How Does Wave Frequency Change with Each Harmonic in a Standing Wave Experiment?

[thunderfvck]

Hi

We just did a lab on waves and one in particular I'm unsure about.

We set up standing waves on a spring and observed the period of each successive harmonic.

It was found that the period was cut in half with each harmonic. Or the frequency was doubled. ie. harmonic 1 produced one standing wave with a frequency of f, harmonic 2 produced two standing waves each having a frequency of 2f, harmonic 3 produced three standing waves each having a frequency of 3f, etc.

It was also found that the wave speed of each wave was (wave length / period) about the same in each case. The spring was 4 m, the first harmonic produced a wave length of 8 m, the second of 4 m, the third of 2.67 m, and the fourth of 2 m.

I am expected to discuss what I have shown by doing this.

Here is what I am thinking, but I am unsure as to whether or not I'm correct:

In each harmonic, the same energy is distributed in the spring and so in each standing wave. Since the same energy is in each standing wave, the frequency of the first harmonic is f having a speed of v. In the second harmonic, there are two standing waves and so the energy is distributed equally, so the frequency of each wave will be twice as much; in other words, the period T will be half as much in each standing wave to make an overall value of T, or 1/f. The wave speed remains the same in each case because each wave is moving with the same speed, because the spring as a whole is moving with the same speed.

?

Another thing, we had to measure the speed of a pulse which we sent down the spring, let's say it was x. How does x apply to what's been performed?

x ~ 5 m/s (4 m, 0.85 s for pulse)
v ~ 8 m/s (wave speed, about the same for each +/- 1 m/s)

T1 ~ 1.2 s (period of first harmonic)
T2 ~ 0.6 s
T3 ~ 0.3 s
T4 ~ 0.15 s

Thank you very much!

 

[Tide]

Are these transverse waves or longitudinal waves? (Just curious!)

Also, you aren't dealing with multiple "standing waves" in any of those cases. You are dealing with a single standing wave with multiple nodes!

Try thinking of each standing wave as being composed of two traveling waves moving in opposite directions.

 

[M98Ranger]

Your understanding is correct! The concept of standing waves and harmonics can be a bit confusing, but it seems like you have a good grasp on it.

To summarize, when a wave travels through a medium, it can reflect off of boundaries and interfere with itself, creating a standing wave pattern. This standing wave pattern has points of maximum and minimum amplitude, called nodes and antinodes, respectively.

In the case of a spring, the first harmonic (also known as the fundamental frequency) produces one standing wave with a node at each end of the spring. This standing wave has a frequency of f and a period of T1. As you observed, the second harmonic produces two standing waves with a node and antinode at each end, and each standing wave has a frequency of 2f and a period of T2 (which is half of T1).

This pattern continues for each successive harmonic, with the frequency doubling and the period halving. This is because the standing wave pattern is determined by the length of the medium (in this case, the spring) and the speed of the wave. As you mentioned, the wave speed remains constant in each harmonic because the spring as a whole is moving with the same speed.

As for the speed of the pulse you measured, it is related to the wave speed, but it is not the same value. The speed of the pulse represents how fast the disturbance is traveling through the medium, while the wave speed represents how fast the wave as a whole is moving. In this case, the pulse speed (x) is slightly less than the wave speed (v), which is why you observed a slight difference in your measurements.

Overall, it seems like you have a good understanding of standing waves and harmonics. Keep up the good work!