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**The frequency spectrum**

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Objectives Investigate and interpret graphical representations of sound waves, including: waveform graphs frequency spectrum graphs spectrograms. Investigate and analyze characteristics of waves: frequency and amplitude.

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Assessment 1. G-sharp has a frequency of 417 Hz and the musical note A has a frequency of 440 Hz. If the two notes are graphed on the same waveform graph, how will the two curves differ? The A curve would be taller. The A curve would be shorter. The crests of the A curve would be closer together. The crests of the A curve would be farther apart.

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Assessment One of these three graphs shows a sound that contains two different frequencies. Which graph is it and how do you know? b. What is the lower frequency in this sound? c. What is the higher frequency in the sound?

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Assessment For which of the following would a spectrogram be able to represent different parts of sound? speech music bird songs all of the above

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Assessment At which frequency listed below is the sound represented on this spectrogram the loudest? A Hz B Hz C Hz D Hz

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**Physics terms microphone frequency spectrum Fourier’s theorem**

spectrogram

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**Visualizing sound waves**

Sound waves are compression waves in air that cannot be seen. Several different kinds of graphs are used to help us visualize sound waves.

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Waveform graphs A waveform graph describes how pressure changes over time. Notice the “zoomed-in” time scale. This graph shows a single frequency of 417 Hz. (12.5 cycles in 0.03 seconds: the musical note G-sharp)

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Waveform graphs When multiple frequencies are present, the wave oscillates in a more complicated pattern. This waveform graph shows the addition of 300 Hz, 400 Hz and 450 Hz waves of the same amplitude.

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“Real” sound Real sounds contain thousands of different frequencies, all with different and changing phases and amplitudes.

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**Interpreting a sound track**

A sound track is a waveform graph that displays complex sounds, such as music.

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**Interpreting a sound track**

A sound track is a waveform graph that displays complex sounds, such as music. The graph shows pressure as a function of time. To see individual oscillations, you have to zoom in on the time axis.

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**Click on the simulation on page 453.**

Investigation In Investigation 16C you will explore different graphical representations of sounds. Click on the simulation on page 453.

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**Investigation: Part 1 Part 1: Multi-frequency sound**

The simulation shows a waveform graph. Set a frequency of 300 Hz and adjust the volume. Set the time axis to display 0.02 s.

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**Investigation: Part 1 Part 1: Multi-frequency sound**

Add a 400 Hz and a 450 Hz sound. Listen to the frequencies separately and together and observe the wave form. Adjust the volume on ONE of the frequencies. Can you hear the changing frequency separately?

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**Investigation: Part 1 Part 1: Multi-frequency sound**

Switch the graph to display a spectrum—a bar chart that shows the frequencies of the sound. Set the same 3 frequencies as before and observe the spectrum as you change the frequency and volume.

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**Investigation: Part 1 Part 1: Multi-frequency sound**

Starting with 300 Hz, use three frequencies in the ratios 1:3:5 to create the best approximation to a square wave. Answer the questions in Part 1 of your student assignment.

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Fourier’s theorem Fourier’s theorem states that any repetitive wave can be reproduced exactly by combining simple sine waves of different frequencies and amplitudes. Fourier’s theorem provides a mathematical formula for determining this combination of waves, which is known as a Fourier series.

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**Fourier’s theorem: an example**

How can this 100 Hz square wave be reproduced from a combination of sine waves?

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**Fourier’s theorem: an example**

How can this 100 Hz square wave be reproduced from a combination of sine waves? The first four sine waves in the Fourier series (100 Hz, 300 Hz, 500 Hz, and 700 Hz) add up to a fairly good approximation. Adding more waves will make the approximation even better!

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**Spectrum of a square wave**

This bar chart shows the relative amplitudes of the first four frequencies in the series.

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**Real spectra Everyday sounds are more complicated than square waves.**

They contain thousands of different frequencies, each with its own amplitude and phase. This frequency spectrum is from an acoustic guitar playing the note E.

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**Multi-frequency sound**

The ear can listen to about 15,000 different frequencies simultaneously!

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**Multi-frequency sound**

The ear can listen to about 15,000 different frequencies simultaneously! The brain assembles a sonic “picture” from the changing patterns of rising and falling amplitudes at many thousands of frequencies.

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**Multi-frequency sound**

This waveform graph shows pressure variations in the 3-frequency sound from the investigation. The waveform graph matches the in-and-out oscillation of your eardrum.

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**Multi-frequency sound**

This waveform graph shows pressure variations in the 3-frequency sound from the investigation. The waveform graph matches the in-and-out oscillation of your eardrum. Is it easy to deduce the original frequencies from the waveform?

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**Multi-frequency sound**

This waveform graph shows pressure variations in the 3-frequency sound from the investigation. The waveform graph matches the in-and-out oscillation of your eardrum. No. The information is here, but it’s not easy to understand. There is another type of graph that lets you see frequency AND amplitude as a function of time. Is it easy to deduce the original frequencies from the waveform?

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**Investigation: Part 2 Part 2: Real-time sound analysis**

Use the spectrogram tool to capture and display your voice. Modulate your voice and watch how the frequency and amplitude vary. Note to teacher: the spectrogram tool is available as a free download for pc computers.

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**Investigation: Part 2 Part 2: Real-time sound analysis**

Repeat for various musical and non-musical sounds. Click the speaker symbols at the bottom of the investigation page to generate the various sounds shown here.

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**Investigation: Part 2 Questions for Part 2**

What characteristics make musical sounds different from other sounds? Describe how the spectrogram represents the three variables of time, frequency, and amplitude.

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**Investigation: Part 2 Questions for Part 2**

Interpret and compare the charts you generated for the frequencies in a voice to the frequencies you combined in Part 1. Are there more or fewer frequencies in the voice? Propose an explanation for how sound carries the information in words and music..

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Spectrogram charts A spectrogram depicts both frequency and loudness over time. Frequency is plotted vertically. Loudness is represented by color Time is plotted on the x-axis.

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Spectrogram charts A spectrogram depicts both frequency and loudness over time. This spectrogram shows: 500 Hz tone that is soft, gets louder, and then soft again

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Spectrogram charts A spectrogram depicts both frequency and loudness over time. This spectrogram shows: 500 Hz tone that is soft, gets louder, and then soft again soft 300 Hz tone (3 to 5 s)

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Spectrogram charts A spectrogram depicts both frequency and loudness over time. This spectrogram shows: 500 Hz tone that is soft, gets louder, and then soft again soft 300 Hz tone (3 to 5 s) loud 200 Hz tone (1 to 3 s)

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**Interpreting spectrogram charts**

This spectrogram is of a human voice. How long does the sound last? Which is louder in this event, the low frequencies or the high frequencies? How do you know?

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**Interpreting spectrogram charts**

This spectrogram is of a human voice. How long does the sound last? about half a second. Which is louder in this event, the low frequencies or the high frequencies? How do you know? The low frequencies are red, indicating that they are louder. Can you infer from the graph if the speaker is a man or a young child?

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**Interpreting spectrogram charts**

This spectrogram is of a human voice. How long does the sound last? about half a second. Which is louder in this event, the low frequencies or the high frequencies? How do you know? The low frequencies are red, indicating that they are louder. Can you infer from the graph if the speaker is a man or a young child? This is a low male voice saying the word “hello”.

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**Digital sound recording**

As the spectrograms show, sound is highly complex and changes rapidly. How do sound engineers capture the sounds of music and voices? And how do we access these stored sounds to replay them later?

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**Digital sound recording**

To record sound, a microphone converts pressure variations in the air into electrical signals. In CD-quality recording the signal is sampled 44,100 times a second by an analog to digital converter (ADC). The sound pressure wave vibrates the thin diaphragm inside the microphone, much as the ear drum is vibrated. This produces a changing voltage (the analog signal) in the microphone’s circuitry. The ADC transforms this voltage reading into a digital string of 0’s and 1’s.

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**Digital sound recording**

The resulting string of numbers is recorded as data on a CD or other digital formats such as MP3.

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Playback To play back the recording, the numbers are read by a laser and converted back into electrical signals by a digital to analog converter. The electrical signal (a time-varying voltage) is amplified until it is strong enough to vibrate the coil in a speaker and reproduce the sound.

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Assessment 1. G-sharp has a frequency of 417 Hz and the musical note A has a frequency of 440 Hz. If the two notes are graphed on the same waveform graph, how will the two curves differ? The A curve would be taller. The A curve would be shorter. The crests of the A curve would be closer together. The crests of the A curve would be farther apart.

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Assessment 1. G-sharp has a frequency of 417 Hz and the musical note A has a frequency of 440 Hz. If the two notes are graphed on the same waveform graph, how will the two curves differ? The A curve would be taller. The A curve would be shorter. The crests of the A curve would be closer together. The crests of the A curve would be farther apart.

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Assessment One of these three graphs shows a sound that contains two different frequencies. Which graph is it and how do you know? b. What is the lower frequency in this sound? What is the higher frequency in the sound?

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Assessment One of these three graphs shows a sound that contains two different frequencies. Which graph is it and how do you know? Graph C is more complex. b. What is the lower frequency in this sound? 40 Hz What is the higher frequency in the sound? 80 Hz: It has two peaks for every one period of the lower frequency.

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Assessment For which of the following would a spectrogram be able to represent different parts of sound? speech music bird songs all of the above

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Assessment For which of the following would a spectrogram be able to represent different parts of sound? speech music bird songs all of the above

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Assessment At which frequency listed below is the sound represented on this spectrogram the loudest? 500 Hz 1000 Hz 3000 Hz 4000 Hz

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Assessment At which frequency listed below is the sound represented on this spectrogram the loudest? 500 Hz 1000 Hz 3000 Hz 4000 Hz

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