Solved: Sound intensity from a localized source decreases as the inverse square of the

What distinguishes a wave from an oscillation?

Red light has a longer wavelength than blue light. Compare their frequencies.

Consider a light wave and a sound wave with the same wavelength. Which has the higher frequency?

As a wave propagates on a string, the string moves back and forth sideways. Is the string speed related to the wave speed? Explain.

If you doubled the tension in a string, what would happen to the speed of waves on the string?

A heavy cable is hanging vertically, its bottom end free. How will the speed of transverse waves near the top and bottom of the cable compare? Why?

The intensity of light from a localized source decreases as the inverse square of the distance from the source. Does this mean that the light loses energy as it propagates?

Medical ultrasound uses frequencies around 107 Hz, far above the range of the human ear. In what sense are these waves sound?

If you double the pressure of a gas while keeping its density the same, what happens to the sound speed?

Water is about a thousand times more dense than air, yet the speed of sound in water is greater than in air. Why might this be?

If you place a perfectly clear piece of glass in perfectly clear water, you can still see the glass. Why?

When a wave source moves relative to the medium, a stationary observer measures changes in both wavelength and frequency. But when the observer moves and the source is stationary, only the frequency changes. Why the difference?

Why can a boat easily produce a shock wave on the water surface, while only a very high-speed aircraft can produce a sonic boom?

Ocean waves with 18-m wavelength travel at 5.3 m/s. Whats the time interval between wave crests passing a boat moored at a fixed location?

Ripples in a shallow puddle propagate at 34 cm/s. If the wave frequency is 5.2 Hz, find (a) the period and (b) the wavelength.

An 89.5-MHz FM radio wave propagates at the speed of light. Whats its wavelength?

Calculate the wavelengths of (a) a 1.0-MHz AM radio wave, (b) a channel 9 TV signal (190 MHz), (c) a police radar (10 GHz), (d) infrared radiation from a hot stove 14*1013 Hz2, (e) green light 16.0*1014 Hz2, and (f) 1.0*1018@Hz X rays. All are electromagnetic waves that propagate at 3.0*108 m/s.

A seismograph located 1250 km from an earthquake detects seismic waves 5.12 min after the quake occurs. The seismograph oscillates in step with the waves, at 3.21 Hz. Find the wavelength.

Medical ultrasound waves travel at about 1500 m/s in soft tissue. Higher frequencies provide clearer images but dont penetrate to deeper organs. Find the wavelengths of (a) 8.0-MHz ultrasound used in fetal imaging and (b) 3.5-MHz ultrasound used to image an adults kidneys.

An ocean wave has period 4.1 s and wavelength 10.8 m. Find its (a) wave number and (b) angular frequency.

Find the (a) amplitude, (b) wavelength, (c) period, and (d) speed of a wave whose displacement is given by y = 1.3 cos10.69x + 31t2, where x and y are in centimeters and t in seconds. (e) In which direction is the wave propagating?

Ultrasound used in a medical imager has frequency 4.86 MHz and wavelength 0.313 mm. Find (a) the angular frequency, (b) the wave number, and (c) the wave speed.

A simple harmonic wave of wavelength 18.7 cm and amplitude 2.34 cm is propagating in the negative x-direction at 38.0 cm/s. Find its (a) angular frequency and (b) wave number. (c) Write a mathematical expression describing the displacement y of this wave (in centimeters) as a function of position and time. Assume the maximum displacement occurs when t = 0.

Analysis of waves in shallow water (depth much less than wavelength) yields the following wave equation: 0 2 y 0x 2 = 1 gh 0 2 y 0t 2 where h is the water depth and g the gravitational acceleration. Give an expression for the wave speed.

The main cables supporting New Yorks George Washington Bridge have a mass per unit length of 4100 kg/m and are under 250-MN tension. At what speed would a transverse wave propagate on these cables?

A transverse wave 1.2 cm in amplitude propagates on a string; its frequency is 44 Hz. The string is under 21-N tension and has mass per unit length 15 g/m. Determine its speed.

A transverse wave with 3.0-cm amplitude and 75-cm wavelength propagates at 6.7 m/s on a stretched spring with mass per unit length 170 g/m. Find the spring tension.

A rope is stretched between supports 18.3 m apart; its tension is 78.6 N. If one end of the rope is tweaked, the resulting disturbance reaches the other end 585 ms later. Find the ropes mass.

A rope with 280 g of mass per meter is under 550-N tension. Find the average power carried by a wave with frequency 3.3 Hz and amplitude 6.1 cm propagating on the rope.

Show that 1P/r from Equation 14.9 has the units of speed.

Find the sound speed in air under standard conditions with pressure 101 kN/m2 and density 1.20 kg/m3 .

Timers in sprint races start their watches when they see smoke from the starting gun, not when they hear the sound. Why? How much error would be introduced by timing a 100-m race from the sound of the gun?

The factor g for nitrogen dioxide 1NO22 is 1.29. Find the sound speed in NO2 at 4.8*104 @N/m2 pressure and 0.35@kg/m3 density.

A gas with density 1.0 kg/m3 and pressure 81 kN/m2 has sound speed 368 m/s. Are the gas molecules monatomic or diatomic?

Divers in an underwater habitat breathe a special mixture of oxygen and neon to prevent the possibly fatal effects of nitrogen in ordinary air. With pressure 6.2*105 N/m2 and density 4.5 kg/m3 , the effective g value for the mixture is 1.61. Find the frequency in this mixture for a 50-cm-wavelength sound wave, and compare with its frequency in air under normal conditions.

Youre flying in a twin-engine turboprop aircraft, with its two propellers turning at 985 and 993 rpm, respectively. How often to you hear a peak in the engine sound?

Whats the wavelength of the ocean waves in Example 14.5 if the calm water you encounter at 33 m is the second calm region on your voyage from the center line?

A 2.0-m-long string is clamped at both ends. (a) Find the longestwavelength standing wave possible on this string. (b) If the wave speed is 56 m/s, whats the lowest standing-wave frequency?

When a stretched string is clamped at both ends, its fundamental frequency is 140 Hz. (a) Whats the next higher frequency? If the same string, with the same tension, is now clamped at one end and free at the other, what are (b) the fundamental and (c) the next higher frequency?

A string is clamped at both ends and tensioned until its fundamental frequency is 85 Hz. If the string is then held rigidly at its midpoint, whats the lowest frequency at which it will vibrate?

A crude model of the human vocal tract treats it as a pipe closed at one end. Find the effective length of a vocal tract whose fundamental tone is 620 Hz.Take Vsound = 354m/s at body temperature.

A car horn emits 380-Hz sound. If the car moves at 17 m/s with its horn blasting, what frequency will a person standing in front of the car hear?

A fire stations siren is blaring at 85 Hz. Whats the frequency perceived by a firefighter racing toward the station at 120 km/h?

A fire trucks siren at rest wails at 1400 Hz; standing by the roadside as the truck approaches, you hear it at 1600 Hz. How fast is the truck going?

Red light emitted by hydrogen atoms at rest in the laboratory has wavelength 656 nm. Light emitted in the same process on a distant galaxy is received at Earth with wavelength 708 nm. Describe the galaxys motion relative to Earth.

Figure 14.36 shows a simple harmonic wave at time t = 0 and later at t = 2.6 s. Write a mathematical description of this wave. t = 0 s 2 4 6 8 10 12 14 16 x (cm) t = 2.6 s 1.5 y (cm) 1 0.5 0 -0.5 -1 -1.5 Figure 14.36 Problem 46

Transverse waves propagate at 18 m/s on a string under 14-N tension. What will be the wave speed if the tension is increased to 40 N?

A uniform cable hangs vertically under its own weight. Show that the speed of waves on the cable is given by v = 1yg, where y is the distance from the bottom of the cable.

Figure 14.37 shows a wave train consisting of two sine wave cycles propagating along a string. Obtain an expression for the total energy BIo CH l A Figure 14.37 Problem 49 in this wave train, in terms of the string tension F, the wave amplitude A, and the wavelength l.

A loudspeaker emits energy at the rate of 50 W, spread in all directions. Find the intensity of sound 18 m from the speaker.

Light intensity 3.3 m from a lightbulb is 0.73 W/m2 . Find the bulbs power output, assuming it radiates equally in all directions.

Light emerges from a 5.0-mW laser in a beam 1.0 mm in diameter. The beam shines on a wall, producing a spot 3.6 cm in diameter. What is the beams intensity (a) at the laser and (b) at the wall?

Two waves have the same angular frequency v, wave number k, and amplitude A, but they differ in phase: y1 = A cos1kx - vt2 and y2 = A cos1kx - vt + f2. Show that their superposition is also a simple harmonic wave, and determine its amplitude as a function of the phase difference f.

A wire is under 32.8-N tension, carrying a wave described by y = 1.75 sin10.211x - 466t2, where x and y are in centimeters and t is in seconds. What are (a) the wave amplitude, (b) the wavelength, (c) the wave period, (d) the wave speed, and (e) the power carried by the wave?

A spring of mass m and spring constant k has an unstretched length L0. Find an expression for the speed of transverse waves on this spring when its been stretched to a length L.

When a 340-g spring is stretched to a total length of 40 cm, it supports transverse waves propagating at 4.5 m/s. When its stretched to 60 cm, the waves propagate at 12 m/s. Find (a) the springs unstretched length and (b) its spring constant.

At a point 15 m from a source of spherical sound waves, you measure the intensity 750 mW/m2 . How far do you need to walk, directly away from the source, until the intensity is 270 mW/m2 ?

Figure 14.38 shows two observers 20 m apart on a line that connects them to a spherical light source. If the observer nearer the source measures a light intensity 50% greater than the other observer, how far is the nearer observer from the source? 20 m x = ? Figure 14.38 Problem 58

An ideal spring is stretched to a total length L1. When that length is doubled, the speed of transverse waves on the spring triples. Find an expression for the unstretched length of the spring.

Show that the time it takes a wave to propagate up the cable in Problem 48 is t = 21L/g, where L is the cable length.

You see an airplane 5.2 km straight overhead. Sound from the plane, however, seems to be coming from a point back along the planes path at 35 to the vertical. Whats the planes speed, assuming an average sound speed of 330 ms?

What are the intensities in W/m2 of sound with intensity levels of (a) 65 dB and (b) -5 dB?

Show that a doubling of sound intensity corresponds to approximately a 3-dB increase in the decibel level.

Sound intensity from a localized source decreases as the inverse square of the distance, according to Equation 14.8. If the distance from the source doubles, what happens to (a) the intensity and (b) the decibel level?

At 2.0 m from a localized sound source you measure the intensity level as 75 dB. How far away must you be for the perceived loudness to drop in half (i.e., to an intensity level of 65 dB)?

The A-string (440 Hz) on a piano is 38.9 cm long and is clamped at both ends. If the string tension is 667 N, whats its mass?

Show that the standing-wave condition of Equation 14.13 is equivalent to the requirement that the time it takes a wave to make a round trip from one end of the medium to the other and back be an integer multiple of the wave period.

Youre designing an organ for a new concert hall; the lowest note is to be 22 Hz. The architects have asked you to minimize the lengths of the organ pipes. How long will the longest pipe be if its (a) closed at one end and (b) open at both ends?

Show by differentiation and substitution that a wave described by Equation 14.3 satisfies the wave equation (Equation 14.5), with wave speed v = v/k.

Show by differentiation and substitution that any function of the form y = f1x { vt2 satisfies the wave equation (Equation 14.5).

Youre a marine biologist concerned with the effect of sonic booms on plankton, and you need to estimate the altitude of a supersonic aircraft flying directly over you at 2.2 times the speed of sound. You hear its sonic boom 19 s later. Assuming a constant 340 m/s sound speed, find the planes altitude.

A 2.25-m-long pipe has one end open. Among its possible standing-wave frequencies is 345 Hz; the next higher frequency is 483 Hz. Find (a) the fundamental frequency and (b) the sound speed.

A wave source recedes from you at 8.2 m/s, and the wavelength you measure is 20% greater than what you would measure if the source were at rest. Whats the wave speed?

Obstetricians use ultrasound to monitor fetal heartbeat. If 5.0-MHz ultrasound reflects off the moving heart wall with a 100-Hz frequency shift, whats the speed of the heart wall? (Hint: You have two shifts to consider.)

Youre in court, trying to argue your way out of a speeding ticket. You were stopped going 120 km/h in a 90-km/h zone. A technical expert testifies that the 70-GHz police radar signal underwent a 15.6-kHz frequency shift when it reflected off your car. You claim that corresponds to an impossible 240 km/h, so the radar must be defective. How should the judge rule?

You move at speed u toward a wave source thats stationary with respect to the medium in which waves of wavelength l propagate with speed v. Your speed relative to the wave crests is therefore v + u. Show that for you, the time between wave crests is T = l/1v + u2, and from this show that you perceive a frequency given by Equation 14.16, with the + sign.

Youre a meteorologist specifying a new Doppler radar system that determines the velocity of distant raindrops by reflecting radar signals (which travel at the speed of light) off them and measuring the Doppler shift. You need a system that will measure speeds as low as 2.5 km/h. A vendor offers a 5.0-GHz radar that can detect a frequency shift of only 50 Hz. Is that sufficient?

Use a computer to form the sum implied in the caption of Figure 14.17, taking v = 1 s -1 and using (a) the three terms shown and (b) 10 terms (note that only odd harmonics appear in the sum). Plot your result over one cycle (t from 0 to 2p) and compare with the square wave shown in the figure.

Your little sister and her friend build treehouses and stretch a rope between them for sending messages. They hang a 1.4-kg mass on one end of the rope that passes over a pulley. The other end is tied to the second treehouse. When your sister plucks the rope, a wave propagates at 18 m/s. The girls deem this too slow; they want to increase the wave speed to 30 m/s. Your sister asks, What mass should I use? What do you reply?

An airport neighborhood is concerned about the basing of the new F-35 jet fighter. Theyve got the data below for the sound intensity level measured at different distances from the plane as it takes off. Theyd like to know the total sound power emitted by the plane. As a physics student, youre called to help. First, convert the sound intensity levels to actual intensity. Then find a quantity which, when you plot intensity against it, should give a straight line. Make your plot, determine the best-fit line, and use its slope to report the total sound power. Distance (m) 1000 1200 1500 2000 3000 4000 Sound intensity level (dB) 80.7 79.4 76.9 74.2 71.6 68.8

Tsunamis are ocean waves generally produced when earthquakes suddenly displace the ocean floor, and with it a huge volume of water. Unlike ordinary waves on the ocean surface, a tsunami involves the entire water column, from surface to bottom. To a tsunami, the ocean is shallowand that makes tsunamis shallow-water waves, whose speed is v = 1gd, where d is the water depth and g the gravitational acceleration. Tsunamis can travel thousands of kilometers across an ocean to reach the shore with their initial energy nearly intact; when they do, they can cause massive damage and loss of life (Fig. 14.39). As a tsunami approaches shore, it a. speeds up. b. slows down. c. maintains its speed.

Tsunamis are ocean waves generally produced when earthquakes suddenly displace the ocean floor, and with it a huge volume of water. Unlike ordinary waves on the ocean surface, a tsunami involves the entire water column, from surface to bottom. To a tsunami, the ocean is shallowand that makes tsunamis shallow-water waves, whose speed is v = 1gd, where d is the water depth and g the gravitational acceleration. Tsunamis can travel thousands of kilometers across an ocean to reach the shore with their initial energy nearly intact; when they do, they can cause massive damage and loss of life (Fig. 14.39). For a tsunami to behave as a shallow-water wave, its wavelength a. must be comparable to or longer than the ocean depth. b. must be shorter than the ocean depth. c. can have any value.

Tsunamis are ocean waves generally produced when earthquakes suddenly displace the ocean floor, and with it a huge volume of water. Unlike ordinary waves on the ocean surface, a tsunami involves the entire water column, from surface to bottom. To a tsunami, the ocean is shallowand that makes tsunamis shallow-water waves, whose speed is v = 1gd, where d is the water depth and g the gravitational acceleration. Tsunamis can travel thousands of kilometers across an ocean to reach the shore with their initial energy nearly intact; when they do, they can cause massive damage and loss of life (Fig. 14.39). A tsunami is traveling at 450 km/h when the ocean depth abruptly doubles. Its new speed is roughly a. 225 km/h. b. 320 km/h. c. 640 km/h. d. 900 km/h.

Tsunamis are ocean waves generally produced when earthquakes suddenly displace the ocean floor, and with it a huge volume of water. Unlike ordinary waves on the ocean surface, a tsunami involves the entire water column, from surface to bottom. To a tsunami, the ocean is shallowand that makes tsunamis shallow-water waves, whose speed is v = 1gd, where d is the water depth and g the gravitational acceleration. Tsunamis can travel thousands of kilometers across an ocean to reach the shore with their initial energy nearly intact; when they do, they can cause massive damage and loss of life (Fig. 14.39). On the open ocean, a tsunami has relatively small amplitude typically 1 m or less. As the tsunami approaches shore, its amplitude increases and its wavelength decreases. As a result, a. its total energy increases. b. the rate at which it carries energy shoreward increases. c. the wave frequency increases. d. none of these quantities changes.