What is the velocity of the given wave based on the wave equation?

[sweep123]

[SOLVED] Satisfying wave equation

Homework Statement

Confirm that the following wave satisfies the wave equation and obtain an expression for the velocity of a wave

Y=Asin(2x-5t)*e^(-2t)

Homework Equations

the wave equation is

(d^2y/dt^2)=(V^2)*(d^2y/dx^2)

The Attempt at a Solution

I assumed that I had to differentiate Y with respect to 't' twice and the differentiate Y with respect to 'x' twice and then substitute these into the equation.

This left me with

-21Ae^(-2t)sin(2x-t)+20Ae^(-2t)cos(2x-5t)=(V^2)(-4Ae^(-2t)sin(2x-5t))

but this doesn't really prove that the wave satisfies the equation. Does it?

I can then rearrange to get V the wave velocity. Am I on the right track?

 

[Rainbow Child]

Is A constant or is it A(x)? Because with A constant, your function $$y(x,t)$$, does not satisfy the wave equation.

 

[Astronuc]

One would have to demonstrate that both sides of the wave equation are equal when using the proposed solution.

The general wave equation is
$$\frac{\partial^2 u} {\partial t^2} = c^2 \nabla^2 u$$, where c is the wave velocity. That constant, c, would be found in the solution.

So then, what is the value of V based on the given function?

I would expect A is a constant coefficient of amplitude.