- 14.1 Speed of Sound, Frequency, and Wavelength
- Introduction
- 1.1 Physics: Definitions and Applications
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- 13.1 Types of Waves
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- A | Reference Tables

## Section Learning Objectives

By the end of this section, you will be able to do the following:

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

## Teacher Support

The learning objectives in this section will help your students master the following standards:

- (A) examine and describe oscillatory motion and wave propagation in various 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.

## Section Key Terms

## Properties of Sound Waves

## The Speed of Sound

## Misconception Alert

## The Relationship Between the Speed of Sound and the Frequency and Wavelength of a Sound Wave

## Teacher Demonstration

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

## Virtual Physics

## Tips For Success

- Because, intensity of the sound wave changes with the frequency.
- Because, the speed of the sound wave changes when the frequency is changed.
- Because, loudness of the sound wave takes time to adjust after a change in frequency.
- Because it takes time for sound to reach the listener, so the listener perceives the new frequency of sound wave after a delay.
- Yes, the speed of propagation depends only on the frequency of the wave.
- Yes, the speed of propagation depends upon the wavelength of the wave, and wavelength changes as the frequency changes.
- No, the speed of propagation depends only on the wavelength of the wave.
- No, the speed of propagation is constant in a given medium; only the wavelength changes as the frequency changes.

## Voice as a Sound Wave

- Suspend a sheet of paper so that the top edge of the paper is fixed and the bottom edge is free to move. You could tape the top edge of the paper to the edge of a table, for example.
- Gently blow air near the edge of the bottom of the sheet and note how the sheet moves.
- Speak softly and then louder such that the sounds hit the edge of the bottom of the paper, and note how the sheet moves.
- Interpret the results.

## Grasp Check

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

## Worked Example

What are the wavelengths of audible sounds.

To find wavelength from frequency, we can use v = f λ v = f λ .

(1) Identify the knowns. The values for v and f are given.

(2) Solve the relationship between speed, frequency and wavelength for λ λ .

(3) Enter the speed and the minimum frequency to give the maximum wavelength.

(4) Enter the speed and the maximum frequency to give the minimum wavelength.

## Practice Problems

- 5\times 10^3\,\text{m}/\text{s}
- 3.2\times 10^2\,\text{m}/\text{s}
- 2 \times 10^{-4}\,\text{m/s}
- 8 \times 10^2\,\text{m}/\text{s}
- 2.0\times 10^7\,\text{m}
- 1.5\times 10^7\,\text{m}
- 1.4\times 10^2\,\text{m}
- 7.4 \times 10^{-3}\,\text{m}

## Links To Physics

## Check Your Understanding

- Rarefaction is the high-pressure region created in a medium when a longitudinal wave passes through it.
- Rarefaction is the low-pressure region created in a medium when a longitudinal wave passes through it.
- Rarefaction is the highest point of amplitude of a sound wave.
- Rarefaction is the lowest point of amplitude of a sound wave.

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

What does the speed of sound depend on?

- The wavelength of the wave
- The size of the medium
- The frequency of the wave
- The properties of the medium

What property of a gas would affect the speed of sound traveling through it?

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Access for free at https://openstax.org/books/physics/pages/1-introduction

- Authors: Paul Peter Urone, Roger Hinrichs
- Publisher/website: OpenStax
- Book title: Physics
- Publication date: Mar 26, 2020
- Location: Houston, Texas
- Book URL: https://openstax.org/books/physics/pages/1-introduction
- Section URL: https://openstax.org/books/physics/pages/14-1-speed-of-sound-frequency-and-wavelength

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## 17.3: Speed of Sound

## Learning Objectives

- Explain the relationship between wavelength and frequency of sound
- Determine the speed of sound in different media
- Derive the equation for the speed of sound in air
- Determine the speed of sound in air for a given temperature

\[v = f \lambda, \label{17.3}\]

## Speed of Sound in Various Media

\[v = \sqrt{\frac{\text{elastic property}}{\text{inertial property}}} \ldotp\]

Also, sound waves satisfy the wave equation derived in Waves ,

\[v = \sqrt{\frac{B}{\rho}} \ldotp \label{17.4}\]

The speed of sound in a solid the depends on the Young’s modulus of the medium and the density,

\[v = \sqrt{\frac{Y}{\rho}} \ldotp \label{17.5}\]

In an ideal gas (see The Kinetic Theory of Gases ), the equation for the speed of sound is

\[v = \sqrt{\frac{\gamma RT_{K}}{M}}, \label{17.6}\]

\[v_{rms} = \sqrt{\frac{3k_{B}T}{m}}.\]

## Derivation of the Speed of Sound in Air

\[\rho_{in} A_{in}v_{in} = \rho_{out} A_{out}v_{out}.\]

\[\rho Av = (\rho + d \rho)A(v + dv) \ldotp\]

From the continuity equation \(\rho\) dv = −vd\(\rho\), we obtain

If the air can be considered an ideal gas, we can use the ideal gas law:

Here M is the molar mass of air:

Since the speed of sound is equal to v = \(\sqrt{\frac{dp}{d \rho}}\), the speed is equal to

\[v = \sqrt{\frac{\gamma RT}{M}} \ldotp\]

## Example \(\PageIndex{1}\): Calculating Wavelengths

To find wavelength from frequency, we can use \(v = f \lambda\).

- Identify knowns. The value for \(v\) is given by \[v = 331\; m/s \sqrt{\frac{T}{273\; K}} \ldotp \nonumber\]
- Convert the temperature into kelvins and then enter the temperature into the equation \[v = 331\; m/s \sqrt{\frac{303\; K}{273\; K}} = 348.7\; m/s \ldotp \nonumber\]
- Solve the relationship between speed and wavelength for \(\lambda\): $$\lambda = \frac{v}{f} \ldotp \nonumber$$
- Enter the speed and the minimum frequency to give the maximum wavelength: \[\lambda_{max} = \frac{348.7\; m/s}{20\; Hz} = 17\; m \ldotp \nonumber\]
- Enter the speed and the maximum frequency to give the minimum wavelength: \[\lambda_{min} = \frac{348.7\; m/s}{20,000\; Hz} = 0.017\; m = 1.7\; cm \ldotp \nonumber\]

## Exercise \(\PageIndex{1}\)

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## Factors Affecting Wave Speed

## The Speed of Sound in Air

v = 331 m/s + (0.6 m/s/C)•(20 C)

## Look It Up!

Using wave speed to determine distances.

## The Wave Equation Revisited

Using the symbols v , λ , and f , the equation can be rewritten as

## Check Your Understanding

a. What is the frequency in Hertz of the sound wave? b. Assuming the sound wave moves with a velocity of 350 m/s, what is the wavelength of the wave?

3. Doubling the frequency of a wave source doubles the speed of the waves.

a. True b. False

Answer: 0.0173 meters (rounded)

7. Determine the speed of sound on a cold winter day (T=3 degrees C).

v = 331 m/s + (0.6 m/s/C) • T

where T is the temperature in Celsius. Substitute and solve.

v = 331 m/s + (0.6 m/s/C) • 3 C v = 331 m/s + 1.8 m/s v = 332.8 m/s

a. one-ninth b. one-third c. the same as d. three times larger than

## Speed of Sound Formula

## What is the Speed of Sound?

The speed of sound equation is expressed as,

c = \[\frac{\gamma \times P}{\rho}\]

## Speed of Sound in Air Formula

The speed of sound in air formula is,

γ = \[\sqrt{\frac{\gamma \times R \times T}{M}}\]

For air, γ = 1.4, R = 8.31 J/mol, and M = 0.02897 kg/mol.

## Speed of Sound in Solid Formula

C\[_{solid}\] = \[\frac{E}{\rho}\]

## Speed of Sound in Water Formula

In conclusion, the sound speed in a fluid (water) is given by

C\[_{fluid}\] = \[\frac{K}{\rho}\]

## Speed of Sound in Gas Formula

The speed of sound in the fluid is,

K = bulk modulus of the fluid.

γ = \[\sqrt{\frac{\gamma \times R \times T}{M}}\]

R = 8.314 J/mol - k universal gas constant

## Wavelength Sound Formula

The wavelength formula sound is given by,

Wavelength = \[\frac{\text{Speed of sound}}{Frequency}\]

## Solved Examples

By using the wavelength formula sound we get,

Rearranging the wavelength sound formula we get,

Density ρ = 0.037 Kg/m\[^{3}\]

The specific heat in air = 1.4

The speed of sound equation is given by,

c = \[\sqrt{\gamma \times \frac{P}{\rho}}\]

c = \[\sqrt{1.4 \times \frac{4000}{0.037}}\]

c = \[\sqrt{1.4 \times 108, 108.1081}\]

## FAQs on Speed of Sound Formula

Q.1) In Which Material Does Sound Travel the Fastest?

Q.2) What is the Highest Sound Frequency?

Answer: The highest sound frequency is 20 kHz.

Q.3) Can Sound Waves Travel in a Vacuum?

Q.4) What is the Relationship Between the Speed of Sound Temperature Formula?

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## 5th Grade Physical Science Worksheet

## View aligned standards

## Speed of Sound Calculator

Where Ratio of Specific Heats ( γ ) = 1.4

Enter the unknown value as ‘ x ‘

Speed of sound ( c ) = m s − 1

## How to Use the Speed of Sound Calculator?

The procedure to use the speed of sound calculator is as follows:

Step 1: Enter the pressure, density and x for the unknown value in the input field

Step 2: Now click the button “Calculate x” to get the sound speed

Step 3: Finally, the speed of the sound will be displayed in the output field

## What is Meant by Speed of Sound?

\(\begin{array}{l}v =\sqrt{\frac{\gamma P}{\rho }}\end{array} \)

\(\begin{array}{l}\rho\end{array} \) is the density

\(\begin{array}{l}\gamma\end{array} \) is the adiabatic expansion coefficient

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The speed of sound in a medium is determined by 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 sound in air is low, because air is compressible.

If the temperature is T C = 20 °C (T = 293 K), the speed of sound is v = 343 m/s. The equation for the speed of sound in air v = √γRT M can be simplified to give the equation for the speed of sound in air as a function of absolute temperature: v = √γRT M = √γRT M (273 K 273 K) = √(273 K)γR M √ T 273 K ≈ 331 m / s√ T 273 K.

overhead at an altitude of 15 km at Mach 2.5. Calculate the speed of the plane in km/h if the ground temperature is 23oC. 5. Calculate the wavelength in the following substances if the frequency is 1000s-1 and the speed of sound in the medium is given: a) Helium (1230 m/s) b) Hydrogen (1267 m/s) c) Steel (5130 m/s) d) Glass (4700 m/s) 6.

sound: • Isothermal Speed of Sound, cT =(RT) 1 2 • Adiabatic Speed of Sound, cA =(γRT) 1 2 We note that in air cA is 18.3% larger than cT. It is appropriate to detail some examples of the magnitude of the speed of sound. In air with γ =1.4and R = 280 m2/s2 Ko, the adiabatic speed of sound at a temperature of T = 293oK is 339m/s.Interms

2. The speed of sound in air depends on the _____ of the air. At 0oC, the speed of sound in air is _____ m/s. For every degree above 0oC, the speed _____ by 0.6 m/s. For every degree below 0oC, the speed _____by 0.6 m/s. The equation is: 3. What is the speed of sound at 35oC? _____ 4.

The speed ( v) at which sound travels through air is dependent upon the temperature of the air and seems to follow the equation v = 331 m/s + 0.6 m/s/°C * T where T is the Celsius temperature of the air. Determine the speed of sound … a. … on a cold day when the outdoor temperature is 4°C. b. … inside the school where the temperature is 24°C.

The speed of sound in water and air is 1450 and 330m/sm/s respectively. Sound ... Let us begin solving this problem by drawing the free body diagrams for both masses. Since we know that both masses are at rest that means that the forces on each mass must balance. Focusing our attention on the tension in the wire gives us these two equations

cally (less than the speed of sound). As the source approaches the speed of sound, you can see from the picture that the wave-fronts in the forward direction start bunching up. When the speed of sound is hit, they all come together. Remember for sound these are wavefronts of pressure. The large increase in pressure

7. A wave has a wavelength of 125 meters is moving at a speed of 20 m/s. What is it's frequency? 8. A wave has a frequency of 900 Hz and a wavelength of 200 m. At what speed is this wave traveling? 9. A wave has a wavelength of 0.5 meters and a frequency of 120 Hz. What is the wave's speed? 10. Radio waves travel at a speed of 300,000,000 m/s.

DISTANCE, TIME, SPEED PRACTICE PROBLEMS YOU MUST SHOW YOUR WORK. ... 11. If you shout into the Grand Canyon, your voice travels at the speed of sound (340 m/s) to the bottom of the canyon and back, and you hear an echo. ... CHALLENGE PROBLEM Bill and Amy want to ride their bikes from their neighborhood to school which is 14.4 kilometers away ...

Problem Solving Exercises - Pearson Education

Using this equation to determine the speed of a sound wave in air at a temperature of 20 degrees Celsius yields the following solution. v = 331 m/s + (0.6 m/s/C)•T v = 331 m/s + (0.6 m/s/C)• (20 C) v = 331 m/s + 12 m/s v = 343 m/s

9. If the limits of human hearing are 20 Hz. to 20,000 Hz, what are the sound wavelengths that are associated with both of these two extremes, assuming the speed of sound is 345 m/s. Frequency = 20 Hz : Wavelength = 1.725 1× 10 m 2 m Speed "c" (m/s) Frequency "ν" Wavelength (Hz) "λ" (m)

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Find out the speed of the sound. Solution: Given, Temperature T = 276 K Density ρ = 0.037 Kg/m 3 Pressure p = 4kPa = 4000 Pa The specific heat in air = 1.4 The speed of sound equation is given by, c = γ × P ρ c = 1.4 × 4000 0.037 c = 1.4 × 108, 108.1081 c = 151351.3513 c = 389.0390 m/s Is this page helpful?

Calculate the speed of the sound. Solution: As given here: Temperature, T = 2° C Density, ³ and, pressure P = 2k Pa i.e. P = 2000 Pa as we know that specific heat ratio in air is, Now the speed of sound formula is given by c = c = c= c = 286.97 Therefore, speed of sound = 286.97 m/s.

In non-humid air at 20 degrees Celsius, the speed of sound is about 343 meters per second or 767 miles per hour. We can also watch the speed of sound of a repeating simple harmonic wave. The speed of the wave can again be determined by the speed of the compressed regions as they travel through the medium.

Solving for λ and T yields λ = 11.4 m and T = 0.100 s. We know that frequency is given by f = 1/T = 10.0 Hz. As well, the speed of the wave is given by v = λ/T = 114 m/s. To find the tension in the string, we take and rewrite it as . For the given values, Ftension = 260 N. The rate of energy flow, or energy per unit time, or power, is given ...

Sound speed, acoustic impedance, and attenuation coefficient Acoustic properties from Zagzebski (1996) and Shung (2006) Material Sound speed (m/s) Air 330 Water 1480 Fat 1450-1460 Liver 1555-1570 Blood 1550-1560 Muscle 1550-1600 Skull bone 3360-4080 Sound speeds •highest : in solids •lowest : in gases Sound speed in soft tissues is

Choose 1 answer: Sound waves can propagate as longitudinal or transverse waves, depending on the transmitting medium. A. Sound waves can propagate as longitudinal or transverse waves, depending on the transmitting medium. Sound waves are transverse waves and they propagate perpendicular to the transmitting medium. B.

Entire Library Worksheets Fifth Grade Science Speed of Sound. 5th Grade Physical Science Worksheet Speed of Sound. How fast does sound travel? Help your child figure out the physics of sound waves with this informative worksheet. He'll learn about the speed of sound through different materials, and even try a tough math challenge.

The procedure to use the speed of sound calculator is as follows: Step 1: Enter the pressure, density and x for the unknown value in the input field. Step 2: Now click the button "Calculate x" to get the sound speed. Step 3: Finally, the speed of the sound will be displayed in the output field.