Partial differential wave (d'Alembert) solution check please

[CannonSLX]

Homework Statement

Homework Equations

General wave solution y=f(x+ct)+g(x-ct) [/B]

The Attempt at a Solution

[/B]

Graphical sketch

 

[haruspex]

Your solution to the first part is essentially right, but the logic flow is backwards. You have shown that if the differential equation has solutions of the given form then the parameter must be ±2. You were asked to show that with a parameter of either of those values a function of the proposed form is a solution to the equation.

For the second part, I'm not sure what is wanted, so cannot comment.

In the last part, did you check that your answer satisfies the boundary conditions?

 

[CannonSLX]

Your solution to the first part is essentially right, but the logic flow is backwards. You have shown that if the differential equation has solutions of the given form then the parameter must be ±2. You were asked to show that with a parameter of either of those values a function of the proposed form is a solution to the equation.

For the second part, I'm not sure what is wanted, so cannot comment.

In the last part, did you check that your answer satisfies the boundary conditions?

How do I check my answer satisfies the boundary conditions please?

 

[haruspex]

How do I check my answer satisfies the boundary conditions please?

Plug in t=0.

 

[CannonSLX]

Plug in t=0.

So is that substituting t=0 into the final solution I have at the bottom right to achieve what exactly, please?

 

[haruspex]

So is that substituting t=0 into the final solution I have at the bottom right to achieve what exactly, please?

To see whether it yields y(x,0)=sin(x).

 

[CannonSLX]

To see whether it yields y(x,0)=sin(x).

when t=0
The following occurs

 

[haruspex]

when t=0
The following occurs

View attachment 209329

Which gives 0, not the required sin(x). So find your error.

 

[CannonSLX]

Which gives 0, not the required sin(x). So find your error.

Dont see my error from my workings

 

[haruspex]

Dont see my error from my workings

Look at the equation where you introduce the constant B. Check the sign of the sin(x) term.

 

[CannonSLX]

Look at the equation where you introduce the constant B. Check the sign of the sin(x) term.

I see, it should be + sin(x)/2

But I don't see how substituting t=0 will provide the result of y(x,0)=sin(x), as its sin(x)/2 along with an exponential function.

 

[haruspex]

I see, it should be + sin(x)/2

But I don't see how substituting t=0 will provide the result of y(x,0)=sin(x), as its sin(x)/2 along with an exponential function.

Take a closer look at your post #7. There is some cancellation. As it is there, everything cancels, but with the corrected sign it will give the desired result.