Standing Waves

Anyone who wants to “hear the ocean” knows they just have to cup their ears(if you have never done this, try it out!). The sounds you hear are not actually those of the ocean, but rather are select frequencies of standing waves contained within the air chamber created by your body. Standing waves are also the source of sounds that emerge from stringed instruments– guitar, piano, harp, etc. While anyone can make sounds with these by plucking the strings, it takes a skilled musician to create the pleasant music we enjoy.

In this lab, your group will work to observe and measure standing waves on a string. By tuning a mechanical oscillator, you will be able to determine specific frequencies– harmonics of the fundamental frequency – that generate standing waves. You will also work towards drawing relationships between several parameters that determine the fundamental frequency of any string system. Through investigating these relationships, you and your group are tasked with developing a model that relates them to the fundamental frequency of the system.Should you desire to relate these investigations to music, several questions are featured to help guide these discussions.

All objects have a resonant frequency where waves within the medium constructively interfere with one another. While at resonance, the medium will vibrate with maximum amplitude. This is easy to observe with a string and a mechanical resonator. For a standing wave on a string, every point along the string oscillates with constant amplitude. Therefore, some points oscillate very rapidly at maximum amplitude, called an anti-­‐node, while other points don't seem to move at all, called nodes.

For a string (length L) that is bound on both ends (i.e., the ends are fixed in place),the longest length that can oscillate as a standing wave has one anti-­‐node, or two nodes. This represents the fundamental wavelength of the system (λ0), which occurs at the which is given as !=2 This can be related to the_fundamental frequency_ (f0) and the speed of the wave © as

! = !

There are many further equations that define standing wave systems, but they will be left to your investigations and modeling.

In order to investigate standing waves, it would be best to ensure your understanding of:

• Nodes and anti-­‐nodes, as well as how to count the number of them on a string(i.e., does n depend on number of nodes? Anti-­‐nodes? Something else?)

• How the frequency, wavelength, and speed of a wave are related

• How waves interact and interfere, called_superposition_

• Harmonics

• How to model data in Excel

Part 1 – Observing Waves on a String

With the equipment provided, observe the result of oscillating a string with a mechanical resonator. In doing so, be sure to explore a wide range of frequencies to help determine how both large and small changes affect the system. It may be helpful to consider:

• Can you find frequencies that result in standing waves?

• Can you change the amplitude of the vibrations without changing the frequency?

• What affect does changing various parameters of your system have on the string's oscillation?

Part 2 – Measuring Harmonics

By now, you have likely noticed that you are able to obtain standing waves at several frequencies. Without changing any other parameters,model the relationship between each standing-­‐wave frequency and the fundamental frequency. To do so, it may be helpful to consider:

• What is the fundamental frequency?

• How does the number of nodes and anti-­‐nodes depend on frequency?

The relationship between the fundamental frequency and the first harmonic is the same relationship between octaves. Can you relate the observation of harmonics to octaves in music? Why do the notes periodically repeat on a piano? (i.e., why are there multiple “C” notes and how do their frequencies relate?)

Part 3 – Length Dependence

As discussed in the theory section, the length of the string determines the wavelength of a standing wave on it. Therefore, it would be an appropriate conclusion that the frequencies observed also depend on the length of string. Without changing other parameters, model the relationship between the fundamental frequency and the length of the string.

• From these measurements and observations, can you develop an understanding for why the pitch of a guitar or violin depends on the fret held by the musician? Why a harp or piano has a specific shape?

Part 4 – Tension Dependence

You may have already adjusted the amount of mass on the end of the string in Part 1 , but if not, adjust this parameter to observe its affect on the system.

• What affect does changing the mass on the hanger have on the system physically?

• What changes in the oscillations do you observe when adjusting this mass?

Without changing other parameters, model the relationship between the fundamental frequency and the tension on the string.

• From these measurements, can you draw conclusions as to the mechanisms behind how instruments like guitars and violins are tuned?

Part 5 – Challenges

Option A – String Density

A useful parameter of physical systems is its density. A string could be considered one-­‐dimensional, and thus we refer to its density as its_linear mass density_ (μ), or the mass per length.

By switching the string with other options, model the relationship between the fundamental frequency and string density.

With your understanding of tension in Part 4 , why does the material or thickness of a string matter (i.e., why do stringed instruments have different string thicknesses as you change notes)?

Option A, continued – Putting it all Together

You have by now modeled the frequency on a number of important parameters. By considering all of these parameters, can you determine the complete model to determine the fundamental frequency of resonance for a given physical system (i.e., the equation that relates physical parameters to the fundamental frequency)?

Option B – Creating an Instrument

By now, you should be familiar with all of the necessary parameters that determine the pitch of a guitar string. Replacing the string with a guitar string, determine the fundamental frequency. From there, can you adjust the parameters of the system such that it's fundamental frequency is an “E”? (If successful, plucking the string with the mechanical oscillator off should result in an “E” tone.)

While conducting the experiment, consider the following questions:

• How many data points are worth taking to obtain a clear relationship between parameters?

• How are you determining a reasonable fit of your data?

• What are sources of uncertainty that arise?

• How can you minimize the effect of these sources?

• How can you consider these sources when analyzing your data?