Problem 5

[Miranda Jarvis]

I was wondering if for problem 5 if we have to show all the steps to getting answers or if we can simply apply the applicable formulae from lecture notes? (example can we simply use the D'Alembert formula?)

[Laurie Deratnay]

Just wondering if in part c) of problem 5 the question should read: Consider (7) in (x<3t, x>-3t) ?
(instead of x>3t ....)

[Victor Ivrii]

I was wondering if for problem 5 if we have to show all the steps to getting answers or if we can simply apply the applicable formulae from lecture notes? (example can we simply use the D'Alembert formula?)

What problem D'Alembert formula solves? Is it the same problem as here?

[Victor Ivrii]

Just wondering if in part c) of problem 5 the question should read: Consider (7) in (x<3t, x>-3t) ?
(instead of x>3t ....)

Just wondering whether you have any reason to prefer $\{x<3t, x>-3t)\}$ to $\{x>3t, x>-3t\}$ or just want to boost the number of posts

[Levon Avanesyan]

I will expand this question a little bit:
So, can we use the formula of general solution for wave equation or we should prove/derive it?

P.S. This is my first post here, so I am not sure if I am asking an appropriate question

[Victor Ivrii]

I will expand this question a little bit:
So, can we use the formula of general solution for wave equation or we should prove/derive it?

Yes, you are allowed to use every appropriate (relevant) formula given in the class without rederiving it (unless specifically asked to derive it first). Definitely you need to ask yourself: "can I apply a formula for a general solution?", "Can I apply D'Alembert formula?"

There is a subtle difference between allowed and  can: you can use any appropriate formula but it may happen that you are explicitly asked to use some specific approach in which case you are allowed to use only some specific formulae.

P.S. This is my first post here, so I am not sure if I am asking an appropriate question

Yes, it is a completely appropriate question.

[Laurie Deratnay]

My reason for the asking the above question is that the examples that I have managed to find regarding Goursat problems and the wave equation all have the x<3t which made more sense to me for the integration - after thinking about it I am guessing #5 has x>3t due to the initial velocity?  The examples I was looking at only included initial positions - I was just trying to get a picture of what was happening.

[Victor Ivrii]

My reason for the asking the above question is that the examples that I have managed to find regarding Goursat problems and the wave equation all have the x<3t which made more sense to me for the integration - after thinking about it I am guessing #5 has x>3t due to the initial velocity?  The examples I was looking at only included initial positions - I was just trying to get a picture of what was happening.

Both problems have exactly the same properties. In fact if one considers one spatial dimension then $x$ and $t$ could be permuted (and equation multiplied by $-1$) and the type of equation would not change; so in fact $x$ could be a time ant $t$ a spatial coordinate.

Of course, it would not be a case if there were 2 or more spatial variables, in
$u_{tt}-c^2 u_{xx}-c^2u_{yy}=0$  $t$ and $x$ are not on equal footing (as presence $u_{yy}$ prevents from multiplying by $-1$).

[James McVittie]

For the initial conditions for Problem 5 (c) should they both be at x = 3t or is it not a typo?
Thanks

[Victor Ivrii]

For the initial conditions for Problem 5 (c) should they both be at x = 3t or is it not a typo?
Thanks

There are no initial conditions in 5(c). Note a special name of the problem (Goursat). It is not IVP!

[Laurie Deratnay]

Ok great - does this mean that I can understand the 2nd auxiliary condition of 5 c) as corresponding physically to a boundary condition on the velocity?

[James McVittie]

My question was with the Goursat problem, should it be of the form:

$$u_{tt}-c^{2}u_{xx}=0$$

$$u(t=\tau)=g(x)$$

$$u_{t}(t=\tau)=h(x)$$

like the example from class?

[Victor Ivrii]

My question was with the Goursat problem, should it be of the form:

$$u_{tt}-c^{2}u_{xx}=0$$

$$u(t=\tau)=g(x)$$

$$u_{t}(t=\tau)=h(x)$$

like the example from class?

Then it would be not a Goursat, but Cauchy problem

[Djirar]

I can't really see how part C of problem 5 is any different then a Cauchy problem, except for a change in coordinates and in the notes the initial conditions of the Goursat problem are given with respect to U only and not U_t .

Could someone please explain the differences between Goursat and Cauchy problems? thanks in advance.

[Jinlong Fu]

I guess that there is a typo in the assignment for 5.(c) about the Goursat problem:

the formula (9) should be as below to be a Goursat problem
\begin{equation}  u|_{x=3t}=t, \quad u|_{x=-3t}=2t. \end{equation}

instead of that given in the assignment: (this is the IVP)
\begin{equation} u|_{x=3t}=t,\quad u_t|_{x=-3t}=2t. \end{equation}