# 3.2: Activities

- Page ID
- 25274

## Things You Will Need

Besides the computer that you will need to run this PhET simulator (go ahead and run this now), you will also need a separate stopwatch timer – this comes standard in most smart phones and tablets.

## Using the Simulator

For this lab, the computer that runs the simulator must be able to play sound, and the sound must be turned on and audible. We will be using only the simplest of the selections in the simulator:

**select the window "Waves"****set the "Frequency" slider to its "min"****setting****set the "Amplitude" slider to its "max" setting****click****the sound source button****select the "Play Tone" check box****leave the continuous wave button selected**

## Properties of the Simulator's Sound

The speed of sound can vary greatly depending upon the medium through which it is traveling. The simulator does not specify what the medium is, so we will measure the speed directly. The way to make the most accurate measurements possible is to **select the "Graph" checkbox**. As we saw with a previous simulator, a powerful tool for precise time measurements is to pause the simulation, start the timer (it will not run until the simulation is restarted), and then continue the simulation one step at a time with the step simulator button.

- Use the simulator to find the speed of the sound wave in the simulator. Compare it with the standardly-accepted value of the speed of sound in air.
- Compute the frequency of the sound, using each of the two methods given below. [
*There is a trick we have used in past labs to reduce the percentage error by increasing its denominator. You should use that trick here.*]- Use the step feature of the simulator and measure the time for each full oscillation at a fixed point using the timer tool .
- Use the wavelength and the speed of the wave. You will find the tape measure tool useful here.

## Beats

Next we will examine an interference effect for sound. To do this, we need *two* sources that are at slightly different frequencies, so:

**open a second browser window, and run the simulator in it as well, configuring it the same as the simulator you already have running****move the "Frequency" slider of one of the simulators off the minimum by the smallest amount that you can.**

- Compute the frequency of the sound coming from the second simulator, using whichever method you prefer (but use the "high denominator trick" again to keep the percentage error low!).

Now **run both simulators simultaneously**. You should hear pulsing "beats."

- Let's check what we know about beats...
- Use the timer on your smartphone to measure the frequency of the beats directly, keeping the percentage error low.
- Compare this measurement from what you expect the beat frequency to be based on the individual sound waves.

## Graphing

Let's show how this phenomenon works graphically. Navigate your browser to this site, where you will find an online graphing calculator. When you are listening to the sound coming from your computer speakers, you are measuring the sound at a fixed point in space. This simplifies the harmonic wave equations of two variables to an equation in time only, as stated in the text reference in the Background Material. Our two browser windows with simulators therefore each provide independent harmonic pressure waves that satisfy the following equation at our eardrums:

\[ P\left(t\right)=P_o\cos\left(2\pi f t + \phi\right) \]

Both browsers were set at the same amplitude, so \(P_o\) is the same for both (we'll just call it '1' for our graphing purposes). We don't know what the phase angle \(\phi\) is for each browser, nor do we even know if they are the same. It is dangerous to assume that they are, so we will not.

**use the desmos online graphing calculator to graph the superposition of the two sound waves created by the two simulators**(note: you can put \(\pi\) into your formula by simply typing "pi")**create different "slider" variables for the phase angles of the two waves****use the graph settings button to make the range of the horizontal axis smaller, so that you can make out some basic features**(the graph doesn't need to show detail of the carrier wave, but you may want to have a look at it before you're done)

- Do the following with your graph:
- Submit a screen capture of the graph with your lab report, including the formula panel.
- Use the graph to show/explain how it confirms the beat frequency you measured.
- Describe the effects of changing ("sliding") the phase angles of the two waves. In particular, is the beat frequency affected?

## Lab Report

Download, print, and complete this document, then upload your lab report to Canvas. [*If you don't have a printer, then two other options are to edit the pdf directly on a computer, or create a facsimile of the lab report format by hand.*]