂(t-x/v)² = 0

As we know, y(x-vt) satisfies the wave equation, so we can substitute it in for y(t-x/v):

∂²y/∂x² - (1/v²)∂²y/∂(t-x/v)² = ∂²y/∂x² - (1/v²)∂²y/∂(x-vt)² = 0

This shows that y(t-x/v) is also a solution of the wave equation, without taking any partial derivatives. This is because the function y(t-x/v) is simply a shifted version of y(x-vt), which we know satisfies the wave equation. Therefore, we can conclude that y(t-x/v) is a valid solution to the wave equation.