Experiment 2:Adding Sine Waves of Different Frequency Introduction Two sine wave of the same amplitude and phase can be added mathematically using the formula: sin(2 t)+sin(2mv2t)=2cosa(v -v2 )tsin a(v +v2 )t If v and v,are greatly different,then the wave described by the right side of this equation is even more difficult to visualize than the wave described by the left side. However of v and v,are nearly the same,the cosine term will have a much lower frequency than the sine term.Then the wave can be visualized as a sine wave of frequency,with the amplitude of this sine wave varying slowly with a frequency of(v -v,)/2. If this were a sound wave,you would hear what is called beats,the relatively slow pulsing of the tone as the amplitude rises and falls. Procedure NOTE: a.This procedure assumes you are using a dual trace scope with an external trigger. b.A speaker output,in addition to the oscilloscope,is a valuable addition to this lab 1.Hook up the Fourier Synthesizer to your oscilloscope as described in the SETUP section. Connect one of the oscilloscope inputs to the 10k Output connector of the 9th harmonic. Connect the other oscilloscope input to the 10k Output connector of the summed waveform. 2.Switch the 8th harmonic into the summing amplifier.Make sure all the other waveforms are switched out.Examine the 8th and 9th hammonics at the same time.Adjust their amplitudes and phases to the same values. 3.Now add the two waves by switching the 9th harmonic into the summing amplifier(be sure to flip the 0-180 switch at the same time to offset the 180"phase shift caused by the summing amplifier).Describe the resulting waveform.Do you see beats,If so,what is the frequency of the beats?What is the frequency of the modulated wave? 4.Describe what happens as you vary the amplitude or phase ofeither harmonic. 5.Repeat the above steps using different combinations of harmonics,such as the 7th and the 8th, the 7th and the 9th,the,Ist and the 2nd,the 2nd and the 9th.Try any combinations that you think might be interesting.In each case,describe your results.Is the resulting waveform periodic?If so,what is the period?Do beats occur? 6.From your observations,what generalizations can you make about adding sine waves of different frequencies?Under what conditions do you expect beats?2 Experiment 2: Adding Sine Waves of Different Frequency Introduction Two sine wave of the same amplitude and phase can be added mathematically using the formula: sin( 2 v t) sin( 2 v t) 2cos (v v )tsin (v v )t  1 +  2 =  1 − 2  1 + 2 If v 1 and v 2 are greatly different, then the wave described by the right side of this equation is even more difficult to visualize than the wave described by the left side. However of v 1 and v 2 are nearly the same, the cosine term will have a much lower frequency than the sine term. Then the wave can be visualized as a sine wave of frequency, with the amplitude of this sine wave varying slowly with a frequency of (v 1 -v 2 )/2. If this were a sound wave, you would hear what is called beats, the relatively slow pulsing of the tone as the amplitude rises and falls. Procedure NOTE: a. This procedure assumes you are using a dual trace scope with an external trigger. b. A speaker output, in addition to the oscilloscope, is a valuable addition to this lab. 1. Hook up the Fourier Synthesizer to your oscilloscope as described in the SETUP section. Connect one of the oscilloscope inputs to the 10kΩ Output connector of the 9th harmonic. Connect the other oscilloscope input to the 10kΩ Output connector of the summed waveform. 2. Switch the 8th harmonic into the summing amplifier. Make sure all the other waveforms are switched out. Examine the 8th and 9th harmonics at the same time. Adjust their amplitudes and phases to the same values. 3. Now add the two waves by switching the 9th harmonic into the summing amplifier (be sure to flip the 0-180 switch at the same time to offset the 180" phase shift caused by the summing amplifier). Describe the resulting waveform. Do you see beats, If so, what is the frequency of the beats? What is the frequency of the modulated wave? 4. Describe what happens as you vary the amplitude or phase of either harmonic. 5. Repeat the above steps using different combinations of harmonics, such as the 7th and the 8th, the 7th and the 9th, the, 1st and the 2nd, the 2nd and the 9th. Try any combinations that you think might be interesting. In each case, describe your results. Is the resulting waveform periodic? If so, what is the period? Do beats occur? 6. From your observations, what generalizations can you make about adding sine waves of different frequencies? Under what conditions do you expect beats?