How Do You Calculate the Speed of Transverse Waves in a Vibrating Wooden Bar?

[jmm5872]

A wooden bar when struck vibrates as a transverse standing wave with three antinodes and two nodes. The lowest frequency note is 43.6 Hz, produced by a bar 55.4 cm long. Find the speed of transverse waves on the bar.


I assumed that 3 antinodes and 2 nodes means the eigenfrequency f=3/2(v/L). I also assumed that 43.6 Hz was the fundamental frequency. Since I want f₃, I multiplied 43.6 by 3 and got 130.8 Hz.

From here I plugged into the first equation 130.8=(3/2)(v/.554) and solved for v.
v=48.3088 m/s.

But this answer was wrong, so I am not sure what I did wrong.

I would appreciate any advice, Thanks,
Jason

 

[inutard]

For a wooden bar with anti-nodes on both sides, the formula for wavelength is:
Wavelength = 2L/n, where L is the length of the bar and n is the harmonics number.

For three antinodes and 2 nodes, the bar is in its second harmonic and so wavelength is:
2L/2 = L

Since Frequency*Wavelength = Velocity,
Velocity = (43.6)(0.554) = 24.15 m/s

**Its been a while since I did this, so i may be wrong.

 

[jmm5872]

Thanks, that was correct.

I guess i still don't understant how it is the second harmonic though. I thought the antinodes were the max points, and three max points means 3/2 of a wavelength. For example, two positive maximums, and one negative maximum.

 

[inutard]

Ah but think about it. Say we're looking at one interval of a cosine curve (0 to 2pi). How many max points are there? How many points cross y = 0? The points that cross y = 0 are like the nodes (when you multiply the cosine curve by -1 they remain invariant) while the antinodes are the points where y = +/- 1. (This results in 3 antinodes and 2 nodes which means 3 antinodes and 2 nodes = 1 wavelength of a cosine curve).

Its a weird way of thinking about it. But try drawing a picture, it might help.

 

[jmm5872]

You're right, I was picturing a sine function. I didn't notice that the sine is opposite, three nodes and two antinodes.

Thanks again