Testing functions as solutions to the vibrating string wave equation

[Feynmanfan]

Dear friends,

I need some help with transversal waves, to be precise: the vibrating string. I’ve been given many mathematical representations of what can be a wave (e.g: 10(x^2-v^2*t^2)
or this one, 5 Sinx cosv*t)

I have to argue which of them can be a solution of the vibrating string wave equation. And if it’s a solution I’ve been asked to write it in the D’Alembert form (that’s f(x-vt)+g(x+vt)).

Just tell me if what I think is correct: I insert the possible solution in the wave equation to see if both sides of the equation match (is that all or am I missing something?).

Thanks

 

[dextercioby]

Yeah,it needs to satisfy 1D d'Alembert's equation.

Daniel.

 

[Feynmanfan]

Thanks Daniel.

Now I have a more specific question on this vibrating string problem. Given u(x,t)=5Senx*Cos(vt), I've proved that it's a solution. However, while trying to write it in the D'Alembert form I get this: 5i cos(x+vt)cos(x-vt)

Does it make sense that it is imaginary? Aren't string waves supposed to be real. I don't know if I'm mixing up things.

 

[dextercioby]

It can't be complex (with a nonzero imaginary part,that is).U should use a trigonometric identity

$$\sin x\cos y\equiv \frac{1}{2}\left[\sin\left(x+y\right)+\sin\left(x-y\right)\right]$$

Daniel.