SOUND INTENSITY: The human ear is able to hear sound over a very wide range of intensities. The loudest sound a healthy person can hear without damage to the eardrum has an intensity 1 trillion (1,000,000,000,000) times that of the softest sound a person can hear. If we were to use these intensities as a scale for measuring volume, we would be stuck using numbers from zero all the way to the trillions, which seems cumbersome, if not downright silly. In the last section, we saw that logarithmic functions increase very slowly. We can take advantage of this to create a scale for sound intensity that is much more condensed, and therefore more manageable.
The decibel scale for sound intensity is an example of such a scale. The decibel, named after the inventor of the telephone, Alexander Graham Bell (1847–1922), is defined as follows:
where D is the decibel level of the sound, I is the intensity of the sound measured in watts per square meter (W/m2) and is the intensity of the least audible sound that an average healthy young person can hear. The latter is standardized to be I0=10-12 watts per square meter. Table 1 lists some typical sound intensities from familiar sources. In Example 1 and Problems 5 and 6 in Exercises 5-4, we will calculate the decibel levels for these sounds.
EXAMPLE 1 Sound Intensity
(A) Find the number of decibels from a whisper with sound intensity 5.2×10-10 watts per square meter, then from heavy traffic at 8.5×10-4 watts per square meter. Round your answers to two decimal places. (B) How many times larger is the sound intensity of heavy traffic compared to a whisper?
EARTHQUAKE INTENSITY: The energy released by the largest earthquake recorded, measured in joules, is about 100 billion (100,000,000,000) times the energy released by a small earthquake that is barely felt. In 1935 the California seismologist Charles Richter devised a logarithmic scale that bears his name and is still widely used in the United States. The magnitude of an earthquake M on the Richter scale* is given as follows:
EXAMPLE 2 Earthquake Intensity
The 1906 San Francisco earthquake released approximately 5.96×1016of energy. Another quake struck the Bay Area just before game 3 of the 1989 World Series, releasing 1.12×1015 joules of energy.
(A) Find the magnitude of each earthquake on the Richter scale. Round your answers to two decimal places.
(B) How many times more energy did the 1906 earthquake release than the one in 1989?
EXAMPLE 3 Earthquake Intensity
If the energy release of one earthquake is 1,000 times that of another, how much larger is the Richter scale reading of the larger than the smaller?
ROCKET FLIGHT: The theory of rocket flight uses advanced mathematics and physics to show that the velocity v of a rocket at burnout (depletion of fuel supply) is given by
where c is the exhaust velocity of the rocket engine, Wt is the takeoff weight (fuel, structure, and payload), and Wb is the burnout weight (structure and payload).
Because of the Earth’s atmospheric resistance, a launch vehicle velocity of at least 9.0 kilometers per second is required to achieve the minimum altitude needed for a stable orbit. Formula (3) indicates that to increase velocity v, either the weight ratio Wt/Wb must be increased or the exhaust velocity c must be increased. The weight ratio can be increased by the use of solid fuels, and the exhaust velocity can be increased by improving the fuels, solid or liquid.
EXAMPLE 4 Rocket Flight Theory
A typical single-stage, solid-fuel rocket may have a weight ratio Wt/Wb =18.7and an exhaust velocity c=2.38 kilometers per second. Would this rocket reach a launch velocity of 9.0 kilometers per second?
The velocity of the launch vehicle is far short of the 9.0 kilometers per second required to achieve orbit. This is why multiple-stage launchers are used—the deadweight from a preceding stage can be jettisoned into the ocean when the next stage takes over.