The wave function for a certain standing wave on a string

Two rectangular wave pulses are traveling in opposite directions along a string. At the two pulses are as shown in Figure 16-29. Sketch the wave functions for and 3.0 s. SSM

Repeat Problem 1 for the case in which the pulse on the right is inverted.

Beats are produced by the superposition of two harmonic waves if (a) their amplitudes and frequencies are equal, (b) their amplitudes are the same but their frequencies differ slightly, (c) their frequencies are equal but their amplitudes differ slightly.

Two tuning forks are struck and the sounds from each reach your ears at the same time. One sound has a frequency of 256 Hz, and the second sound has a frequency of 258 Hz. The underlying hum frequency that you hear is (a) 2 Hz, (b) 256 Hz, (c) 258 Hz, (d) 257 Hz.

In Problem 4, the beat frequency is (a) \(2 \mathrm{~Hz}\), (b) \(256 \mathrm{~Hz}\), (c) \(258 \mathrm{~Hz}\),(d) \(257 \mathrm{~Hz}\).

CONTEXT-RICH As a graduate student, you are teaching your first physics lecture while the professor is away. To demonstrate interference of sound waves, you have set up two speakers that are driven coherently and in phase by the same frequency generator on the front desk. Each speaker generates sound with a 2.4-m wavelength. One student in the front row says she hears a very low volume (loudness) of the sound from the speakers compared to the volume of the sound she hears when only one speaker is generating sound. What could be the difference in the distance between her and each of the two speakers? (a) 1.2 m, (b) 2.4 m, (c) 4.8 m, (d) You cannot determine the difference in distances from the data given.

In Problem 6, determine the longest wavelength for which a student would hear extra loud sound due to constructive interference, assuming this student is located so that one speaker is 3.0 m farther from her than the other speaker.

Consider standing waves in an organ pipe. True or false: (a) In a pipe open at both ends, the frequency of the third harmonic is three times that of the first harmonic. (b) In a pipe open at both ends, the frequency of the fifth harmonic is five times that of the fundamental. (c) In a pipe that is open at one end and stopped at the other, the even harmonics are not excited. Explain your choices.

Standing waves result from the superposition of two waves that have (a) the same amplitude, frequency, and direction of propagation, (b) the same amplitude and frequency and opposite directions of propagation, (c) the same amplitude, slightly different frequencies, and the same direction of propagation, (d) the same amplitude, slightly different frequencies, and opposite directions of propagation.

If you blow air over the top of a fairly large drinking straw you can hear a fundamental frequency due to a standing wave being set up in the straw. What happens to the fundamental frequency, (a) if while blowing, you cover the bottom of the straw with your fingertip? (b) if while blowing you cut the straw in half with a pair of scissors? (c) Explain your answers to Parts (a) and (b).

An organ pipe that is open at both ends has a fundamental frequency of 400 Hz. If one end of this pipe is now stopped, the fundamental frequency is (a) 200 Hz, (b) 400 Hz, (c) 546 Hz, (d) 800 Hz.

A string fixed at both ends resonates at a fundamental frequency of 180 Hz. Which of the following will reduce the fundamental frequency to 90 Hz? (a) Double the tension and double the length. (b) Halve the tension and keep the length and the mass per unit length fixed. (c) Keep the tension and the mass per unit length fixed and double the length. (d) Keep the tension and the mass per unit length fixed and halve the length.

ENGINEERING APPLICATION Explain how you might use the resonance frequencies of an organ pipe to estimate the temperature of the air in the pipe.

In the fundamental standing-wave pattern of an organ pipe stopped at one end, what happens to the wavelength, frequency, and speed of the sound needed to create the pattern if the air in the pipe becomes significantly colder? Explain your reasoning.

(a) When a guitar string is vibrating in its fundamental mode, is the wavelength of the sound it produces in air typically the same as the wavelength of the standing wave on the string? Explain. (b) When an organ pipe is in any one of its standing-wave modes, is the wavelength of the traveling sound wave it produces in air typically the same as the wavelength of the standing sound wave in the pipe? Explain.

Figure 16-30 is a photograph of two pieces of very finely woven silk placed one on top of the other. Where the pieces overlap, a series of light and dark lines are seen. This moir pattern can also be seen when a scanner is used to copy photos from a book or newspaper. What causes the moir pattern, and how is it similar to the phenomenon of interference?

When a musical instrument consisting of drinking glasses, each partially filled to a different height with water, is struck with a small mallet, each glass produces a different frequency of sound wave. Explain how this instrument works.

ENGINEERING APPLICATION During an organ recital, the air compressor that drives the organ pipes suddenly fails. An enterprising physics student in the audience tries to help by replacing the compressor with a pressurized tank of nitrogen gas. What effect, if any, will the nitrogen gas have on the frequency output of the organ pipes? What effect, if any, would helium gas have on the frequency output of the organ pipes?

The constant for helium (and all monatomic gases) is 1.67. If a man inhales helium and then speaks, his voice has a high pitch and becomes cartoon-like. Why?

It is said that a powerful opera singer can hit a high note with sufficient intensity to shatter an empty wine glass by causing the air in it to resonate at the frequency of her voice. Estimate the frequency necessary to obtain a standing wave in an 8.0-cm-high glass. (The 8.0 cm does not include the height of the stem.) Approximately how many octaves above middle C (262 Hz) is this? Hint: To go up one octave means to double the frequency.

Estimate how accurately you can tune a piano string to a tuning fork of known frequency using only your ears, the tuning fork, and a wrench. Explain your answer.

The shortest pipes used in organs are 7.5 cm long. (a) Estimate the fundamental frequency of a pipe this long that is open at both ends. (b) For such a pipe, estimate the harmonic number of the highest-frequency harmonic that is within the audible range. (The audible range of human hearing is about 20 to 20,000 Hz.)

BIOLOGICAL APPLICATION Estimate the resonant frequencies that are in the audible range of human hearing of the human ear canal. Treat the canal as an air column open at one end, stopped at the other end, and with a length of 1.00 in. How many resonant frequencies lie in this range? Human hearing has been found experimentally to be the most sensitive at frequencies of about 3, 9, and 15 kHz. How do these frequencies compare to your calculations?

Two harmonic waves traveling on a string in the same direction both have a frequency of 100 Hz, a wavelength of 2.0 cm, and an amplitude of 0.020 m. In addition, they overlap each other. What is the amplitude of the resultant wave if the original waves differ in phase by (a) and (b)

Two harmonic waves having the same frequency, wave speed, and amplitude are traveling in the same direction and in the same propagating medium. In addition, they overlap each other. If they differ in phase by and each has an amplitude of 0.050 m, what is the amplitude of the resultant wave?

Two audio speakers facing in the same direction oscillate in phase at the same frequency. They are separated by a distance equal to one-third of a wavelength. Point is in front of both speakers, on the line that passes through their centers. The amplitude of the sound at due to either speaker acting alone is What is the amplitude (in terms of ) of the resultant wave at point

Two compact sources of sound oscillate in phase with a frequency of 100 Hz. At a point 5.00 m from one source and 5.85 m from the other, the amplitude of the sound from each source separately is (a) What is the phase difference of the two waves at that point? (b) What is the amplitude (in terms of ) of the resultant wave at that point?

With a drawing program or a compass, draw circular arcs of radius 1 cm, 2 cm, 3 cm, 4 cm, 5 cm, 6 cm, and 7 cm centered at each of two points a distance apart. Draw smooth curves through the intersections corresponding to points centimeters farther from than from for and and label each curve with the corresponding value of There are two additional such curves you can draw, one for and one for If identical sources of coherent in-phase 1.0-cmwavelength waves were placed at points and the waves would interfere constructively along each of the smooth curves.

Two speakers separated by some distance emit sound waves of the same frequency. At some point the intensity due to each speaker separately is The distance from to one of the speakers is longer than that from to the other speaker. What is the intensity at if (a) the speakers are coherent and in phase, (b) the speakers are incoherent, and (c) the speakers are coherent and 180 out of phase? SSM

Two speakers separated by some distance emit sound waves of the same frequency. At some point the intensity due to each speaker separately is The distance from to one of the speakers is one wavelength longer than that from to the other speaker. What is the intensity at if (a) the speakers are coherent and in phase, (b) the speakers are incoherent, and (c) the speakers are coherent and out of phase?

A transverse harmonic wave with a frequency equal to 40.0 Hz propagates along a taut string. Two points 5.00 cm apart are out of phase by (a) What is the wavelength of the wave? (b) At a given point on the string, how much does the phase change in 5.00 ms? (c) What is the wave speed?

BIOLOGICAL APPLICATION It is thought that the brain determines the direction of the source of a sound by sensing the phase difference between the sound waves striking the eardrums. A distant source emits sound of frequency 680 Hz. When you are directly facing a sound source there is no phase difference. Estimate the phase difference between the sounds received by your ears when you are facing away from the direction of the source.

Sound source A is located at and sound source B is located at The two sources radiate coherently and in phase. An observer at notes that as he takes a few steps from in either the or direction, the sound intensity diminishes. What is the lowest frequency, and the next to lowest frequency of the sources that can account for that observation?

Suppose that the observer in Problem 33 finds himself at a point of minimum intensity at What is then the lowest frequency and next to lowest frequency of the sources that can account for this observation?

SPREADSHEET Two harmonic water waves of equal amplitudes but different frequencies, wave numbers, and speeds are traveling in the same direction. In addition, they are superposed on each other. The total displacement of the wave can be written as where (the speed of the first wave) and (the speed of the second wave). (a) Show that can be written in the form where and The factor is called the envelope of the wave. (b) Let and Using a spreadsheet program or graphing calculator, make a plot of versus at (c) Using a spreadsheet program or graphing calculator, make three plots of versus for on the same graph. Make one plot for the second for and the third for Estimate the speed at which the envelope moves from the three plots, and compare this estimate with the speed obtained using

Two coherent point sources are in phase and are separated by a distance An interference pattern is detected along a line parallel to the line through the sources and a large distance from the sources, as shown in Figure 16-31. (a) Show that the path difference from the two sources to some point on the line at an angle is given, approximately, by Hint: Assume that so the lines from the sources to P are approximately parallel (Figure 16-31b). (b) Show that the two waves interfere constructively at if where (That is, show there is an interference maximum at if where (c) Show that the distance from the central maximum (at ) to the th interference maximum at is given by where d sin um _ ml.

Two sound sources radiating in phase at a frequency of 480 Hz interfere such that maxima are heard at angles of and from a line perpendicular to that joining the two sources. The listener is at a large distance from the line through both sources, and no additional maxima are heard at angles in the range Find the separation between the two sources, and any other angles at which intensity maxima will be heard. (Use the result of Problem 36.)

Two loudspeakers are driven in phase by an audio amplifier at a frequency of 600 Hz. The speakers are on the axis, one at and the other at A listener, starting at where walks in the direction along the line (See Problem 36.) (a) At what angle will she first hear a minimum in the sound intensity? ( is the angle between the positive axis and the line from the origin to the listener.) (b) At what angle will she first hear a maximum in the sound intensity (after (c) How many maxima can she possibly hear if she keeps walking in the same direction?

Two sound sources driven in phase by the same amplifier are 2.00 m apart on the axis, one at and the other at At points large distances from the axis, constructive interference is heard at angles with the axis of and and at no angles in between (see Figure 16-31). (a) What is the wavelength of the sound waves from the sources? (b) What is the frequency of the sources? (c) At what other angles is constructive interference heard? (d) What is the smallest angle for which the sound waves cancel?

The two sound sources from Problem 39 are now driven out-of-phase, but at the same frequency as in Problem 39. At what angles are constructive and destructive interference heard?

ENGINEERING APPLICATION An astronomical radio telescope consists of two antennas separated by a distance of 200 m. Both antennas are tuned to the frequency of 20 MHz. The signals from each antenna are fed into a common amplifier, but one signal first passes through a phase selector that delays its phase by a chosen amount so that the telescope can look in different directions (Figure 16-32). When the phase delay is zero, plane radio waves that are incident vertically on the antennas produce signals that add constructively at the amplifier. What should the phase delay be so that signals coming from an angle with the vertical (in the plane formed by the vertical and the line joining the antennas) will add constructively at the amplifier? Hint: Radio waves travel at 3.00 _ 108 m>s. u _ 10 90 SSM

When two tuning forks are struck simultaneously, 4.0 beats per second are heard. The frequency of one fork is 500 Hz. (a) What are the possible values for the frequency of the other fork? (b) A piece of wax is placed on the 500-Hz fork to lower its frequency slightly. Explain how the measurement of the new beat frequency can be used to determine which of your answers to Part (a) is the correct frequency of the second fork.

ENGINEERING APPLICATION A stationary police radar gun emits microwaves at 5.00 GHz. When the gun is aimed at a car, it superposes the transmitted and reflected waves. Because the frequencies of these two waves differ, beats are generated, with the speed of the car proportional to the beat frequency. The speed of the car, appears on the display of the radar gun. Assuming the car is moving along the line-of-sight of the police officer, and using the Doppler-shift equations, (a) show that, for a fixed radar-gun frequency, the beat frequency is proportional to the speed of the car. Hint: Car speeds are tiny compared to the speed of light. (b) What is the beat frequency in this case? (c) What is the calibration factor for this radar gun? That is, what is the beat frequency generated per of speed?

A string fixed at both ends is 3.00 m long. It resonates in its second harmonic at a frequency of 60.0 Hz. What is the speed of transverse waves on the string?

A string 3.00 m long and fixed at both ends is vibrating in its third harmonic. The maximum displacement of any point on the string is 4.00 mm. The speed of transverse waves on this string is (a) What are the wavelength and frequency of this standing wave? (b) Write the wave function for this standing wave.

Calculate the fundamental frequency for an organ pipe, with an effective length equal to 10 m, that is (a) open at both ends, and (b) stopped at one end.

A5.00-g, 1.40-m-long flexible wire has a tension of 968 N and is fixed at both ends. (a) Find the speed of transverse waves on the wire. (b) Find the wavelength and frequency of the fundamental. (c) Find the frequencies of the second and third harmonics.

A taut, 4.00-m-long rope has one end fixed and the other end free. (The free end is attached to a long, light string.) The speed of waves on the rope is \(20.0 \mathrm{~m} / \mathrm{s}\). (a) Find the frequency of the fundamental. (b) Find the second harmonic. (c) Find the third harmonic.

A steel piano wire without windings has a fundamental frequency of 200 Hz. When it is wound with copper wire, its linear mass density is doubled. What is its new fundamental frequency, assuming that the tension is unchanged?

What is the greatest length that an organ pipe can have in order that its fundamental note be in the audible range (20 to 20,000 Hz) if (a) the pipe is stopped at one end, and (b) it is open at both ends?

The wave function for a certain standing wave on a string that is fixed at both ends is given by where and are in centimeters and is in seconds. A standing wave can be considered as the superposition of two traveling waves. (a) What are the wavelength and frequency of the two traveling waves that make up the specified standing wave? (b) What is the speed of these waves on this string? (c) If the string is vibrating in its fourth harmonic, how long is it?

The wave function for a certain standing wave on a string that is fixed at both ends is given by A standing wave can be considered as the superposition of two traveling waves. (a) What are the speed and amplitude of the two traveling waves that result in the specified standing wave? (b) What is the distance between successive nodes on the string? (c) What is the shortest possible length of the string?

A 1.20-m-long pipe is stopped at one end. Near the open end, there is a loudspeaker that is driven by an audio oscillator whose frequency can be varied from 10.0 to 5000 Hz. (Neglect any end corrections.) (a) What is the lowest frequency of the oscillator that will produce resonance within the tube? (b) What is the highest frequency of the oscillator that will produce resonance within the tube? (c) How many different frequencies of the oscillator will produce resonance within the tube?

A 460-Hz tuning fork causes resonance in the tube depicted in Figure 16-33 when the length L of the air column above the water is 18.3 and 55.8 cm. (a) Find the speed of sound in air. (b) What is the end correction to adjust for the fact that the antinode does not occur exactly at the open end of the tube?

An organ pipe has a fundamental frequency of 440.0 Hz at What will the fundamental frequency of the pipe be if the temperature increases to (assuming the length of the pipe remains constant)? Would it be better to construct organ pipes from a material that expands substantially as the temperature increases, or should the pipes be made of material that maintains the same length at all normal temperatures?

According to theory, the end correction for a pipe is approximately where is the pipe diameter. Find the actual length of a pipe open at both ends that will produce a middle C (256 Hz) as its fundamental mode for pipes of diameter and 30.0 cm.

Assume a 40.0-cm-long violin string has a mass of 1.20 g and is vibrating in its fundamental mode* at a frequency of 500 Hz. (a) What is the wavelength of the standing wave on the string? (b) What is the tension in the string? (c) Where should you place your finger to increase the fundamental frequency to 650 Hz?

The G string on a violin is 30.0 cm long. When played without fingering, it vibrates in its fundamental mode* at a frequency of 196 Hz. The next higher notes on its C-major scale are A (220 Hz), B (247 Hz), C (262 Hz), and D (294 Hz). How far from the end of the string must a finger be placed to play each of these notes?

Astring that has a linear mass density of is under a tension of 360 N and is fixed at both ends. One of its resonance frequencies is 375 Hz. The next higher resonance frequency is 450 Hz. (a) What is the fundamental frequency of this string? (b) Which harmonics have the given frequencies? (c) What is the length of the string?

A string fixed at both ends has successive resonances with wavelengths of 0.54 m for the th harmonic and 0.48 m for the harmonic. (a) Which harmonics are these? (b) What is the length of the string?

The strings of a violin are tuned to the tones G, D, A, and E, which are separated by a fifth from one another. That is, and The distance between the bridge at the scroll and the bridge over the body, the two fixed points on each string, is 30.0 cm. The tension on the E string is 90.0 N. (a) What is the linear mass density of the E string? (b) To prevent distortion of the instrument over time, it is important that the tension on all strings be the same. Find the linear mass densities of the other strings.

On a cello, like most other stringed instruments, the positioning of the fingers by the player determines the fundamental frequencies of the strings. Suppose that one of the strings on a cello is tuned to play a middle C (262 Hz) when played at its full length. By what fraction must that string be shortened in order to play a note that is the interval of a third higher (namely, an E (330 Hz)? How about a fifth higher or a G (392 Hz)?

To tune your violin, you first tune the A string to the correct pitch of 440 Hz, and then you bow both it and an adjoining string simultaneously, all the while listening for beats. While bowing the A and E strings, you hear a beat frequency of 3.00 Hz and note that the beat frequency increases as the tension on the E string is increased. (The E string is to be tuned to 660 Hz.) (a) Why are beats produced by these two strings when bowed simultaneously? (b) What is the frequency of the E string vibration when the beat frequency is 3.00 Hz?

A 2.00-m-long string fixed at one end and free at the other end (the free end is fastened to the end of a long, light thread) is vibrating in its third harmonic with a maximum amplitude of 3.00 cm and a frequency 100 Hz. (a)Write the wave function for this vibration. (b)Write a function for the kinetic energy of a segment of the string of length at a point a distance from the fixed end, as a function of time At what times is this kinetic energy maximum? What is the shape of the string at these times? (c) Find the maximum kinetic energy of the string by integrating your expression for Part (b) over the total length of the string.

CONTEXT-RICH A commonly used physics experiment that examines resonances of transverse waves on a string is shown in Figure 16-34. A weight is attached to the end of a string draped over a pulley; the other end of the string is attached to a mechanical oscillator that moves up and down at a frequency that remains fixed throughout the demonstration. The length between the oscillator and the pulley is fixed, and the tension is equal to the gravitational force on the weight. For certain values of the tension, the string resonates. Assume the string does not stretch or shrink as the tension is varied. You are in charge of setting up this apparatus for a lecture demonstration. (a) Explain why only certain discrete values of the tension result in standing waves on the string. (b) Do you need to increase or decrease the tension to produce a standing wave with an additional antinode? Explain. (c) Prove your reasoning in Part (b) by showing that the values for the tension F for the nth standing-wave mode are given by and thus the is inversely proportional to (d) For your particular setup to fit onto the lecture table, you chose and Calculate how much tension is needed to produce each of the first three modes (standing waves) of the string.

Aguitar string is given a light pluck at its midpoint. Amicrophone on your computer detects the sound and a program on the computer determines that most of the subsequent sound consists of a 100-Hz tone accompanied by a bit of sound with a 300-Hz tone. What are the two dominant standing-wave modes on the string?

A tuning fork with natural frequency begins vibrating at time and is stopped after a time interval, The waveform of the sound at some later time is shown (Figure 16-35) as a function of Let be an estimate of the number of cycles in this waveform. (a) If is the length in space of this wave packet, what is the range in wave numbers of the packet? (b) Estimate the average value of the wavelength in terms of and (c) Estimate the average wave number in terms of and (d) If is the time it takes the wave packet to pass a point in space, what is the range in angular frequencies of the packet? (e) Express in terms of and ( f) The number is uncertain by about cycle. Use Figure 16-35 to explain why. ( g) Show that the uncertainty in the wave number due to the uncertainty in N is 2p>x. SSM

A 35-m-long string has a linear mass density of and is under a tension of 18 N. Find the frequencies of the lowest four harmonics (a) if the string is fixed at both ends, and (b) if the string is fixed at one end and free at the other. (That is, if the free end is attached to a long string of negligible mass.)

CONTEXT-RICH, ENGINEERING APPLICATION Working for a small gold mining company, you stumble across an abandoned mine shaft that, because of decaying wood shoring, looks too dangerous to explore in person. To measure its depth, you employ an audio oscillator of variable frequency. You determine that successive resonances are produced at frequencies of 63.58 and 89.25 Hz. Estimate the depth of the shaft.

A5.00-m-long string that is fixed at one end and attached to a long string of negligible mass at the other end is vibrating in its fifth harmonic, which has a frequency of 400 Hz. The amplitude of the motion at each antinode is 3.00 cm. (a) What is the wavelength of this wave? (b) What is the wave number? (c) What is the angular frequency? (d) Write the wave function for this standing wave.

The wave function for a standing wave on a string is described by where and are in meters and is in seconds. Determine the maximum displacement and maximum speed of a point on the string at (a) (b) (c) and (d)

A 2.5-m-long string that has a mass of 0.10 kg is fixed at both ends and is under a tension of 30 N. When the harmonic is excited, there is a node 0.50 m from one end. (a) What is (b) What are the frequencies of the first three harmonics of this string?

An organ pipe is such that under normal conditions its fundamental frequency is 220 Hz. It is placed in an atmosphere of sulfur hexafluoride at the same temperature and pressure. The molar mass of air is and the molar mass of is What is the fundamental frequency of the organ pipe when it is in an atmosphere of

During a lecture demonstration of standing waves, one end of a string is attached to a device that vibrates at 60 Hz and produces transverse waves of that frequency on the string. The other end of the string passes over a pulley, and the tension is varied by attaching weights to that end. The string has approximate nodes next to both the vibrating device and the pulley. (a) If the string has a linear mass density of and is 2.5 m long from the vibrating device to the pulley, what must be the tension for the string to vibrate in its fundamental mode? (b) Find the tension necessary for the string to vibrate in its second, third, and fourth harmonics.

Three successive resonance frequencies in an organ pipe are 1310, 1834, and 2358 Hz. (a) Is the pipe closed at one end or open at both ends? (b) What is the fundamental frequency? (c) What is the effective length of the pipe?

During an experiment studying the speed of sound in air using an audio oscillator and a tube that is open at one end and stopped at the other end, a particular resonant frequency is found to have nodes roughly 6.94 cm apart. The oscillators frequency is increased, and the next resonant frequency found has nodes 5.40 cm apart. (a) What are the two resonant frequencies? (b) What is the fundamental frequency? (c) Which harmonics are these two modes? The speed of sound is

A standing wave on a rope is represented by the wave function where and are in meters and is in seconds. (a) Write wave functions for two traveling waves that, when superimposed, produce this standing-wave pattern. (b) What is the distance between the nodes of the standing wave? (c) What is the maximum speed of the rope at (d) What is the maximum acceleration of the rope at

SPREADSHEET Two traveling-wave pulses on a string are represented by the wave functions where is in meters and is in seconds. (a) Using a spreadsheet program or graphing calculator, make a separate graph of each wave function as a function of x at t _ 0 and again at t _ 1.0 s, and make your plot for (b) Graph the resultant wave function at at and at

Three waves that have the same frequency, wavelength, and amplitude are traveling along the x axis. The three waves are described by the following wave functions: \(y_1(x, t)=(5.00 \mathrm{~cm})\) \(\sin \left(k x-\omega t-\frac{1}{3} \pi\right), y_2(x, t)=(5.00 \mathrm{~cm}) \sin (k x-\omega t)\), and \(y_3(x, t)=\) \((5.00 \mathrm{~cm}) \sin \left(k x-\omega t+\frac{1}{3} \pi\right)\), where x is in meters and t is in seconds. The resultant wave function is given by \(y_3(x, t)=\) \(A \sin (k x-\omega t+\delta)\). What are the values of A and \(\delta ?\)

A harmonic pressure wave produced by a distant source is traveling through your vicinity, and the wavefronts that travel through your vicinity are vertical planes. Let the direction be to the east and the direction be toward the north. The wave function for the wave is Show that the direction in which the wave is traveling makes an angle with the direction and that the wave speed is

The speed of sound in air is proportional to the square root of the absolute temperature (Equation 15-5). (a) Show that if the air temperature changes by a small amount, the fractional change in the fundamental frequency of an organ pipe is approximately equal to half the fractional change in the absolute temperature. That is, show that where is the frequency at absolute temperature and is the change in frequency when the temperature changes by (Ignore any change in the length of the pipe due to thermal expansion.) (b) Suppose that an organ pipe that is stopped at one end has a fundamental frequency of 200.0 Hz when the temperature is Use the approximate result from Part (a) to determine the pipes fundamental frequency when the temperature is (c) Compare your Part (b) result to what you would get using exact calculations. (Ignore any change in the length of the pipe due to thermal expansion.)

The pipe in Figure 16-36 is kept filled with natural gas . The pipe is punctured by a line of small holes 1.00 cm apart down its entire 2.20-m length. A speaker forms the closure on one end of the pipe, and a solid piece of metal closes the other end. What frequency is being played in this picture? The speed of sound in low-pressure methane at room temperature is about 460 m>s.

CONTEXT-RICH Assume that your clarinet is entirely filled with helium and that before you start to play you fill your lungs with helium. You pick up the clarinet and play it as though you were trying to play a B-flat, which has a frequency of 277 Hz. The frequency of 277 Hz is the natural resonance frequency of this clarinet with all finger holes closed and when filled with air. What frequency do you actually hear?

A 2.00-m-long wire that is fixed at both ends is vibrating in its fundamental mode. The tension in the wire is 40.0 N and the mass of the wire is 0.100 kg. At the midpoint of the wire, the am plitude is 2.00 cm. (a) Find the maximum kinetic energy of the wire. (b) What is the kinetic energy of the wire at the instant the transverse displacement is given by where is in meters if is in meters, for (c) For what value of is the average value of the kinetic energy per unit length the greatest? (d) For what value of does the elastic potential energy per unit length have its maximum value?

SPREADSHEET In principle, a wave with almost any arbitrary shape can be expressed as a sum of harmonic waves of different frequencies. (a) Consider the function defined by Write a spreadsheet program to calculate this series using a finite number of terms, and make three graphs of the function in the range to To create the first graph, for each value of that you plot, approximate the sum from to with the first term of the sum. To create the second and third graphs, use only the first five terms and the first ten terms, respectively. This function is sometimes called the square wave. (b) What is the relation between this function and Leibnitzs series for

SPREADSHEET Write a spreadsheet program to calculate and graph the function for Use only the first 25 terms in the sum for each value of that you plot.

SPREADSHEET If you clap your hands at the end of a long, cylindrical tube, the echo you hear back will not sound like the handclap; instead, you will hear what sounds like a whistle, initially at a very high frequency, but descending rapidly down to almost nothing. This culvert whistler is easily explained if you think of the sound from the clap as a single compression radiating outward from the hands. The echoes of the handclap arriving at your ear have traveled along different paths through the tube, as shown in Figure 16-37. The first echo to arrive travels straight down and straight back along the tube, while the second echo reflects once off of the center of the tube going out, and again going back, the third echo reflects twice at points and of the distance, and so on. The tone of the sound you hear reflects the frequency at which these echoes reach your ears. (a) Show that the time delay between the echo and the echo is where is the speed of sound, is the length of the tube, and is the tubes radius. (b) Using a spreadsheet program or graphing calculator, graph versus for (These values are the approximate length and radius of the long tube in the San Francisco Exploratorium.) Go to at least (c) From your graph, explain why the frequency decreases over time. What are the highest and lowest frequencies you will hear in the whistler?