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Faculty of Science Department of Physics Laboratory Report Cover Page Course Number PCS 125 Course Title Physics and Waves Semester/Year Summer 2022 Instructor Jian Yuan TA Name Shweta Mistry INTRODUCTION The fluctuations in air pressure that make up sound waves are arranged in a certain pattern. These fluctuations in pressure result in your eardrum to vibrate, which your brain then interprets as the sensation of hearing sound. A Microphone diaphragm works in a similar way, recording changes in pressure and converting them into an electrical signal that can subsequently be seen to an oscilloscope. Any two waves that are travelling through the same area of space have the potential to either add up (constructive interference) or cancel each other out (destructive interference). The term beats refer to the rhythmic pulses that may be heard when two sound waves of slightly differing frequencies are played simultaneously. These pulses are the result of the waves continually shifting from being in-phase to being out-of-phase and back again, which causes constructive and destructive interference to occur in sequential fashion. THEORY Starting with two equations: ∆ P 1 ( t ) = Asin ( 2 π f 1 t ) , and ∆ P 2 ( t ) = Asin ( 2 π f 2 t ) , we can use a trigonometric identity to represent the sum of the two functions. The sum of the two functions can be written as ∆ P total ( t ) = Asin ( 2 π f 1 t ) + Asin ( 2 π f 2 t ) , and further simplified to ∆ P total ( t ) = A ( sin ( 2 π f 1 t ) + sin ( 2 π f 2 t ) ) . The following trigonometric identity can be applied: sin a + sin b = 2cos ( a − b 2 ) sin ( a + b 2 ) . Substituting our previous equation in, we get the resulting equation: ∆ P total ( t ) = A 2cos ( 2 π f 1 t − 2 π f 2 t 2 ) sin ( 2 π f 1 t + 2 π f 2 t 2 ) . 1

Simplifying this equation gives us a final equation of f 1 − f 2 πt ( ¿ ) ¿ f 1 + f 2 πt ( ¿ ) ¿ ∆ P total ( t ) = 2 A cos ¿ . This equation represents the sum of two sound wave functions. PROCEDURE A microphone was attached to the computer that would be used to capture data for this experiment. In the first stage of the experiment, a rotating fork was hit with a rubber mallet to make a sound that was recorded for 0.05 seconds to demonstrate the rapid pressure changes. The wave's period was calculated using the captured waveform and Logger Pro software. During the experiment, the average time difference was calculated using Excel while accounting for uncertainty. Lastly, the amplitude of the wave was determined using the same technique as before, and a sine curve was fitted to this data. The data was stored along with a screenshot of the graph. In the second portion of the experiment, tone generator beats were evaluated. Ten seconds of a tone with a frequency of 300 Hz were produced. Observations were made while a second tone of 310 Hz was played simultaneously with the first tone for ten seconds. Similarly, two simultaneous tones were produced, one at 300 Hz and the other at 340 Hz, and any observations were recorded. Using a microphone, the last two tones were recorded in Logger Pro, the period was established, and a sine curve was fit to the data. Similar to parts one and two of this experiment, two rotating forks were hit together, and the sound was recorded and examined for the third and final question. All of the stored data and screenshots were forwarded to both parties for future reference. RESULTS AND CALCULATION 2

Figure 2: The following is a graphical representation of the sound wave that was produced when a 300 Hz and a 340 Hz were played simultaneously during the second part of the experiment. The graph shows a relation of sound pressure (pa) on the y-axis as a function of time (s) on the x-axis. Table 3: Summary table showing the sine function parameters which were found by fitting a sine function on the data in figure 2. A B C D 0.03944 122.0 4.194 2.674 4

Figure 3 : The following is a graphical representation of the sound wave that was produced when two turning forks were struck together to produce beat frequencies. The graph shows a relation of sound pressure (pa) on the y-axis as a function of time (s) on the x-axis. Table 4: Summary table showing the sine function parameters which were found by fitting a sine function on the data in figure 3. A B C D 0.013458 95 3.293 2.676 Figure 4: Calculations for wrap up question 1: Figure 5: Calculations for wrap up question 2: 5

experienced musician who can tune by ear to find the ideal pitch for their instrument. 5. Using active noise control, noise-canceling headphones block out undesirable ambient noise. This contrasts with passive headphones, which, if they suppress ambient sounds at all, employ soundproofing techniques. Using active noise control, noise-canceling headphones block out undesirable ambient noise. This contrasts with passive headphones, which, if they suppress ambient sounds at all, employ soundproofing techniques. Noise-cancelling headphones make it possible to listen to sounds without significantly raising the volume. They increase the ratio of signal to noise. They can improve listening sufficiently to totally counteract the effect of a concurrently occurring distraction. They generate a waveform that is the exact opposite of the ambient sound measured by a microphone. Sound is a pressure wave with compression and rarefaction phases. A noise-canceling speaker creates a sound wave with the same amplitude but inverted phase to the interference process. These waves cancel each other out, a phenomenon known as phase cancellation. The signal is then amplified, and a transducer generates a sound wave directly proportionate to the amplitude of the original waveform. This phenomenon is known as creative destructive interference. This reduces the perceived noise level significantly. (1) REFERENCES 1. “How do noise cancelling headphones work?,” SoundGuys , 04-Feb-2022. [Online]. Available: https://www.soundguys.com/how-noise-cancelling- headphones-work-12380/. [Accessed: 05-Aug-2022]. 7