ChemPhys 173/273

Unit 4: Sound and Music

Problem Set A

Overview:

Problem Set A targets your ability to basic mathematical principles associated with the speed of sound and the intensity level of sound. The first five of the 15 problems pertain to the speed of sound. The last 10 of the 15 problems pertain to the intensity level of sound.

Speed of Sound

The speed of sound, like the speed of any wave, is dependent upon the properties of the medium through which it is moving. For a sound wave moving through air, the primary property of air which effects its speed is the temperature of the air. There are a couple of equations which describe the speed (v) - temperature (T) relationship; the following might be the easiest to remember:

v = 331 m/s + 0.6 Cm/s •T

where T is the Celsius temperature of the air through which the sound wave is moving. At 0 degrees Celsius, the speed of sound is 331 m/s. For every degree Celsius above 0, the speed of sound increases by approximately 0.6 m/s. This equation provides a rather accurate estimate of the speed of sound for temperatures upwards towards 50 degrees Celsius.

The speed of a wave (v) is defined as the distance traveled (d) per time of travel (t) and is described by the following equation:

v = d / t

Thus, the distance traveled by a wave is related to the time required for it to travel that distance. Problems #4 and #5 of this set will target your understanding of this relationship.

Intensity Level of Sound

Sound waves are produced by a vibrating object - vocal chords, guitar string, or diaphragm on a speaker. The sound waves begin at a point (or approximately a point) and then propagate through space in three dimensions. As the sound wave propagates, it creates a wave front that fills the surface area of an ever-expanding sphere. A sound wave (like any wave) is often referred to as an energy-transport phenomenon. The rate at which energy is put into the wave is referred to as the power of the source. Power (P) is expressed in units of Watts. As these vibrations propagate through space, they become less intense as the size of the spherical wave front expands. Whatever energy is created by the wave at the source fills the surface area of a sphere some distance R away. Because the sphere is constantly expanding, the energy is becoming diluted with increasing distance from the source. The sound intensity (I) at any given location is defined as the rate at which energy passes through that location. Because the wave is filling a surface area, intensity is often expressed in units of Watts/meter2 and given by the equation

I = P / (4 • ¼ • R2)

The human ear is sensitive enough to detect the smallest of vibrations. The lowest amplitude vibration which most humans can hear is defined as the threshold of hearing (TOH). While such a sound will be different for different people, the intensity associated with this sound is defined as 1.0 x 10-12 W/m2. The range of sound intensities which a typical human can detect is enormous. sound which 100 billion times more intense than the threshold of hearing (1 x 10-1 W/m2) is detectable without pain. Intensities beyond this level begin to cause pain and possibly the risk of hearing loss. Because their is an enormous range of intensities from the threshold of hearing to the threshold of pain, a logarithmic scale - known as the decibel scale - is often used to express sound intensity. This logarithmic scale simply expresses the sound level in terms of how many factors of 10 that an intensity is than the threshold of hearing (1 x 10-12 W/m2). A sound which is 10 times more intense than the TOH is 1 bel. A sound which is 100 (102) times more intense than the TOH is 2 bels. And a sound which is 1000 (103) times more intense than the TOH is 3 bels. More commonly, the sound level is expressed in the smaller unit decibel (1/10-th of a Bel), abbreviated dB. The sound level in decibel can be properly determined from the intensity (I) of a sound using the following equation:

dB = 10 • log ( I / 1 x 10-12 W/m2 )

Often times, the sound level in decibels is known and the the intensity in W/m2 is desired. In such instances the related equation below can be used.

I = 1 x 10-12 W/m2 • 10x where x = dB/10

The following pages from The Physics Classroom tutorial may serve to be useful in assisting you in the understanding of speed of sound waves and the intensity of sound waves.

Speed of Sound | Intensity of Sound

View Sample Problem Set.

 Problem Description Audio Link 1 Routine calculation of speed from temperature. 2 Routine calculation of speed from temperature. 3 Routine calculation of speed from temperature. 4 Referring to problem #1; determination of distance of travel; requires good conceptual understanding. 5 Determination of time of travel; requires good conceptual understanding. 6 Routine calculation of decibel level. 7 Routine calculation of decibel level. 8 Routine calculation of intensity level from decibel rating. 9 Routine calculation of intensity level from decibel rating. 10 Multistep problem involving the use of the inverse square law to determine the decibel level at a different location. 11 Multistep problem involving the use of the inverse square law to determine the decibel level at a different location. 12 Multistep problem involving the use of the inverse square law to determine the decibel level at a different location. 13 Multistep problem involving the use of the inverse square law to determine the decibel level at a different location. 14 Determination of the intensity level from a distance and the power of the sound source. 15 Referring to the previous problem; determination of decibel level from the intensity.

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