# PHYS 2211 Module 15.2

## Speed of Sound 15.2 Speed of Sound

### Learning Objectives

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

• 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

### Speed of Sound in Various Media

This table shows that the speed of sound varies greatly in different media. The speed of sound in a medium depends on how quickly vibrational energy can be transferred through the medium. For this reason, the derivation of the speed of sound in a medium depends on the medium and on the state of the medium. In general, the equation for the speed of a mechanical wave in a medium depends on the square root of the restoring force, or the elastic property, divided by the inertial property:

Recall from Module 14 that the speed of waves on a string is where FT is the restoring force and µ is the inertia.

The speed of sound in a fluid is: where B is the bulk modulus and is the mass density.

The speed of sound in a solid the depends on the Young’s modulus of the medium and the density: . Practice! Discuss!

A popular party trick is to inhale helium and speak in a high-frequency, funny voice. Explain this phenomenon.

You may have used a sonic range finder in lab to measure the distance of an object using a clicking sound from a sound transducer. What is the principle used in this device?

The sonic range finder discussed in the preceding question often needs to be calibrated. During the calibration, the software asks for the room temperature. Why do you suppose the room temperature is required?

Bats are mainly active at night. They have several senses that they use to find their way about, locate prey, avoid obstacles, and “see” in the dark. Besides the usual sense of vision, bats are able to emit high-frequency sound waves and hear the echo that bounces back when these sound waves hit an object. This sonar-like system is called echolocation. Typical frequencies emitted by bats are between 20 and 200 kHz. Note that the human ear is sensitive only to frequencies as high as 20 kHz.

A moth of length 1.0 cm is flying about 1.0 m from a bat when the bat emits a sound wave at 80.0 kHz. The temperature of air is about 10.0 °C. To sense the presence of the moth using echolocation, the bat must emit a sound with a wavelength equal to or less than the length of the insect.

The speed of sound that propagates in an ideal gas is given by , where is the ratio of heat capacities ( = 1.4 for air), T is the absolute temperature in kelvins (which is equal to the Celsius temperature plus 273.15 °C), M is the molar mass of the gas (for air, the average molar mass is M = 28.8×10−3 kg/mol), and R is the universal gas constant (R = 8.314 J⋅mol−1⋅K−1).

(a) Find the wavelength λ of the 80.0-kHz wave emitted by the bat.

(b) Will the bat be able to locate the moth despite the darkness of the night?

(c) How long after the bat emits the wave will it hear the echo from the moth?