The frequency spectrumThe frequency spectrum

ObjectivesObjectives• Investigate and interpret graphical representations

of sound waves, including:

o waveform graphs

o frequency spectrum graphs

o spectrograms.

• Investigate and analyze characteristics of waves: frequency and amplitude.

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?

AssessmentAssessment

A. The A curve would be taller.

B. The A curve would be shorter.

C. The crests of the A curve would be closer together.

D. The crests of the A curve would be farther apart.

2. One of these three graphs shows a sound that contains two different frequencies.

a. 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?

AssessmentAssessment

3. For which of the following would a spectrogram be able to represent different parts of sound?

AssessmentAssessment

A. speech

B. music

C. bird songs

D. all of the above

AssessmentAssessment4. At which frequency listed below

is the sound represented on this spectrogram the loudest?

A. 500 Hz

B. 1000 Hz

C. 3000 Hz

D. 4000 Hz

Physics termsPhysics terms

• microphone

• frequency spectrum

• Fourier’s theorem

• spectrogram

Sound waves are compression waves in air that cannot be seen.

Several different kinds of graphs are used to help us visualize sound waves.

Visualizing sound wavesVisualizing sound waves

A waveform graph describes how pressure changes over time.Notice the “zoomed-in” time scale.

Waveform graphsWaveform graphs

This graph shows a single frequency of 417 Hz. (12.5 cycles in 0.03 seconds: the musical note G-sharp)

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.

Waveform graphsWaveform graphs

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

“Real” sound“Real” sound

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

Interpreting a sound trackInterpreting a sound track

Interpreting a sound trackInterpreting 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.

In Investigation 16C you will explore different graphical representations of sounds.

InvestigationInvestigation

Click on the simulation on page 453.

1. The simulation shows a waveform graph.

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

• Set a frequency of 300 Hz and adjust the volume.

• Set the time axis to

display 0.02 s.

2. Add a 400 Hz and a 450 Hz sound. Listen to the frequencies separately and together and observe the wave form.

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

Adjust the volume on ONE of the frequencies. Can you hear the changing frequency separately?

Investigation: Part 1Investigation: Part 1

3. 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.

Part 1: Multi-frequency sound

4. Starting with 300 Hz, use three frequencies in the ratios 1:3:5 to create the best approximation to a square wave.

Investigation: Part 1Investigation: Part 1

Answer the questions in Part 1 of your student assignment.

Part 1: Multi-frequency sound

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.

Fourier’s theoremFourier’s theorem

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

Fourier’s theorem: an example Fourier’s theorem: an example

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!

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

Fourier’s theorem: an example Fourier’s theorem: an example

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

Spectrum of a square waveSpectrum of a square wave

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.

Real spectraReal spectra

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

Multi-frequency soundMulti-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.

Multi-frequency soundMulti-frequency sound

Multi-frequency soundMulti-frequency sound

The waveform graph matches the in-and-out oscillation of your eardrum.

This waveform graph shows pressure variations in the 3-frequency sound from the investigation.

Is it easy to deduce the original frequencies from the waveform?

Multi-frequency soundMulti-frequency sound

The waveform graph matches the in-and-out oscillation of your eardrum.

This waveform graph shows pressure variations in the 3-frequency sound from the investigation.

This waveform graph shows pressure variations in the 3-frequency sound from the investigation.

Is it easy to deduce the original frequencies from the waveform?

Multi-frequency soundMulti-frequency sound

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.

1. Use the spectrogram tool to capture and display your voice.

Investigation: Part 2Investigation: Part 2

Modulate your voice and watch how the frequency and amplitude vary.

Part 2: Real-time sound analysis

2. Repeat for various musical and non-musical sounds.

Investigation: Part 2Investigation: Part 2

Click the speaker symbols at the bottom of the investigation page to generate the various sounds shown here.

Part 2: Real-time sound analysis

a. What characteristics make musical sounds different from other sounds?

a. Describe how the spectrogram represents the three variables of time, frequency, and amplitude.

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

c. 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?

d. Propose an explanation for how sound carries the information in words and music..

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

A spectrogram depicts both frequency and loudness over time.

Spectrogram chartsSpectrogram charts

•Frequency is plotted vertically.

•Loudness is represented by color

•Time is plotted on the x-axis.

A spectrogram depicts both frequency and loudness over time.

This spectrogram shows:

•500 Hz tone that is soft, gets louder, and then soft again

Spectrogram chartsSpectrogram 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)

Spectrogram chartsSpectrogram 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)

Spectrogram chartsSpectrogram charts

Interpreting spectrogram chartsInterpreting spectrogram chartsThis 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?

Interpreting spectrogram chartsInterpreting spectrogram chartsThis 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 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”.

Interpreting spectrogram chartsInterpreting spectrogram charts

Digital sound recordingDigital sound recordingAs 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?

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).

Digital sound recordingDigital sound recording

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

Digital sound recordingDigital sound recording

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.

PlaybackPlaybackTo play back the recording, the numbers are read by a laser and converted back into electrical signals by a digital to analog converter.

AssessmentAssessment

A. The A curve would be taller.

B. The A curve would be shorter.

C. The crests of the A curve would be closer together.

D. The crests of the A curve would be farther apart.

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?

AssessmentAssessment

A. The A curve would be taller.

B. The A curve would be shorter.

C. The crests of the A curve would be closer together.

D. The crests of the A curve would be farther apart.

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?

2. One of these three graphs shows a sound that contains two different frequencies.

AssessmentAssessment

a. 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?

2. One of these three graphs shows a sound that contains two different frequencies.

AssessmentAssessment

a. 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

c. What is the higher frequency in the sound? 80 Hz: It has two peaks for every one period of the lower frequency.

3. For which of the following would a spectrogram be able to represent different parts of sound?

AssessmentAssessment

A. speech

B. music

C. bird songs

D. all of the above

3. For which of the following would a spectrogram be able to represent different parts of sound?

AssessmentAssessment

A. speech

B. music

C. bird songs

D. all of the above

AssessmentAssessment

A. 500 Hz

B. 1000 Hz

C. 3000 Hz

D. 4000 Hz

4. At which frequency listed below is the sound represented on this spectrogram the loudest?

AssessmentAssessment

A. 500 Hz

B. 1000 Hz

C. 3000 Hz

D. 4000 Hz

4. At which frequency listed below is the sound represented on this spectrogram the loudest?