Sound Waves Legacy Problem #14 Guided Solution
Problem*
In a demonstration, Mr. H stretches a steel wire to a length 1.23 meters and braces both ends so that they are not free to vibrate. He attaches a fancy piece of equipment which he calls a mechanical oscillator to the wire and explains how it works. Then Mr. H turns the oscillator on and tunes the frequency to 588 Hz. To the amazement of the class, the wire begins vibrating in the sixth harmonic wave pattern.
- Determine the speed of waves within the wire.
- Determine the frequency at which the wire will vibrate with the first harmonic wave pattern.
- Determine the frequency at which the wire will vibrate with the second harmonic wave pattern.
Audio Guided Solution
A wire has a set of natural frequencies at which it naturally vibrates with, and when it vibrates at one of its natural frequencies, it vibrates in such a manner that a standing wave pattern is established within the wire. Now, if you click the link back to the Physics Classroom tutorial where they discuss standing wave patterns, you'll see a discussion of standing wave patterns and a collection of animations and diagrams that will help you understand what is meant by a standing wave pattern. Here, what we're told is that it's vibrating in the sixth harmonic wave pattern. That means that within the length of the wire, there are six sections that are vibrating up and down. In the middle of each one of those sections, we have what's called an antinode. Now, if there are six sections vibrating up and down, then along the length of the wire, there should be three complete wavelengths, because each section represents a half of a wavelength. So if the wire is 1.23 meters long, and there are six sections vibrating in three full wavelengths, we can find the wavelength of these waves by dividing the 1.23 meters by three. When we do that, we end up getting 0.41 meters as the wavelength. We also know the frequencies that causes the sixth harmonic wave pattern. It's 588 hertz, and in Part A of the problem, they ask us to calculate the speed of waves within the wire. So I'm going to take the wavelength of 0.41 meters and multiply it by the frequency of 588 hertz, and that gives me the speed of waves within the wire. It comes out to be 241.0800 meters per second, and I can round that to three significant digits. Now, in Part B and Part C of this problem, they ask us to calculate the first and second harmonic frequencies. To do so, you need to understand that for all of the frequencies within the set of frequencies that the wire vibrates with, they're all related to the first harmonic frequency by some integer. In other words, the sixth harmonic frequency is six times the first harmonic frequency. The second harmonic frequency is two times the first harmonic frequency. Sometimes I express it in terms of an equation, f of the nth frequency is equal to n times f of the first frequency, fn equal n times f1. So if I want to find the frequency of the first harmonic, I simply take the sixth harmonic frequency of 588 hertz, and I divide by six. It gives me a value of 98.0 hertz, or 98 hertz. And if I want to find the second harmonic frequency, I just take the first and multiply by two, and that gives me 196 hertz.
Solution
- 241 m/s
- 98 Hz
- 196 Hz
Habbits of an Effective Problem Solver
- Read the problem carefully and develop a mental picture of the physical situation. If necessary, sketch a simple diagram of the physical situation to help you visualize it.
- Identify the known and unknown quantities and record in an organized manner, often times they can be recorded on the diagram itself. Equate given values to the symbols used to represent the corresponding quantity (e.g., \(\descriptive{v}{v,velocity} = 345\unit{\meter\per\second}\), \(\descriptive{λ}{λ,wavelength} = 1.28 \unit{m}\), \(\descriptive{f}{f,frequency} = \colorbox{gray}{Unknown}\)).
- Use physics formulas and conceptual reasoning to plot a strategy for solving for the unknown quantity.
- Identify the appropriate formula(s) to use.
- Perform substitutions and algebraic manipulations in order to solve for the unknown quantity.
Read About It!
Get more information on the topic of Sound Waves at The Physics Classroom Tutorial.