Fourier Analysis Objectives:

The objects of this lab are to 1.) introduce Fourier Analysis , 2.) to investigate sound waves, and 3.) to gain an understanding of the concepts of superposition and interference.

Theory:I. Superposition and Interference

Sound waves, in fact waves in general, are complex. That being said, our study on harmonic motion ( simple harmonic motion, damped harmonic motion and forced, damped harmonic motion ) is appropriate and relevant because most elastic medium that undergo small displacement in response to forcing ( such as air ) exhibit linearity. Waves in such a medium follow the superposition principle, which can be stated as:

If two or more waves travel through the same medium, the resultant wave is the algebraic sum of the individual waves. If a medium can support two waves y1 x , t and y2 x , t , which are solutions to

the wave equation ∂2 y∂ x2−

1v2

∂2 y∂ t 2

, then the medium can support a wave, y x , t = y1 x , t y1 x , t ,which

is the algebraic sum of the two, which is also a solution to the wave equation.

The superposition principle allows us to algebraically add sinusoidal waves which may have different wavelengths, amplitudes, frequencies and may be moving in different directions. The results can be very interesting and complicated. As time progresses, the individual waves appear to pass through one another, unaffected by each other. This phenomenal is known as interference. When two peaks overlap, the two waves reinforce one another and the resultant wave will have a large amplitude at that point. This is known as constructive interference. When a peak and trough overlap, the resultant wave amplitude will be small, it may even be zero. This is known as destructive interference.

Many waves can be modeled as sinusoidal waves. In general y x=A sin 2 x :

where A is the amplitude and is the wavelength. If the wave is moving at a constant velocity, y x ,t =Asin k x− t , where k=2

, =2T , and v=

k .

0 1 2 3 4 5 6

-6

-4

-2

0

2

4

6

x(m)

y(m

)

A λ

Sound waves can also be modeled as sinusoidal waves. They may be modeled as the medium's displacement from and equilibrium position or as a difference in pressure.

s x , t=smax cos k x− tP x , t=P max sin k x−t

The speed of the sound wave depends on the medium, v=elastic property inertail property

= B . The speed on the

sound wave in air is approximated by 343 m/s at “room temperature. A better approximation of the

velocity of sound in air is v=331m/ s T273K .

In this lab you will use a microphone to to record various sounds. The “microphone” uses and electret microphone that has a frequency response covering the range of frequencies that the human ear can hear( 20 Hz to 20,000 Hz ). This microphone will detect differences in pressure and convert those differences into an electric signal. An op-amp circuit amplifies the signal and send it to the British Telecom connector.

The sound signal recorded can be very complicated or it can be rather simple. Consider the two recordings below. The first is of a person saying the word “Hello”. The second is the signal from a tuning fork.

II. Sound Waves and Fourier Analysis

At first it may seem difficult to analyze non-sinusoidal wave forms. If the waveform is periodic, it can be represented as closely as desired by the combination of a sufficiently large number of sinusoidal waves that form a harmonic series. Fourier's theorem suggests that any periodic function can be represented as an algebraic sum of sine and cosine functions called a Fourier Series.

If f x=ao∑k=1

∞ [ak cosk xa bk sin k xa ] on the interval −a≤ x≤a

ao=1

2a∫−aa

f t dt

an=1a∫−a

a

f t cos n ta dtbn=

1a∫−a

a

f t sin n ta dtFor example a sawtooth wave can be represented by a series of cosine waves:

In the lab today you will be using a microphone attached to the Vernier LabPro Data Acquisition System. You will not be required to compute the Fourier coefficients for the waveforms. Instead, you will collect data and graph the change in pressure versus time. From there you will use a Fast Fourier Transform, provided by the Vernier LabPro Software, to find the relative amplitudes of the frequencies present in your sample, which are related to the Fourier coefficients of the Fourier Series.

-6.28 -4.71 -3.14 -1.57 0 1.57 3.14 4.71-0.7500

-0.2500

0.2500

0.7500

1.2500

1.7500

Sawtooth Wave

k=0k=1k=2k=3k=4k=5k=6k=7k=8k=9F(x)

x

F(x)

III. Beats

Many interesting phenomenon can be observed by studying the superposition and interference of sound waves. One of these phenomenon is observed when two sound waves of slightly different frequencies superimpose. The resulting sound produced by the superimposition of the two waves waves is known as “beats”. As shown below, sometimes the two waves are “in phase” and they interfere constructively, sometimes they are out of phase and they interfere destructively. You will investigate this phenomenal in the lab today using tuning forks.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14-20

-10

0

10

20Beats

y1(m)y2(m)

t(s)

y(m

)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14-20

-10

0

10

20Beats

y1+y2

t(s)

y(m

)

Procedure:I. General Instructions for using the Vernier LabPro Data acquisition System

with microphone and LabPro Software

1.) Connect Vernier LabPro Data acquisition System to your computer's USB port.a.) Connect Vernier LabPro to power. (Figure Two).b.) Connect Vernier LabPro to USB port of your computer. (Figure Three).c.) Connect the microphone probe to the Vernier LabPro. (Figure Four).d.) Now a green LED should be lit on the Vernier LabPro. (Figure Five).

Figure One : Vernier LabPro Data Acquisition System with microphone attached. Power cable and USB cables are also attached.

Figure Two : Vernier LabPro Data Acquisition System and Power Cord.

Figure Three : Vernier LabPro Data Acquisition System and Microphone.

Figure Four : Vernier LabPro Data Acquisition System and USB Cord.

2.) Open the Vernier LabPro Software named LoggerPro using the icon or start menu.

3.) Your session should resemble Figure Eight. Add an FFT ( Fast Fourier Transfer ) Graph by using the Insert → Additional Graphs → FFT Graph drop down menu. (See Figure Nine).

4.) Adjust the graphs so you can see both graphs. 5.) Set up a 256 Hz tuning fork on a piece of foam. Only strike the tuning fork with the rubber

mallet provided. Failing to do so will not only damage the tuning fork, but will set up modes in the tuning fork which might, or might not, have a frequency of 256 Hz. Failing to use the rubber mallet provided will give you extremely noisy data.

6.) Take a set of sample data to make sure everything is working properly. While holding the microphone near the 256 Hz. tuning fork, strike the tuning fork with the rubber mallet provided and then click on the green collect button. Your results should resemble Figure Ten. Notice that the FFT Graph displays an amplitude of some magnitude at the 256 Hz frequency. You may also notice two or three extra peaks. What do you suppose the extra peaks represent?

Figure Five : Vernier LabPro Data Acquisition – Green LED shows that system is ready.

Figure Six : Tuning Fork, Foam and Striker.

Figure Seven : Vernier LabPro Logger Pro 3.8 Icon andstart menu.

Figure Eight : Vernier LabPro Logger Pro 3.8 initial screen.

Figure Nine : Vernier LabPro Logger Pro 3.8 adding an FFT graph.

II. A Very Simple Fourier Analysis

1.) Using the general instructions for collecting data, collect a sample of data, using the default settings for 256 Hz tuning fork.

2.) Export the data into a spreadsheet using the File → Export As → Text drop down menu.3.) Open a spreadsheet and import your data that you saved as a text file.4.) Recall:

f t =12ao∑

n=1

∞

[ Ansin 2n t Bn cos 2n t ]

An=2n∗dt ∫0

n∗dt

f t sin 2nt dt

Bn=2n∗dt ∫0

n∗dt

f t cos 2n t dt

Assume an integer number of period which are equal to n*dt.

We can make a very simple approximation.

f t =12ao∑

n=1

∞

[ Ansin 2 f n t Bn cos 2 f n t ]

An≈2n t ∫0

n t

f t sin 2 f n t t

Bn=2n t ∫0

nt

f t cos 2 f n t t

5.) Make an approximation of An for frequencies of 200 Hz – 300 Hz at 10 Hz intervals.

Figure Ten : Vernier LabPro Logger Pro 3.8 Sample data.

6.) Plot An versus f.

III. Fourier Analysis

1.) For each example print out the graph of your data and the FFT graph and export the data as text for each step.

2.) Complete a Fast Fourier Transform (FFT) for your 256 Hz tuning fork.

F 256Hz= _____________ Hz A256Hz= _____________ Hz

3.) Complete a Fast Fourier Transform (FFT) for your first tuning fork of an unknown frequency.

F 1= _____________ Hz A1= _____________ Hz

4.) Complete a Fast Fourier Transform (FFT) for your second tuning fork of an unknown frequency.

F 2= _____________ Hz A2= _____________ Hz

5.) Take tuning forks 1 and 2 and complete an FFT while both are ringing. From your pressure difference versus time graph compute the beat frequency. You may need to change your data collection interval to be about 1.00 seconds. What is the % difference between the computed value and the theoretical value of the beat frequency? In a few words, describe what the FFT recorded. Does this make sense?

F BeatCalculated= _____________ Hz F Beat Theoretical=∣F 1−F 2∣= _____________ Hz difference= _______________________ %

6.) Set the data collection period to be one second. Take an FFT while two of the tuning forks are ringing. Print the FFT.

7.) Take an FFT while two of the tuning forks are ringing AND all the members of your group are trying to talk into the microphone. Print the FFT. Do you notice any difference between this FFT and the one taken in step 8?

8.) Take the two aluminum rods and take a spectrum while you are striking these together. Print the FFT. How does this FFT spectrum compare with any of the others you have taken today?

9.) Set the data collection period at three second, take an FFT while one of the three tuning forks is ringing. Make sure to print a copy of the resultant FFT. Export this data to a text file.

10.) Keep the data collection period at three second. Take an FFT while two tuning forks forks are ringing and then add a third. Print the FFT. Compare this FFT graph with the one taken in step 6. Do you notice any differences?

IV. Modeling Exercise

Open a spreadsheet and import the data collected in step 6. Use the information gathered by the FFT to construct a model of your data. How do they compare?