Interference & Beats

Equipment differences between labs is a bit of an issue

May want to include instructions in resetting fct gen

Mystery freq wasn't coming through sometimes, might be unplugged?

The diagram on the lab page needs tweaking, beams aren't actually parallel

Wording for tuning fork & sine qave parts were vague, should people use beats or scope?


We don't have to go over the mathematics of interference before we try to observe it, but we'll include a little bit here if you want to peek. Maybe try the experiments below and come back here if you need help understanding what's going on?


Interference is simply a consequence of the principle of superposition. This means that at any point in space, the total amplitude of the disturbance due to two separate traveling waves is the sum, or superposition, of the amplitudes of the two separate waves.

Mathematically, we can write two sinusoidal traveling waves of equal amplitude, $y_1$ and $y_2$, at point $x = 0$ as

$y_1 = A\cos (\omega t + \varphi_1)$
$y_2 = A\cos (\omega t + \varphi_2)$,

where $\varphi_i$  may be a function of position, but not of time. The superposition of these waves yields the total wave $y_T$,

$y_T = y_1 + y_2 = 2A\cos\left(\dfrac{\varphi_2-\varphi_1}{2}\right)\cos \left(\omega t + \dfrac{\varphi_2 + \varphi_1}{2}\right)$. (2)

This gives a total intensity of

$I_t = \left< y_T^2\right> = 2A^2\cos^2\left(\dfrac{\varphi_2-\varphi_1}{2}\right) = 2A^2\cos^2\left(\dfrac{\Delta\varphi}{2}\right)$. (3)

Note that the total intensity depends critically on the phase difference $\Delta\varphi$ between the two waves. If $\Delta\varphi$ is zero or an integral multiple of $2\pi$, then the waves are in phase and $I_T$ is twice as large as the sum $I_1+I_2$ (constructive interference). If $\Delta\varphi =(2n+1)\pi$, where $n = 0, 1, 2, \dots$, then the waves are 180 degrees out of phase and $I_T=0$ (destructive interference).

A sinusoidal traveling wave, propagating in the $x$-direction and produced at $x_0$ with phase $\varphi_0$, has the form

$Y = A \cos\left(\omega t - k(x-x_0)+\varphi_0 \right) = A \cos \left(2\pi\left(ft - \dfrac{x-x_0}{\lambda}\right) + \varphi_0\right)$, (4)


$\omega = 2\pi f$
$k = \dfrac{2\pi}{\lambda}$
$v = f\lambda = \dfrac{\omega}{k}$

so that the phase at any point $x$ is $\varphi = \varphi_0 - k(x-x_0)$ If two such waves interfere, $\Delta\varphi$ at the point of interference will depend on the distance traveled by each wave.

For example, suppose two such waves with identical $\omega$ and $\varphi_0$ are produced on the $x$-axis at $x_1$ and $x_2$. Then, at a distant point $x$, the phase difference will be

$\Delta \varphi = k(x_2-x_1) = 2\pi\left(\dfrac{x_2-x_1}{\lambda}\right)$ (6)

and the intensity of the total wave at $x$ will be

$I_T = 2A^2\cos^2\left(\pi\left(\dfrac{x_2-x_1}{\lambda}\right)\right)$. (7)

This analysis applies equally to transverse waves (microwaves and light) and to longitudinal waves (sound). Note that the two waves must have identical frequency and a fixed phase difference at their point of production.

Interference of waves from two parallel sources

One of the simpler ways to observe interference is to consider the strength of the sound wave produced by two parallel audio speakers operated at the same intensity and frequency. An experimental arrangement is shown in Fig. 1.

Since the two identical speakers are driven by the same audio oscillator, the frequency of the waves produced by each will be identical. The phase relationship between the two waves will be fixed as long as the position between the two speakers is held fixed. The microphone will then detect the sum of the two audio signals produced by the two speakers. We will consider the following three cases shown in Fig. 1:

  • Case 1, the speakers side by side;
  • Case 2, the two speakers separated by one-half wavelength; and
  • Case 3, the two speakers separated by one wavelength.

In each case, the waves from each of the speakers is drawn, and the resultant wave is shown below them.

Case 1: Waves in phase

Case 2: Waves out of phase by 180 degrees

Case 3: Waves out of phase by 360 degrees
Figure 1: Addition of waves from two identical sources

It is apparent from Fig. 1 and from Eq. (7), that the signal at the microphone is a maximum (the waves reinforce) when the path lengths traversed by the two component waves differ by some multiple of the wavelength (including zero), i.e. when

$x_2-x_1 = n\lambda\; (\mathrm{for\;} n = \mathrm{\;0,1,2,\dots})$. (8)

A minimum of the signal occurs when the path lengths differ by an odd multiple of the half-wavelength, corresponding to a phase difference between the two waves of 180 degrees.

Interference with a Michelson interferometer

A very well-known device for observing interference between light waves is the Michelson Interferometer (shown in Fig. 2).

Figure 2: Michelson interferometer (rays offset for clarity)

Its operation is as follows: A wave coming from the source impinges on the half-reflecting mirror. Half is reflected to mirror M-1, while the remaining half continues to mirror M-2. These two mirrors in turn reflect the wave back to the half-reflecting mirror. (The rays are shown displaced so you can trace them). The wave reflected from mirror M-2 is now reflected by the half-reflecting mirror, while that from M-1 passes through the half-reflecting mirror, also to the detector. Thus, the original beam was split into two equal parts, traveled different paths, and then recombined. The condition for interference is clearly once again the same, i.e., one gets constructive interference whenever the path differences are integer multiples of the wavelength. The path difference is changed by displacing a mirror along the light path. (Note that when a mirror is displaced the path length is changed by twice the displacement of the mirror.)

The optical Michelson interferometer used in this experiment allows one to detect changes in path length on the order of 0.0001 cm, and is thus a very sensitive instrument.

Experimental procedure

NOTE: There are three experiment stations; they may be completed in any order. Note that each room has only one speed of light apparatus, so complete that (relatively quick) experiment as soon as you notice that the station is available.

Getting started

Lab notebook template

One member of the group should click on the link below to start your group lab notebook. (You may be asked to log into your UChicago Google account if you are not already logged in.) Make sure to share the document with everyone in the group (click the “Share” button in the top right corner of the screen) so each member has access to the notebook after you leave lab.

Experiment 1: The speed of light

NOTE: This experiment does not use interference directly, but we include it here because the result will be useful for the microwave interference section below.

The method we use for measuring the speed of light is illustrated in Fig 3.

  • A small pulsed laser diode flashes periodically.
  • This pulse strikes the beam splitter, sending some of the light to a nearby mirror and some light to a far away mirror.
  • The light reflected off the near mirror passes directly into a light detector (photodiode).
  • The light reflected off the distant mirror returns to the beam splitter where it is then reflected into the photodiode.

Thus, for each light pulse emitted, there will be two pulses detected: the direct beam off the near mirror and the reflected beam off the far mirror.

Figure 3: Speed of light apparatus

Use this apparatus to determine the speed of light.

If you do not see two pulses on the oscilloscope, you may need to adjust one (or more of the following):

  • the alignment of the direct beam through the beam splitter into the detector;
  • the alignment of the pulse onto the far mirror;
  • the alignment of the reflected beam back onto the beam splitter;
  • the time or voltage scale on the oscilloscope; or
  • the triggering on the scope (which can be set to “external” using the square wave direct output from the pulsed laser).

Experiment 2: Interference of sound waves

We provide a Google Colab notebook that you can use to plot and fit your data in the following section (if you need to).

Interference of coherent sound waves

Set up the audio interference experiment shown in Fig 3. Use an audio frequency of about 5 kHz. (Keep unnecessary reflecting surfaces away.) Be sure that both speakers are driven by the same function generator output.

Figure 3a: Two parallel-source interference apparatus
Figure 3b: Physical wiring for parallel-source interference. Note that both speaker cables (outlined in blue and green) connect to the same function generator input.

Change the positions of the two speakers and the sound level meter (microphone) while looking for sound maxima and minima. Consider the following:

  • What causes the response on the sound level meter to reach a maximum? What causes it to reach a minimum?
  • What happens if you reverse the connections to one of your speakers? Why?

Determine how to use this setup (along with the principles of interference) to calculate the velocity of sound (with uncertainties). How does this value compare to the known speed of sound in air?


Now, instead of driving two speakers with an identical source (e.g. creating coherent waves), we will drive two speakers with different sources (e.g. creating incoherent waves).

In this case, the phase difference between the two waves will change continuously. The resulting sum will cycle from maximum amplitude (Case 1 as illustrated in Fig. 1) to minimum amplitude (Case 2) to maximum amplitude (Case 3), etc. This slow variation in the amplitude of the sum of the two frequencies is called beating. This variation in amplitude manifests as a volume which varies periodically with a frequency that is equal to the difference between the two frequencies.

The production of beats is shown in Fig. 4.

Figure 4: Beats formed by the addition of waves with two different frequencies

For this part, you will use two function generators, each outputting a sine wave. The output of each will have a “tee” on it so that you can send some of the signal to one side of a pair of headphones and some of the signal to an oscilloscope.

The “beat box” provided takes the two input signals and sends one signal to the left ear of each headphone and one signal to the right ear of each headphone. In this way, you can hold the headphones up to your ear either individually (to hear one signal at a time) or together (to hear both).

How to connect the function generators to the beat box and the oscilloscope. You'll need a total of 5 BNC cables for this, two for each signal and one for a TTL triggering signal.

Driving one ear

Choose frequencies of about 200 Hz and adjust the loudness to be close to the same for both headphones and at a comfortable listening level. Keep the loudness low enough to avoid distortion in the headphones. Initially place both left and right phones close to one of your ears. Adjust one of the frequencies until beats are clearly audible.

At the same time, look for the signals on the oscilloscope. You should arrange things so that you can see one sine wave on Channel 1, the other sine wave on Channel 2 and the sum of the two channels using the “Math” button. As you bring the two frequencies close together, do you observe beating on the scope?

With both phones at one ear, the mechanism for hearing beats is easily understood: while the two waves are in phase, the air pressure driving the ear drum is larger and the sound is perceived as louder. While out of phase, the pressure is less and is perceived as quieter.

Driving two ears

What happens if we put on the headphones and drive the two ears at different frequencies? In this mode, each ear drum is presented with a single frequency and should, therefore, not experience beats.

Without changing the frequencies used earlier, place one phone on each ear and listen for beats. 

  • In your notebook, describe carefully what you hear.

Repeat the above exercise using frequencies of about 500 Hz and 1000 Hz.

  • Can you hear beats at these higher frequencies? 

The physiology involved here may be summarized as follows: Each eardrum is forced to vibrate by the small pressure changes caused by the vibration of its headphone. The eardrum is mechanically coupled through small bones to another membrane at the entrance to the cochlea, a fluid-filled spiral tube. Inside the cochlea is a membrane containing hair cells which are set in motion by the vibration of the fluid. The hair cells closest to the cochlear entrance are sensitive to high frequencies, while those farther from the entrance are sensitive to lower frequencies. The hair cells convert the mechanical vibrations to electrical signals. The signals from the left and right ears are mixed in each of two sets of neurons in the brain stem. One set of neurons is sensitive to high frequencies (typically kHz.) and detects intensity. At high frequencies, the sound shadow cast by the head gives rise to differences in intensity from left to right and is used to judge direction of the sound source. The other set of neurons, sensitive to lower frequencies (typically 200 Hz.), detects phase or time differences to judge the direction of the sound source. The detection of beats with different frequencies sent to the left and right ears depends on the functioning of the lower frequency, phase-sensing neurons.

It has been observed that some people can hear beats in the two-ear mode while others cannot. According to the above model, the ability to hear beats in this mode should be better at low frequencies.

Is it so for you?

Identifying unknown frequencies

Beats are often used as a way to identify an unknown frequency (or to tune a source to a known frequency). As an example, try using the above beats techniques to identify the frequencies of the following:

  • the tuning fork at your station;
  • the mystery frequency available by attaching one channel of your scope (or one channel of your beat box) to the BNC output on your lab bench.

A tuning fork makes the purest tone if struck against a firm (but not too hard) surface. The sole of your shoe is actually a better surface than the edge of the table.

Experiment 3: Interference of light waves

We provide a Google Colab notebook that you can use to plot and fit your data in the following section (if you need to).

Standing microwaves

Set up the microwave apparatus as shown in Fig. 6. This configuration is equivalent to the standing sound waves in a tube using the plunger as a reflector that you saw the “Wave Motion and Sound” experiment last week. The transmitter horn also acts as a reflector and therefore standing waves will be set up between the transmitter and reflector. The receiver will sample the standing wave energy, i.e., just as your microphone sampled the reflected energy when you were studying the sound waves.

Figure 6: Standing wave interference with microwaves

Consider the following:

  • While moving the detector and emitter around, rotate the reflector to demonstrate whether reflection is mirror-like (the angle of reflection is equal to the angle of incidence) or diffuse (a single ray of light in is scattered in many directions).
  • Now place the detector and emitter side-by-side. Slide the reflector away from the transmitter slowly. Does the intensity of the reflected light (as detected by the receiver) change as a function of position? Why?

Since microwaves are electromagnetic radiation traveling at the speed of light, they obey the relation $c=\lambda f$, where c is the speed of light, $\lambda$ is the wavelength, and $f$ is the frequency of the microwaves.

Use the wavelength (determined here) and the speed of light (determined in Experiment 1) to calculate the microwave frequency and compare it to the frequency given on the microwave transmitter: 10.5 GHz.

Microwave interference using a Michelson interferometer

Set up a microwave Michelson interferometer as shown in Fig. 7. The board with one white side is partially reflective to microwaves and serves as the half-reflecting surface. (Try to place your emitter and receiver both close to the beam splitter.)

  • Move the reflector through several maxima and determine the wavelength.
  • Compare this wavelength with the one you determined for the microwave standing apparatus.
  • Calculate the microwave frequency and compare with what you determined above using standing waves.
Figure 7: Microwave Michelson interferometer.

Optical Michelson interferometer

Here we use an optical Michelson interferometer illuminated with mercury green light. Since the typical wavelengths of visible light is very small (on the order of 0.00005 cm), the adjustment of a light Michelson Interferometer requires some care.

Figure 6: Michelson interferometer geometry

Look at the output of the interferometer. If a strong “bull’s eye” pattern is visible, no adjustment is needed. If not, adjust the mirrors until the multiple images of a pin coincide. Minor adjustment about this point should give you fringes. Further adjust the mirrors until you observe the ring pattern, centered on the field of view. The best adjustment is achieved when the diameters of the fringes do not change as you move your head from side to side.

To take data, we will very carefully move one mirror through a distance corresponding to many wavelengths. As we do so, we will observe rings slowly growing outward and appearing from the center or shrinking and disappearing. (Each time a ring appears/disappears, we have changed the optical path length by one wavelength.) By counting the total number of fringes which appear or disappear for a given mirror displacement, we can determine the wavelength of the light.

Since the distance required to make a single fringe appear or disappear is so small, we will instead count 100 fringes. Take data as follows:

  • Start with the micrometer screw on a clear tick mark and record this starting position.
  • Next, turn the screw while counting the number of bright green fringes which grow from, or collapse into, the center of the interference pattern.
  • Continue turning the micrometer screw until you have counted 100 fringes. Record the ending position of the micrometer, including an estimate of fractional divisions on the micrometer.
Question 12:  How many wavelengths have you moved the mirror while counting 100 fringes?

From Fig. 6, it is apparent that

$\theta \approx \tan\theta = \dfrac{d}{L} = \dfrac{x}{r}$. (9)

For your interferometer, the relevant parameters are as follows:

  • $d = 0.00262 \pm 0.00003 \mathrm{ cm/division}$,
  • $r = 0.2377\pm 0.0003 \mathrm{ cm}$, and
  • $L = 6.005 \pm 0.005 \mathrm{ cm}$.

Use these values in Eq. (9) to show that one division on the micrometer corresponds to a one micron ($1~\mu\textrm{m} = 10^{-6}\textrm{ m}$) displacement of the movable mirror.

Question 13: Calculate $\lambda$. Estimate the uncertainty in your measurement.>
Question 14: Within experimental uncertainties, is your result consistent with the accepted value of 546.1 nm ($1\textrm{ nm} = 10^{-9}\textrm{ m}$)?

Submit your lab notebook

Make sure to submit your lab notebook by the end of the period. Download a copy of your notebook in PDF format and upload it to the appropriate spot on Canvas. Only one member of the group needs to submit to Canvas, but make sure everyone's name is on the document!

When you're finished, don't forget to log out of both Google and Canvas, and to close all browser windows before leaving!

Post-lab assignment

Write your conclusion in a new document (not your lab notebook) and submit that as a PDF to the appropriate assignment on Canvas when you are done. You should write your conclusion by yourself (not in a group), though you are welcome to talk to your group mates to ask questions or to discuss.

The conclusion is your interpretation and discussion of your data. In general, you should ask yourself the following questions as you write (though not every question will be appropriate to every experiment):

  • What do your data tell you?
  • How do your data match the model (or models) you were comparing against, or to your expectations in general? (Sometimes this means using the $t^{\prime}$ test, but other times it means making qualitative comparisons.)
  • Were you able to estimate uncertainties well, or do you see room to make changes or improvements in the technique?
  • Do your results lead to new questions?
  • Can you think of other ways to extend or improve the experiment?

In about one or two paragraphs, draw conclusions from the data you collected today. Address both the qualitative and quantitative aspects of the experiment and feel free to use plots, tables or anything else from your notebook to support your words. Don't include throw-away statements like “Looks good” or “Agrees pretty well”; instead, try to be precise.

REMINDER: Your post-lab assignment is due 24 hours after your lab concludes. Submit a single PDF on Canvas.