For this activity you will be required to use the equations given in lecture for the frequency of the different modes for strings, pipes, and bars to calculate the conditions. The purpose is to give practice using these calculations to make decisions in how these strings/pipes/bars can be design to produce the desired frequencies and explore the relationships between the different variables and the tones produced.
Part 1: String Instruments
For this section you will be using the simulator found here to verify your calculations and help visualize the results (feel free to work through their examples as well for additional practice, but this is not required). You can open the simulation by click on the icon that looks like:
The simulator should appear in the upper left corner of the page (this is a java applet and you may have to allow your browser to open it…don’t worry it is safe). Note that the frequencies that this simulator demonstrates are lower than normal audible frequencies, but the principles are the same as the frequencies are larger.
The fundamental frequency of a string is given by equation 1 below.
where v, is the wave speed, T is the tension in the string, L is the length of the string, and μ is the linear mass density. A more general form for any mode of vibration, n, is given as
If the linear mass density is 100 g/m and the tension is 4.9 N for this 1 m long string, calculate the fundamental frequency (first mode), the frequency of the second harmonic (second mode) and the frequency of the third harmonic (third mode). Be careful of your units! Record these values in your notebook entry and verify them using the simulator. Draw sketches of what the standing waves look like for each of the modes (fundamental frequency and harmonics) and note what happens to the wave when you are far from a harmonic frequency.
In practical applications it is hard to know exactly what the tension is in your string. Instead, tension is used to tune to the correct frequency. However, it is important to know whether a particular tension that is needed in order to create a desired frequency for a given wire will cause the wire to break or significantly deform (stretch). The string will break when the ultimate stress is reached (the string will begin to permanently deform once the yield stress is reached). Both ultimate and yield stress are properties of a material and can be looked up in data bases. Stress is simply a force per unit area or in this case, the stress, σ, is
where A is the cross sectional area. We can also write the linear mass density as
Where m is the mass of the wire and ρ is the density of the wire. Since density of the wire is also a material property that can be looked up, we can combine equations 3 and 4 to find that
For a given wire material, you need to ensure that
In order to prevent permanent deformation in your wire as it is holding a particular fundamental frequency.
Say that you have a wire material that has a density of and a yield stress of 0.4 MPa (note ). Using the simulator, find a tension and a linear mass density of the wire combination that will cause the 1 m long wire to have a fundamental frequency of 2 Hz, 3 Hz, and 4 Hz without failing. Record at least one combination that works in your lab notebook. If it is not possible note that in your notebook.
Part 2: Pipe Wind Instruments
For this section you will be using the simulator found here to verify your calculations and help visualize the results. There is also a java version available at http://www.walter-fendt.de/ph14e/stlwaves.htm This will require you to override some of the java security settings if choose to use java.
The modal frequencies of a pipe with two open ends are given by
which is the same as for a string except that the speed of the wave is just the speed of sound in air (in the simulation it was taken to be 343.5 m/s, but this will vary depending on the temperature of the air).
Similarly, if one end of the pipe is closed off, the node that is created at the end of the pipe causes the relationship between the length of the pipe and frequency to be
where p is the harmonic number and n is the mode number. Now, since there must be node at the closed off end n must be an odd number. This means that only the odd modes will be included in the spectrum. To understand how this happens, you can view the simulation of how standing waves are created from the interaction between the incident and reflected wave found at this website.
Calculate the length of tube needed to create a fundamental frequency of 200 Hz for the case with both sides open and the case where one side is closed off. Put your values into the simulator to check your answers. Record these calculations in your lab notebook.
In your notebook, calculate the length of tube needed to create this frequency while accounting for end effects. For this analysis, use a radius of the pipe of 0.5 in (0.0127 m).
Part 3: Percussion (Bar) Instruments
Calculating the modal frequencies of bars is slightly more difficult due to the number of factors which must be considered, such as material, length, cross-sectional geometry, how the bar is supported, and non-harmonic behavior of oscillations. For this part of the lab you need to find bars with different characteristics and strike them such that they make sound and observe the relative frequencies of the different bars. A hardware store is a great place to find these types of things, just don’t damage them in the store and be prepared to get lots of strange looks. You could alternatively just find things around the house. You should compare drastically different bars and record your observations in your notebook (pick differences that will create significantly different frequencies, low and high). Some examples might include a hollow bar and a solid bar; a short bar and a long bar; a thick bar and a thin bar; metal bar and a wood bar; you could explore how the way you hold it affects the frequency; etc. You should have at least 3 different comparisons (again, just make sure the the frequencies produced are noticeably different when changing the parameter).
The equation which describes the modal frequencies for a simply supported and cantilevered beams is given by
Where, for simply supported beams, L is the length of the bar, v is the speed of sound in the material, m = 3.0112 when n=1, when n=2 m=5 and m=7 when n=3,…(2n+1) and K=bar thickness/3.46 for rectangular bars and for tubes.
For cantilevered beams, the only difference is the values of m which are m=1.194 when n=1, m=2.988 when n=2, and m=5 when n=3,..(2n-1).
For the 3 comparisons that you made, calculate the fundamental frequency for the beams and make sure that the observations correspond to the trends in the calculations (i.e. beam A sounded lower than beam B and beam A’s theoretical frequency was calculated to be lower than beam B’s). We are not concerned about the exact numbers in this set of calculations, but rather just observing general trends. So, just use reasonable values for the speed of sound of the material and dimensions (it doesn’t have to be a perfect prediction, just good enough to make a qualitative comparison). Include these calculations and interpretations in your notebook.