what relationship does the speed of sound, v, have to the wavelength and frequency of the sound?

Department Learning Objectives

Past the cease of this section, you volition be able to do the post-obit:

• Relate the characteristics of waves to properties of sound waves
• Describe the speed of sound and how it changes in various media
• Chronicle the speed of sound to frequency and wavelength of a sound wave

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Teacher Support

The learning objectives in this department will help your students master the post-obit standards:

• (7) Science concepts. The pupil knows the characteristics and behavior of waves. The student is expected to:
  • (A) examine and describe oscillatory move and wave propagation in diverse types of media;
  • (B) investigate and analyze characteristics of waves, including velocity, frequency, amplitude, and wavelength, and calculate using the relationship between wave speed, frequency, and wavelength;
  • (C) compare characteristics and behaviors of transverse waves, including electromagnetic waves and the electromagnetic spectrum, and characteristics and behaviors of longitudinal waves, including sound waves;
  • (F) describe the role of wave characteristics and behaviors in medical and industrial applications.

In addition, the High Schoolhouse Physics Laboratory Transmission addresses content in this section in the lab titled: Waves, as well equally the following standards:

• (7) Scientific discipline concepts. The student knows the characteristics and behavior of waves. The educatee is expected to:
  • (B) investigate and analyze characteristics of waves, including velocity, frequency, amplitude, and wavelength, and calculate using the relationship between wave speed, frequency, and wavelength.

Department Key Terms

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Teacher Support

[BL] [OL] Review waves and types of waves—mechanical and non-mechanical, transverse and longitudinal, pulse and periodic. Review properties of waves—amplitude, period, frequency, velocity and their inter-relations.

Backdrop of Sound Waves

Audio is a wave. More specifically, sound is defined to be a disturbance of matter that is transmitted from its source outward. A disturbance is anything that is moved from its land of equilibrium. Some sound waves can exist characterized as periodic waves, which means that the atoms that brand up the affair experience unproblematic harmonic motion.

A vibrating cord produces a sound wave every bit illustrated in Figure fourteen.2, Figure 14.three, and Figure 14.4. As the string oscillates back and along, part of the string's free energy goes into compressing and expanding the surrounding air. This creates slightly higher and lower pressures. The higher pressure... regions are compressions, and the low pressure level regions are rarefactions. The force per unit area disturbance moves through the air every bit longitudinal waves with the same frequency as the string. Some of the energy is lost in the form of thermal energy transferred to the air. You may recall from the affiliate on waves that areas of pinch and rarefaction in longitudinal waves (such as sound) are coordinating to crests and troughs in transverse waves.

Figure fourteen.2 A vibrating string moving to the correct compresses the air in front of it and expands the air behind information technology.

Figure xiv.3 Equally the string moves to the left, it creates some other compression and rarefaction as the particles on the right move away from the string.

Figure xiv.four After many vibrations, there is a series of compressions and rarefactions that have been transmitted from the string as a sound wave. The graph shows estimate pressure (P_guess) versus distance x from the source. Gauge pressure is the force per unit area relative to atmospheric pressure; it is positive for pressures above atmospheric pressure level, and negative for pressures below it. For ordinary, everyday sounds, pressures vary merely slightly from average atmospheric pressure.

The amplitude of a sound moving ridge decreases with altitude from its source, because the energy of the moving ridge is spread over a larger and larger surface area. Just some of the energy is also absorbed by objects, such as the eardrum in Figure xiv.5, and some of the free energy is converted to thermal energy in the air. Effigy 14.4 shows a graph of judge pressure versus distance from the vibrating cord. From this figure, you lot can see that the compression of a longitudinal wave is coordinating to the pinnacle of a transverse wave, and the rarefaction of a longitudinal wave is analogous to the trough of a transverse wave. Just as a transverse wave alternates between peaks and troughs, a longitudinal wave alternates between compression and rarefaction.

Effigy xiv.5 Sound wave compressions and rarefactions travel up the ear canal and force the eardrum to vibrate. At that place is a net force on the eardrum, since the audio wave pressures differ from the atmospheric pressure found behind the eardrum. A complicated mechanism converts the vibrations to nerve impulses, which are so interpreted past the brain.

The Speed of Sound

Teacher Support

Teacher Support

[BL] Review the fact that sound is a mechanical wave and requires a medium through which it is transmitted.

[OL] [AL] Ask students if they know the speed of sound and if not, ask them to take a guess. Enquire them why the audio of thunder is heard much later the lightning is seen during storms. This phenomenon is also observed during a brandish of fireworks. Through this word, develop the concept that the speed of sound is finite and measurable and is much slower than that of calorie-free.

The speed of audio varies greatly depending upon the medium information technology is traveling through. The speed of sound in a medium is determined past a combination of the medium's rigidity (or compressibility in gases) and its density. The more rigid (or less compressible) the medium, the faster the speed of sound. The greater the density of a medium, the slower the speed of sound. The speed of audio in air is low, because air is compressible. Because liquids and solids are relatively rigid and very hard to compress, the speed of sound in such media is by and large greater than in gases. Table 14.1 shows the speed of audio in various media. Since temperature affects density, the speed of sound varies with the temperature of the medium through which information technology's traveling to some extent, particularly for gases.

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Teacher Support

Misconception Alert

Students might be confused between rigidity and density and how they affect the speed of sound. The speed of sound is slower in denser media. Solids are denser than gases. Withal, they are besides very rigid, and hence sound travels faster in solids. Stress on the fact that the speed of sound always depends on a combination of these two properties of any medium.

Medium	vw (k/s)
Gases at 0 °C
Air	331
Carbon dioxide	259
Oxygen	316
Helium	965
Hydrogen	1290
Liquids at 20 °C
Ethanol	1160
Mercury	1450
Water, fresh	1480
Sea water	1540
Human tissue	1540
Solids (longitudinal or bulk)
Vulcanized rubber	54
Polyethylene	920
Marble	3810
Drinking glass, Pyrex	5640
Lead	1960
Aluminum	5120
Steel	5960

Table 14.1 Speed of Sound in Various Media

Teacher Support

Instructor Support

[BL] Note that in the table, the speed of sound in very rigid materials such as glass, aluminum, and steel ... is quite high, whereas the speed in safety, which is considerably less rigid, is quite depression.

The Relationship Betwixt the Speed of Sound and the Frequency and Wavelength of a Sound Wave

Effigy xiv.half dozen When fireworks explode in the sky, the light energy is perceived before the sound free energy. Audio travels more slowly than light does. (Dominic Alves, Flickr)

Sound, like all waves, travels at certain speeds through different media and has the properties of frequency and wavelength. Audio travels much slower than light—you can observe this while watching a fireworks display (see Figure 14.6), since the wink of an explosion is seen before its sound is heard.

The human relationship between the speed of audio, its frequency, and wavelength is the same as for all waves:

where 5 is the speed of sound (in units of m/s), f is its frequency (in units of hertz), and $\lambda$ is its wavelength (in units of meters). Recall that wavelength is defined as the distance between adjacent identical parts of a moving ridge. The wavelength of a sound, therefore, is the altitude between adjacent identical parts of a audio wave. Just as the altitude between next crests in a transverse wave is one wavelength, the distance betwixt adjacent compressions in a sound moving ridge is likewise ane wavelength, as shown in Figure 14.7. The frequency of a audio wave is the same as that of the source. For example, a tuning fork vibrating at a given frequency would produce sound waves that oscillate at that aforementioned frequency. The frequency of a sound is the number of waves that pass a point per unit fourth dimension.

Figure xiv.7 A audio moving ridge emanates from a source vibrating at a frequency f, propagates at v, and has a wavelength $\lambda$ .

Teacher Support

Teacher Support

[BL] [OL] [AL] In musical instruments, shorter strings vibrate faster and hence produce sounds at higher pitches. Fret placements on instruments such as guitars, banjos, and mandolins, are mathematically adamant to give the correct interval or change in pitch. When the string is pushed against the fret wire, the string is effectively shortened, irresolute its pitch. Ask students to experiment with strings of different lengths and discover how the pitch changes in each case.

1 of the more important backdrop of sound is that its speed is nearly contained of frequency. If this were not the case, and loftier-frequency sounds traveled faster, for example, then the farther you were from a band in a football game stadium, the more the sound from the low-pitch instruments would lag behind the high-pitch ones. But the music from all instruments arrives in cadence independent of distance, and and then all frequencies must travel at nearly the same speed.

Remember that $v=f\lambda$ , and in a given medium nether fixed temperature and humidity, v is constant. Therefore, the relationship betwixt f and $\lambda$ is inverse: The higher the frequency, the shorter the wavelength of a sound wave.

Instructor Back up

Teacher Support

Instructor Demonstration

Agree a meter stick flat on a desktop, with virtually eighty cm sticking out over the edge of the desk. Brand the meter stick vibrate by pulling the tip downwardly and releasing, while property the meter stick tight to the desktop. While it is vibrating, move the stick back onto the desktop, shortening the function that is sticking out. Students will come across the shortening of the vibrating part of the meter stick, and hear the pitch or number of vibrations go up—an increase in frequency.

The speed of sound tin can alter when sound travels from i medium to another. Yet, the frequency unremarkably remains the same considering information technology is like a driven oscillation and maintains the frequency of the original source. If five changes and f remains the aforementioned, then the wavelength $\lambda$ must change. Since $\mathrm{five}=f\lambda$ , the college the speed of a sound, the greater its wavelength for a given frequency.

Teacher Support

Teacher Support

[AL] Ask students to predict what would happen if the speeds of sound in air varied past frequency.

Virtual Physics

Audio

This simulation lets you meet sound waves. Adapt the frequency or amplitude (volume) and y'all can see and hear how the wave changes. Move the listener around and hear what she hears. Switch to the Two Source Interference tab or the Interference by Reflection tab to experiment with interference and reflection.

Tips For Success

Make sure to have sound enabled and fix to Listener rather than Speaker, or else the audio will not vary equally y'all move the listener around.

PhET Explorations: Sound. This simulation lets you meet sound waves. Adjust the frequency or volume and you can encounter and hear how the wave changes. Movement the listener effectually and hear what she hears.

In the first tab, Listen to a Single Source, move the listener every bit far away from the speaker every bit possible, and so change the frequency of the audio wave. You may take noticed that at that place is a filibuster between the time when you change the setting and the fourth dimension when you lot hear the sound become lower or higher in pitch. Why is this?

1. Because, intensity of the audio wave changes with the frequency.

2. Because, the speed of the sound moving ridge changes when the frequency is inverse.

3. Because, loudness of the sound wave takes time to adapt after a change in frequency.

4. Considering it takes time for sound to reach the listener, then the listener perceives the new frequency of sound wave after a delay.

Is there a difference in the amount of delay depending on whether you brand the frequency college or lower? Why?

1. Yes, the speed of propagation depends only on the frequency of the wave.

2. Yep, the speed of propagation depends upon the wavelength of the wave, and wavelength changes as the frequency changes.

3. No, the speed of propagation depends only on the wavelength of the moving ridge.

4. No, the speed of propagation is constant in a given medium; only the wavelength changes as the frequency changes.

Snap Lab

Voice as a Sound Wave

In this lab yous volition observe the effects of blowing and speaking into a piece of newspaper in order to compare and contrast dissimilar sound waves.

• sail of paper
• record
• table

Instructions

Process

1. Suspend a canvas of paper then that the top edge of the paper is fixed and the bottom edge is free to motion. You could tape the top edge of the paper to the border of a table, for case.
2. Gently blow air near the edge of the bottom of the sheet and note how the sheet moves.
3. Speak softly and and so louder such that the sounds hitting the edge of the bottom of the newspaper, and note how the sheet moves.
4. Interpret the results.

Grasp Check

Which audio wave property increases when you are speaking more loudly than softly?

1. amplitude of the wave
2. frequency of the wave
3. speed of the moving ridge
4. wavelength of the wave

Worked Example

What Are the Wavelengths of Aural Sounds?

Summate the wavelengths of sounds at the extremes of the audible range, twenty and 20,000 Hz, in atmospheric condition where audio travels at 348.7 m/due south.

Strategy

To notice wavelength from frequency, nosotros can use $\mathrm{five}=f\lambda$ .

Word

Considering the product of f multiplied past $\lambda$ equals a constant velocity in unchanging atmospheric condition, the smaller f is, the larger $\lambda$ must exist, and vice versa. Note that you tin as well easily rearrange the same formula to find frequency or velocity.

Practice Bug

1 .

What is the speed of a sound wave with frequency 2000\,\text{Hz} and wavelength 0.iv\,\text{m}?

1. 5\times 10^3\,\text{m}/\text{s}

2. 3.ii\times 10^ii\,\text{m}/\text{south}

3. 2 \times x^{-4}\,\text{thousand/s}

4. 8 \times x^2\,\text{chiliad}/\text{southward}

2 .

Dogs can hear frequencies of up to 45\,\text{kHz}. What is the wavelength of a sound wave with this frequency traveling in air at 0^{\circ}\text{C}?

1. 2.0\times 10^7\,\text{m}

2. one.5\times 10^7\,\text{m}

3. 1.iv\times 10^2\,\text{m}

4. 7.4 \times 10^{-three}\,\text{1000}

Links To Physics

Echolocation

Figure 14.viii A bat uses sound echoes to discover its way about and to take hold of prey. The time for the echo to render is straight proportional to the distance.

Echolocation is the utilise of reflected sound waves to locate and identify objects. It is used by animals such every bit bats, dolphins and whales, and is as well imitated by humans in SONAR—Audio Navigation and Ranging—and echolocation technology.

Bats, dolphins and whales use echolocation to navigate and detect food in their environs. They locate an object (or obstruction) by emitting a sound and and then sensing the reflected sound waves. Since the speed of audio in air is constant, the time information technology takes for the sound to travel to the object and back gives the brute a sense of the altitude betwixt itself and the object. This is chosen ranging. Effigy 14.eight shows a bat using echolocation to sense distances.

Echolocating animals identify an object by comparing the relative intensity of the sound waves returning to each ear to figure out the angle at which the sound waves were reflected. This gives information almost the direction, size and shape of the object. Since there is a slight distance in position betwixt the two ears of an animal, the audio may return to i of the ears with a flake of a filibuster, which also provides data most the position of the object. For example, if a acquit is direct to the correct of a bat, the echo volition return to the bat'due south left ear afterwards than to its right ear. If, however, the bear is direct alee of the bat, the echo would return to both ears at the same fourth dimension. For an animal without a sense of sight such every bit a bat, it is important to know where other animals are as well as what they are; their survival depends on information technology.

Principles of echolocation have been used to develop a variety of useful sensing technologies. SONAR, is used by submarines to detect objects underwater and measure water depth. Dissimilar brute echolocation, which relies on just one transmitter (a rima oris) and ii receivers (ears), manmade SONAR uses many transmitters and beams to get a more than accurate reading of the environs. Radar technologies use the echo of radio waves to locate clouds and storm systems in weather condition forecasting, and to locate aircraft for air traffic control. Some new cars use echolocation technology to sense obstacles around the car, and warn the commuter who may be well-nigh to hit something (or fifty-fifty to automatically parallel park). Echolocation technologies and training systems are being adult to assist visually impaired people navigate their everyday environments.

If a predator is directly to the left of a bat, how will the bat know?

1. The echo would return to the left ear commencement.

2. The repeat would return to the correct ear beginning.

Check Your Understanding

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Instructor Support

Utilize these questions to appraise student achievement of the section's Learning Objectives. If students are struggling with a specific objective, these questions volition help identify which and directly students to the relevant content.

3 .

What is a rarefaction?

1. Rarefaction is the high-force per unit area region created in a medium when a longitudinal wave passes through information technology.

2. Rarefaction is the low-force per unit area region created in a medium when a longitudinal wave passes through it.

3. Rarefaction is the highest point of amplitude of a sound wave.

4. Rarefaction is the lowest betoken of aamplitude of a sound wave.

iv .

What sort of motion do the particles of a medium experience when a sound wave passes through it?

1. Simple harmonic move
2. Circular motion
3. Random motion
4. Translational motion

5 .

What does the speed of audio depend on?

1. The wavelength of the wave
2. The size of the medium
3. The frequency of the wave
4. The properties of the medium

vi .

What property of a gas would affect the speed of audio traveling through information technology?

1. The volume of the gas
2. The flammability of the gas
3. The mass of the gas
4. The compressibility of the gas

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Source: https://openstax.org/books/physics/pages/14-1-speed-of-sound-frequency-and-wavelength#:~:text=Since%20v%20%3D%20f%20%CE%BB%20v,wavelength%20for%20a%20given%20frequency.