(a) Compute the derivative of the speed of a wave on a

A rope hangs vertically from the ceiling. A pulse is sent up the rope. Does the pulse travel faster, slower, or at a constant speed as it moves toward the ceiling? Explain your answer.

A pulse on a horizontal taut string travels to the right. If the ropes mass per unit length decreases to the right, what happens to the speed of the pulse as it travels to the right? (a) It slows down. (b) It speeds up. (c) Its speed is constant. (d) You cannot tell from the information given.

As a sinusoidal wave travels past a point on a taut string the arrival time between successive crests is measured to be 0.20 s. Which of the following is true? (a) The wavelength of the wave is 5.0 m. (b) The frequency of the wave is 5.0 Hz. (c) The velocity of propagation of the wave is (d) The wavelength of the wave is 0.20 m. (e) There is not enough information to justify any of these statements.

Two harmonic waves on identical strings differ only in amplitude. Wave A has an amplitude that is twice the amplitude of wave B. How do the energies of these waves compare? (a) (b) (c) (d) There is not enough information to compare their energies.

ENGINEERING APPLICATION To keep all of the lengths of the treble strings (unwrapped steel wires) in a piano about the same order of magnitude, wires of different linear mass densities are employed. Explain how this allows a piano manufacturer to use wires with lengths that are the same order of magnitude.

Musical instruments produce sounds of widely varying frequencies. Which sound waves have the longer wavelengths? (a) The lower frequencies. (b) The higher frequencies. (c) All frequencies have the same wavelength. (d) There is not enough information to compare the wavelengths of the different frequency sounds.

In Problem 6, which sound waves have the larger speeds? (a) The lower frequency sounds. (b) The higher frequency sounds. (c) All frequencies have the same wave speed. (d) There is not enough information to compare their speeds.

Sound travels at in air and in water. A sound of 256 Hz is made under water, but you hear the sound while walking along the side of the pool. In the air, the frequency is (a) the same, but the wavelength of the sound is shorter, (b) higher, but the wavelength of the sound stays the same, (c) lower, but the wavelength of the sound is longer, (d) lower, and the wavelength of the sound is shorter, (e) the same, and the wavelength of the sound stays the same.

While out on patrol, the battleship Rodger Young hits a mine, begins to burn, and ultimately explodes. Sailor Abel jumps into the water and begins swimming away from the doomed ship, while Sailor Baker gets into a life raft. Comparing their experiences later, Abel tells Baker, I was swimming underwater, and heard a big explosion from the ship. When I surfaced, I heard a second explosion. What do you think it could be? Baker says, I think it was your imaginationI only heard one explosion. Explain why Baker only heard one explosion, while Abel heard two.

True or false: A 60-dB sound has twice the intensity of a 30-dB sound.

At a given location, two harmonic sound waves have the same amplitude, but the frequency of sound Ais twice the frequency of sound B. How do their average energy densities compare? (a) The average energy density of A is twice the average energy density of B. (b) The average energy density of Ais four times the average energy density of B. (c) The average energy density of Ais 16 times the average energy density of B. (d) You cannot compare the average energy densities from the data given. SSM

At a given location, two harmonic sound waves have the same frequency, but the amplitude of sound A is twice the amplitude of sound B. How do their average energy densities compare? (a) The average energy density of Ais twice the average energy density of B. (b) The average energy density of A is four times the average energy density of B. (c) The average energy density of A is 16 times the average energy density of B. (d) You cannot compare the average energy densities from the data given.

What is the ratio of the intensity of normal conversation to the sound intensity of a soft whisper (at a distance of 5.0 m)? (a) (b) 2, (c) (d) Hint: See Table 15-1.

What is the ratio of the intensity level of normal conversation to the sound intensity level of a soft whisper (at a distance of 5.0 m)? (a) (b) 2, (c) (d) Hint: See Table 15-1.

To increase the sound intensity level by 20 dB requires the sound intensity to increase by what factor? (a) 10, (b) 100, (c) 1000, (d) 2

You are using a hand-held sound level meter to measure the intensity level of the roars produced by a lion prowling in the high grass. To decrease the measured sound intensity level by 20 dB requires the lion move away from you until its distance from you has increased by what factor? (a) 10, (b) 100, (c) 1000, (d) You cannot tell the required distance from the data given.

One end of a very light (but strong) thread is attached to an end of a thicker and denser cord. The other end of the thread is fastened to a sturdy post and you pull the other end of the cord so the thread and cord are taut. A pulse is sent down the thicker, denser cord. True or false: (a) The pulse that is reflected back from the thread-cord attachment point is inverted compared to the initial incoming pulse. (b) The pulse that continues past the thread-cord attachment point is not inverted compared to the initial incoming pulse. (c) The pulse that continues past the thread-cord attachment point has an amplitude that is smaller than the pulse that is reflected.

Light traveling in air strikes a glass surface at a incident angle. True or false: (a) The angle between the reflected light ray and the incident ray is (b) The angle between the reflected light ray and the refracted light ray is less than

Sound waves in air encounter an open 1.0-m-wide door into a classroom. Due to the effects of diffraction, the sound of which frequency is least likely to be heard by all the students in the roomassuming the room is full? (a) 600 Hz, (b) 300 Hz, (c) 100 Hz, (d) All of the sounds are equally likely to be heard in the room. (e) Diffraction depends on wavelength not frequency, so you cannot tell from the data given.

Microwave radiation in modern microwave ovens has a wavelength on the order of centimeters. Would you expect significant diffraction if this radiation was aimed at a 1.0-m-wide door? Explain.

Stars often occur in pairs revolving around their common center of mass. If one of the stars is a black hole, it is invisible. Explain how the existence of such a black hole might be inferred by measuring the Doppler frequency shift of the light observed from the other, visible star.

Figure 15-30 shows a wave pulse at time moving to the right. (a) At this particular time, which segments of the string are moving up? (b) Which segments are moving down? (c) Is there any segment of the string at the pulse that is instantaneously at rest? Answer these questions by sketching the pulse at a slightly later time and a slightly earlier time to see how the segments of the string are moving.

Make a sketch of the velocity of each string segment versus position for the pulse shown in Figure 15-30.

An object of mass hangs on a very light rope that is connected to the ceiling. You pluck the rope just above the object, and a wave pulse travels up to the ceiling and back. Compare the round-trip time for such a wave pulse to the round-trip time of a wave pulse on the same rope if an object of mass is hung on the rope instead. (Assume that the rope does not stretch, that is, that the mass-to-ceiling distance is the same in each case.)

The explosion of a depth charge beneath the surface of a body of water is recorded by a helicopter hovering above the waters surface, as shown in Figure 15-31. Along which pathA, B, or Cwill the sound wave take the least time to reach the helicopter? Explain why you chose the path you did.

Does a speed of Mach 2 at an altitude 60,000 feet mean the same as a speed of Mach 2 near ground level? Explain clearly.

Many years ago, Olympic 100-m dashes were started by the sound from a starters pistol, with the starter positioned several meters down the track, just on the inside of the track. (Today, the pistol that is used is often only a trigger, which is used to electronically activate speakers behind each sprinters starting blocks. This method avoids the problem of one runner hearing the sound before the other runners.) Estimate the time advantage the runner at the inside lane (relative to the runner at the outside lane of 8 runners) would have if all runners started when they heard the sound from the starters pistol.

Estimate the speed of the bullet as it passes through the helium balloon in Figure 15-32. Hint: A protractor would be beneficial.

The new student townhouses at a local college are in the form of a semicircle half-enclosing the track field. To estimate the speed of sound in air, an ambitious physics student stood at the center of the semicircle and clapped his hands rhythmically at a frequency at which he could not hear the echo of the clap, because the echo reached him at the same time as his next clap. This frequency was about Once he established this frequency, he paced off the distance to the townhouses, which was 30 double strides. Assuming that the length of each stride is equal to half his height (5 ft 11 in.), estimate the speed of sound in air using these data. How far off is your estimation from the commonly accepted value of ?

(a) The bulk modulus of water is Use this value to find the speed of sound in water. (b) The speed of sound in mercury is What is the bulk modulus of mercury

Calculate the speed of sound waves in hydrogen gas ( and at

A 7.00-m-long guitar string has a mass of 100 g and is under a tension of 900 N. What is the speed of a transverse wave pulse on this string?

(a) Compute the derivative of the speed of a wave on a string with respect to the tension and show that the differentials and obey (b) A wave moves with a speed of on a string that is under a tension of 500 N. Using the differential approximation, estimate how much the tension must be changed to increase the speed to (c) Calculate exactly and compare it to the differential approximation result in Part (b). Assume that the string does not stretch with the increase in tension.

(a) Compute the derivative of the speed of sound in air with respect to the absolute temperature, and show that the differentials and obey (b) Use this result to estimate the percentage change in the speed of sound when the temperature changes from to (c) If the speed of sound is at estimate its value at using the differential approximation. (d) How does this approximation compare with the result of an exact calculation?

Derive a convenient formula for the speed of sound in air at temperature t in degrees Celsius. Begin by writing the temperature as where and corresponds to and which is the Celsius temperature. The speed of sound is a function of To a first-order approximation, you can write where is the derivative evaluated at Compute this derivative, and show that the result leads to

Show explicitly that the following functions satisfy the wave equations : (a) (b) where and are constants and and (c)

Show that the function satisfies the wave equation.

One end of a 6.0-m-long string is moved up and down with simple harmonic motion at a frequency of 60 Hz. If the wave crests travel the length of the string in 0.50 s, find the wavelength of the waves on the string.

A harmonic wave on a string that has a mass per unit length of and a tension of 80 N has an amplitude of 5.0 cm. Each point on the string moves with simple harmonic motion at a frequency of 10 Hz. What is the power carried by the wave propagating along the string?

A 2.00-m-long rope has a mass of 0.100 kg. The tension is 60.0 N. An oscillator at one end sends a harmonic wave with an amplitude of 1.00 cm down the rope. The other end of the rope is terminated so all of the energy of the wave is absorbed and none is reflected. What is the frequency of the oscillator if the power transmitted is 100 W?

The wave function for a harmonic wave on a string is (a) In what direction does this wave travel, and what is the waves speed? (b) Find the wavelength, frequency, and period of this wave. (c) What is the maximum speed of any point on the string?

A harmonic wave on a string with a frequency of 80 Hz and an amplitude of 0.025 m travels in the direction with a speed of (a) Write a suitable wave function for this wave. (b) Find the maximum speed of a point on the string. (c) Find the maximum acceleration of a point on the string.

A 200-Hz harmonic wave with an amplitude equal to 1.2 cm moves along a 40-m-long string that has a mass of 0.120 kg and a tension of 50 N. (a) What is the average total energy of the waves on a 20-m-long segment of string? (b) What is the power transmitted past a given point on the string?

On a real string, some of the energy of a wave dissipates as the wave travels down the string. Such a situation can be described by a wave function whose amplitude depends on where What is the power transported by the wave as a function of where

Power is to be transmitted along a taut string by means of transverse harmonic waves. The wave speed is 10 m/s and the linear mass density of the string is 0.010 kg/m. The power source oscillates with an amplitude of 0.50 mm. (a) What average power is transmitted along the string if the frequency is 400 Hz? (b) The power transmitted can be increased by increasing the tension in the string, the frequency of the source, or the amplitude of the waves. By how much would each of these quantities have to increase to cause an increase in power by a factor of 100 if it is the only quantity changed?

Two very long strings are tied together at the point x=0. In the region x<0, the wave speed is \(v_1\), while in the region x>0, the speed is \(v_2\). A sinusoidal wave is incident on the knot from the left (x<0); part of the wave is reflected and part is transmitted. For x<0, the displacement of the wave is described by \(y(x, t)=A \sin \left(k_1 x-\omega t\right) +B \sin \left(k_1 x+\omega t\right)\), while for \(x>0, y(x, t)=C \sin \left(k_2 x-\omega t\right)\), where \(\omega / k_1=v_1\) and \(\omega / k_2=v_2\). (a) If we assume that both the wave function y and its first spatial derivative \(\partial y / \partial x\) must be continuous at x=0, show that \(C / A=2 v_2 /\left(v_1+v_2\right)\), and that \(B / A=\left(v_1-v_2\right) /\left(v_1+v_2\right)\). (b) Show that \(B^2+\left(v_1 / v_2\right) C^2=A^2\).

A sound wave in air produces a pressure variation given by where is in pascals, is in meters, and is in seconds. Find (a) the pressure amplitude, (b) the wavelength, (c) the frequency, and (d) the wave speed.

(a) Middle C on the musical scale has a frequency of 262 Hz. What is the wavelength of this note in air? (b) The frequency of the C an octave above middle C is twice that of middle C. What is the wavelength of this note in air?

(a) What is the displacement amplitude for a sound wave with a frequency of 100 Hz and a pressure amplitude of (b) The displacement amplitude of a sound wave of frequency 300 Hz is Assuming the density of air is what is the pressure amplitude of this wave?

(a) What is the displacement amplitude of a sound wave that has a frequency of 500 Hz at the pain-threshold pressure amplitude of 29.0 Pa? (b) Assuming the density of air is what is the displacement amplitude of a sound wave that has the same pressure amplitude as the wave in Part (a), but has a frequency of 1.00 kHz?

Atypical loud sound wave that has a frequency of 1.00 kHz has a pressure amplitude of about (a) At the pressure is a maximum at some point What is the displacement at that point at (b) Assuming the density of air is what is the maximum value of the displacement at any time and place?

An octave represents a change in frequency by a factor of 2. Over how many octaves can a typical person hear?

BIOLOGICAL APPLICATION In the oceans, whales communicate by sound transmission through the water. A whale emits a sound of 50.0 Hz to tell a wayward calf to catch up to the pod. The speed of sound in water is about (a) How long does it take the sound to reach the calf if he is 1.20 km away? (b) What is the wavelength of this sound in the water? (c) If the whales are close to the surface, some of the sound energy might refract out into the air. What would be the frequency and wavelength of the sound in the air?

A spherical sinusoidal source radiates sound uniformly in all directions. At a distance of 10.0 m, the sound intensity level is (a) At what distance from the source is the intensity (b) What power is radiated by this source?

ENGINEERING APPLICATION A loudspeaker at a rock concert generates a sound that has an intensity level equal to at 20.0 m and has a frequency of 1.00 kHz Assume that the speaker spreads its energy uniformly in three dimensions. (a) What is the total acoustic power output of the speaker? (b) At what distance will the sound intensity be at the pain threshold of (c) What is the sound intensity at 30.0 m?

When a pin of mass 0.100 g is dropped from a height of 1.00 m, 0.050 percent of its energy is converted into a sound pulse that has a duration of 0.100 s. (a) Estimate how far away the dropped pin can be heard if the minimum audible intensity is (b) Your result in Part (a) is much too large in practice, because of background noise. If you assume the intensity must be at least for the sound to be heard, estimate how far away the dropped pin can be heard. (In both parts, assume that the intensity is

What is the intensity level in decibels of a sound wave that has an intensity equal to (a) and (b)

What is the intensity of a sound wave if, at a particular location, the intensity level is (a) and (b)

At a certain distance, the sound intensity level of a dogs bark is 50 dB. At that same distance, the sound intensity level of a rock concert is 10,000 times that of the dogs bark. What is the sound intensity level of the rock concert?

What fraction of the acoustic power of a noise would have to be eliminated to lower its sound intensity level from 90 to 70 dB?

A spherical source radiates sound uniformly in all directions. At a distance of 10 m, the sound intensity level is 80 dB. (a) At what distance from the source is the intensity level 60 dB? (b) What power is radiated by this source?

Harry and Sally are sitting on opposite sides of a circus tent when an elephant trumpets a loud blast. If Harry experiences a sound intensity level of 65 dB and Sally experiences only 55 dB, what is the ratio of the distance between Sally and the elephant to the distance between Harry and the elephant?

Three noise sources produce intensity levels of 70 dB, 73 dB, and 80 dB, when acting separately. When the sources act together, the resultant intensity is the sum of the individual intensities. (a) Find the sound intensity level in decibels when the three sources act at the same time. (b) Discuss the effectiveness of eliminating the two less intense sources in reducing the intensity level of the noise.

Show that if two people are different distances away from a sound source, the difference between the intensity levels reaching the people, in decibels, will always be the same, no matter the power radiated by the source.

Everyone at a party is talking equally loudly. One person is talking to you and the sound intensity level at your location is 72 dB. Assuming that all 38 people at the party are at the same distance from you as the person who you are talking to, find the sound intensity level at your location.

When a violinist pulls the bow across a string, the force with which the bow is pulled is fairly small, about 0.60 N. Suppose the bow travels across the A string, which vibrates at 440 Hz, at 0.50 m/s. A listener 35 m from the performer hears a sound of 60-dB intensity. Assuming that the sound radiates uniformly in all directions, with what efficiency is the mechanical energy of bowing converted to sound energy?

The noise intensity level at some location in an empty classroom is 40 dB. When 100 students are writing during an exam, the noise level at that location increases to 60 dB. Assuming that the noise produced by each student contributes an equal amount of acoustic power, find the noise intensity level in the room after 50 students have left.

A 3.00-m-long piece of string, with a mass of 25.0 g, is tied to 4.00 m of heavy twine with a mass of 75.0 g, and the combination is put under a tension of 100 N. If a transverse pulse is sent down the less dense string, determine the reflection and transmission coefficients at the junction point.

Consider a taut string, with a mass per unit length carrying transverse wave pulses that are incident upon a point where the string connects to a second string, with a mass per unit length (a) Show that if then the reflection coefficient equals zero and the transmission coefficient equals (b) Show that if then and (c) Show that if then and

Verify the validity of (Equation 15-36) by substituting the expressions for and into it.

Consider a taut string that has a mass per unit length carrying transverse wave pulses of the form that are incident upon a point where the string connects to a second string with mass per unit length Derive by equating the power incident on point to the power reflected at plus the power transmitted at P.

A sound source is moving at toward a stationary listener that is standing in still air. (a) Find the wavelength of the sound in the region between the source and the listener. (b) Find the frequency heard by the listener.

Consider the situation described in Problem 72 from the reference frame of the source. In this frame, the listener and the air are moving toward the source at and the source is at rest. (a) At what speed, relative to the source, is the sound traveling in the region between the source and the listener? (b) Find the wavelength of the sound in the region between the source and the listener. (c) Find the frequency heard by the listener.

A sound source is moving away from the stationary listener at (a) Find the wavelength of the sound waves in the region between the source and the listener. (b) Find the frequency heard by the listener.

The listener is moving at away from the stationary source that is at rest relative to the air. Find the frequency heard by the listener.

CONTEXT-RICH You have made the trek to observe a Space Shuttle landing. Near the end of its descent, the ship is traveling at Mach 2.50 at an altitude of 5000 m. (a) What is the angle that the shock wave makes with the line of flight of the shuttle? (b) How far are you from the shuttle by the time you hear its shock wave, assuming the shuttle maintains both a constant heading and a constant 5000-m altitude after flying directly over your head?

ENGINEERING APPLICATION The SuperKamiokande neutrino detector of Japan is a water tank the size of a 14-story building. When neutrinos collide with electrons in the water, most of their energy is transferred to the electrons. As a consequence, the electrons then fly off at speeds that approach c. The neutrino is counted by detecting the shock wave, called Cerenkov radiation, that is produced when the high-speed electrons travel through the water at speeds greater than the speed of light in water. If the maximum angle of the Cerenkov shock-wave cone is what is the speed of light in water?

ENGINEERING APPLICATION, CONTEXT-RICH You are in charge of calibrating the radar guns for a local police department. One such device emits microwaves at a frequency of 2.00 GHz. During the trials, these waves are reflected from a car moving directly away from the stationary emitter. You detect a frequency difference (between the received microwaves and the ones sent out) of 293 Hz. Find the speed of the car.

ENGINEERING APPLICATION, CONTEXT-RICH The Doppler effect is routinely used to measure the speed of winds in storm systems. As the manager of a weather monitoring station in the Midwest, you are using a Doppler radar system that has a frequency of to bounce a radar pulse off of the raindrops in a swirling thunderstorm system 50 km away. You measure the reflected radar pulse to be up-shifted in frequency by 325 Hz. Assuming the wind is headed directly toward you, how fast are the winds in the storm system moving? Hint: The radar system can only measure the component of the wind velocity along its line of sight.

ENGINEERING APPLICATION A stationary destroyer is equipped with sonar that sends out 40 MHz pulses of sound. The destroyer receives reflected pulses back from a submarine directly below with a time delay of 80 ms at a frequency of 39.958 MHz. If the speed of sound in seawater is (a) what is the depth of the submarine? (b) What is its vertical speed?

A police radar unit transmits microwaves of frequency and their speed in air is Suppose a car is receding from the stationary police car at a speed of (a) What is the frequency difference between the transmitted signal and the signal received from the receding car? (b) Suppose the police car is, instead, moving at a speed of in the same direction as the other vehicle. What is the difference in frequency between the emitted and the reflected signals?

BIOLOGICAL APPLICATION, CONTEXT-RICH In modern medicine, the Doppler effect is routinely used to measure the rate and direction of blood flow in arteries and veins. High-frequency ultrasound (sound at frequencies above the human hearing range) is typically employed. Suppose you are in charge of measuring the blood flow in a vein (located in the lower leg of an older patient) that returns blood upward to the heart. Her varicose veins indicate that perhaps the one-way valves in the vein are not working properly and that the blood is pooling in the veins and perhaps even that the blood flow is backward toward her feet. Employing sound that has a frequency of 50.0 kHz, you point the sound source from above her thigh region down toward her feet and measure the sound reflected from that vein area to be lower than 50.0 kHz. (a) Was your diagnosis of the valve condition correct? If so, explain. (b) Estimate the instruments frequency difference capability to enable you to measure speeds down to Take the speed of sound in flesh to be the same as that in water,

A sound source of frequency moves with speed relative to still air toward a receiver who is moving away from the source with speed relative to still air. (a) Write an expression for the received frequency (b) Use the result that to show that if both and are small compared to then the received frequency is approximately where is the velocity of the source relative to the receiver.

To study the Doppler shift on your own, you take an electronic tone generating device that is set to a frequency of middle C (262 Hz) to a campus wishing well known as The Abyss. When you hold the device at arms length (1.0 m), you measure its intensity level to be 80.0 dB. You then drop the tuner down the hole, listening to its sound as it falls. (a) After the tuner has fallen for 5.50 s, what frequency do you hear? (b) Estimate the time at which you can no longer hear the tuner.

You are in a hot-air balloon carried along by a wind and have a sound source with you that emits a sound of 800 Hz as it approaches a tall building. (a) What is the frequency of the sound heard by an observer at the window of this building? (b) What is the frequency of the reflected sound heard by you?

A car is approaching a reflecting wall. A stationary observer behind the car hears a sound of frequency 745 Hz from the car horn and a sound of frequency 863 Hz from the wall. (a) How fast is the car traveling? (b) What is the frequency of the car horn? (c) What frequency does the car driver hear reflected from the wall?

The driver of a car traveling at toward a vertical wall briefly sounds the horn. Exactly 1.00 s later she hears the echo and notes that its frequency is 840 Hz. How far from the wall was the car when the driver sounded the horn and what is the frequency of the horn?

You are on a transatlantic flight traveling due west at An experimental plane flying at Mach 1.6 and 3.0 km to the north of your plane is also on an east-to-west course. What is the distance between the two planes when you hear the sonic boom from the experimental plane?

The Hubble space telescope has been used to determine the existence of planets orbiting around distant stars. A planet orbiting a star will cause the star to wobble with the same period as the planets orbit. Because of this wobble, light from the star will be Doppler-shifted up and down periodically. Estimate the maximum and minimum wavelengths of light of nominal wavelength 500 nm emitted by the Sun that is Doppler-shifted by the motion of the Sun due to the planet Jupiter.

At time the shape of a wave pulse on a string is given by the function where is in meters. (a) Sketch versus (b) Give the wave function at a general time if the pulse is moving in the with a speed of and if the pulse is moving in the with a speed of

A whistle that has a frequency of 500 Hz moves in a circle of radius 1.00 m at What are the maximum and minimum frequencies heard by a stationary listener in the plane of the circle and 5.00 m away from its center?

Ocean waves move toward the beach with a speed of and a crest-to-crest separation of 15.0 m. You are in a small boat anchored off shore. (a) At what frequency do the wave crests reach your boat? (b) You now lift anchor and head out to sea at a speed of At what frequency do the wave crests reach your boat now?

A 12.0-m-long wire that has an 85.0-g mass is under a tension of 180 N. Apulse is generated at the left end of the wire, and 25.0 ms later a second pulse is generated at the right end of the wire. Where do the pulses first meet?

You are parked on the shoulder of a highway. Find the speed of a car in which the tone of the car’s horn drops by 10 percent as it passes you. (In other words, the total drop in frequency between the “approach” value and the “recession” value is 10%.)

A loudspeaker driver 20.0 cm in diameter is vibrating at 800 Hz with an amplitude of 0.0250 mm. Assuming that the air molecules in the vicinity have the same amplitude of vibration, find (a) the pressure amplitude immediately in front of the driver, (b) the sound intensity, and (c) the acoustic power being radiated by the front surface of the driver.

A plane, harmonic, sound wave in air has an amplitude of has an intensity of What is the frequency of the wave?

Water flows at in a pipe of radius 5.0 cm. A plate with area equal to the cross-sectional area of the pipe is suddenly inserted to stop the flow. Find the force exerted on the plate. Take the speed of sound in water to be Hint: When the plate is inserted, a pressure wave propagates through the water at the speed of sound, The mass of water brought to a stop in time is the water in a length of pipe equal to

A high-speed flash photography setup meant to capture a picture of a bullet exploding a soap bubble is shown in Figure 15-33. The shock wave from the bullet is to be detected by a microphone that will trigger the flash. The microphone is placed on a track that is parallel to and 0.350 m below the path of the bullet. The track is used to adjust the position of the microphone. If the bullet is traveling at 1.25 times the speed of sound, and the distance between the lab bench and the track is 0.350 m, how far back from the soap bubble must the microphone be set to trigger the flash? (Assume that the flash itself is instantaneous once the microphone is triggered.)

A column of precision marchers keeps in step by listening to the band positioned at the head of the column. The beat of the music is for A television camera shows that only the marchers at the front and the rear of the column are actually in step. The marchers in the middle section are striding forward with the left foot when those at the front and rear are striding forward with the right foot. The marchers are so well trained, however, that they are all certain that they are in proper step with the music. How long is the column?

BIOLOGICAL APPLICATION A bat flying toward a stationary obstacle at emits brief, high-frequency sound pulses at a repetition frequency of 80.0 Hz. What is the interval between the arrival times of the reflected pulses heard by the bat?

Laser ranging to the moon is done routinely to accurately determine the Earth-moon distance. However, to determine the distance accurately, corrections must be made for the average speed of light in Earths atmosphere, which is 99.997 percent of the speed of light in vacuum. Assuming that Earths atmosphere is 8.00 km high, estimate the length of the correction.

A tuning fork attached to a taut string generates transverse waves. The vibration of the fork is perpendicular to the string. Its frequency is 400 Hz and the amplitude of its oscillation is 0.50 mm. The string has a linear mass density of and is under a tension of 1.0 kN. Assume that there are no waves reflected at the far end of the string. (a) What are the period and frequency of waves on the string? (b) What is the speed of the waves? (c) What are the wavelength and wave number? (d) What is a suitable wave function for the waves on the string? (e) What is the maximum speed and acceleration of a point on the string? (f) At what minimum average rate must energy be supplied to the fork to keep it oscillating at a steady amplitude?

A long rope with a mass per unit length of is under a constant tension of 10.0 N. A motor drives one end of the rope with transverse simple harmonic motion at 5.00 cycles per second and an amplitude of 40.0 mm. (a) What is the wave speed? What is the wavelength? (c) What is the maximum transverse linear momentum of a 1.00-mm segment of the rope? (d) What is the maximum net force on a 1.00-mm segment of the rope?

In this problem, you will derive an expression for the potential energy of a segment of a string carrying a traveling wave (Figure 15-34). The potential energy of a segment equals the work done by the tension in stretching the string, which is where is the tension, is the length of the stretched segment, and is its original length. (a) Use the binomial expansion to show that and therefore (b) Compute from the wave function (Equation 15-15) and show that _ _4(x)2 _ (y)2 _ x[1 _ (y>x)2]1>2.