Tapping a steel rod with a hammer-standing waves

[redbeard]

Homework Statement

so the rod is of length l, and the speed of a transverse wave is v_w. It asks for the possible frequencies f_i. The rod is held in the centre so a node exists here.

Homework Equations

look below

The Attempt at a Solution

I know that v_w = $$\lambda$$ * f_i

So it follows that i must find an expression relating wavelength to length. Here is where I run into my problem, it being conceptual. I don't know what occurs at the ends of the rods. I saw a picture somewhere and it appeared to be antinodes but I don't understand why. Could someone give me an explanation?

I would personally think that there could be nodes and antinodes but I'm not sure of this.

Thanks, Redbeard

 

[ehild]

Nodes are at the fixed end and antinodes at the free end. Presumably, the ends of the rod are free, as the problem says that it is held in the centre.

ehild

 

[redbeard]

Yes, but that doesn't quite explain it for me. Why are the ends anti-nodes?

Like if you think of longitudinal waves in an air tube being made at one end and going towards an open end, the pressure difference at the open end is 0 and so the displacement is at a max (an anti-node). This makes physically intuitive sense.

I was kinda hoping there was some sort of similar explanation for anti-nodes at the ends of the rod? What is a physical explanation of what is happening here?

Redbeard