Solving Waves on a Metal Rod with kx + Φ

[LCSphysicist]

Homework Statement

I will post an image right below.

Relevant Equations

All below.

I am trying to solve this question by ξ = A*cos(ωt + θ)*sin(kx + Φ)
Anyway, the two initial terms of the product helps nothing (i think), what matters is sin(kx + Φ)
So, i tried by two ways:

First:

The stress is essentially zero on the ends, that is, something like cte*∂ξ/∂x (strain) would be zero, so
L is the length of the rod.

cos(kx + Φ) need to be 0
Φ = (2n-1)*π
kL + Φ = (2n-1)*π

It is not good.

Second:

The half length position would carry a node, so
ξ = A*cos(ωt + θ)*sin(kL/2 + Φ) = 0
KL/2 + Φ = (n)*π
and with this i can not solve without Φ in the expression.

 

[haruspex]

I assume ξ is displacement. You would find it easier with x measured from the centre of the rod.

 

[LCSphysicist]

I assume ξ is displacement. You would find it easier with x measured from the centre of the rod.

Definitively is easier deal with x measured from the center of the rod, but what is wrong in the above approach?

 

[haruspex]

Definitively is easier deal with x measured from the center of the rod, but what is wrong in the above approach?

It seems to me you should need to use the knowledge of the state both at the clamped point and one end, so they are not alternative methods. Rather, they should be combined into a single method.
Also

cos(kx + Φ) need to be 0
Φ = (2n-1)*π

I assume you meant cos(kL + Φ).
I would have thought the free end would have maximum amplitude.
On the other hand, cos((2n-1)*π)=-1, not 0, so you have effectively taken it as max.