Problems

Consider a sound wave modeled with the equation$s\left(x,t\right)=4.00\text{nm}$
$\text{cos}\left(3.66{\text{m}}^{\mathrm{-1}}x-1256{\text{s}}^{\mathrm{-1}}t\right).$ What is the maximum displacement, the wavelength, the frequency, and the speed of the sound wave?

Consider a diagnostic ultrasound of frequency 5.00 MHz that is used to examine an irregularity in soft tissue. (a) What is the wavelength in air of such a sound wave if the speed of sound is 343 m/s? (b) If the speed of sound in tissue is 1800 m/s, what is the wavelength of this wave in tissue?

A sound wave is modeled with the wave function $\text{Δ}P=1.20\text{Pa}\text{sin}\left(kx-6.28\times {10}^4{\text{s}}^{\mathrm{-1}}t\right)$ and the sound wave travels in air at a speed of $v=343.00\text{m/s}\text{.}$ (a) What is the wave number of the sound wave? (b) What is the value for $\text{Δ}P\left(3.00\text{m},20.00\text{s}\right)$?

A speaker is placed at the opening of a long horizontal tube. The speaker oscillates at a frequency f, creating a sound wave that moves down the tube. The wave moves through the tube at a speed of $v=340.00\text{m/s}\text{.}$ The sound wave is modeled with the wave function $s\left(x,t\right)=s_{\text{max}}\text{cos}\left(kx-\omega t+\varphi \right).$ At time $t=0.00\text{s}$, an air molecule at $x=3.5\text{m}$ is at the maximum displacement of 7.00 nm. At the same time, another molecule at $x=3.7\text{m}$ has a displacement of 3.00 nm. What is the frequency at which the speaker is oscillating?

A sound wave produced by an ultrasonic transducer, moving in air, is modeled with the wave equation $s\left(x,t\right)=4.50\text{nm}\text{cos}(9.15\times {10}^4{\text{m}}^{\mathrm{-1}}x-$
$2\pi \left(5.00\text{MHz}\right)t).$ The transducer is to be used in nondestructive testing to test for fractures in steel beams. The speed of sound in the steel beam is $v=5950\text{m/s}\text{.}$ Find the wave function for the sound wave in the steel beam.

Bats use sound waves to catch insects. Bats can detect frequencies up to 100 kHz. If the sound waves travel through air at a speed of $v=343\text{m/s,}$ what is the wavelength of the sound waves?

Consider the graph shown below of a compression wave. Shown are snapshots of the wave function for $t=0.000\text{s}$ (blue) and $t=0.005\text{s}$ (orange). What are the wavelength, maximum displacement, velocity, and period of the compression wave?

A guitar string oscillates at a frequency of 100 Hz and produces a sound wave. (a) What do you think the frequency of the sound wave is that the vibrating string produces? (b) If the speed of the sound wave is
$v=343\text{m/s,}$ what is the wavelength of the sound wave?

What frequency sound has a 0.10-m wavelength when the speed of sound is 340 m/s?

(a) What is the speed of sound in a medium where a 100-kHz frequency produces a 5.96-cm wavelength? (b) Which substance in Table 17.1 is this likely to be?

Air temperature in the Sahara Desert can reach $56.0\text{°}\text{C}$ (about $134\text{°}\text{F}$). What is the speed of sound in air at that temperature?

A sonar echo returns to a submarine 1.20 s after being emitted. What is the distance to the object creating the echo? (Assume that the submarine is in the ocean, not in fresh water.)

Ultrasonic sound waves are often used in methods of nondestructive testing. For example, this method can be used to find structural faults in a steel I-beams used in building. Consider a 10.00 meter long, steel I-beam with a cross-section shown below. The weight of the I-beam is 3846.50 N. What would be the speed of sound through in the I-beam? $\left(Y_{\text{steel}}=200\text{GPa},{\beta }_{\text{steel}}=159\text{GPa}\right)$.

During a 4th of July celebration, an M80 firework explodes on the ground, producing a bright flash and a loud bang. The air temperature of the night air is $T_{\text{F}}=95.00\text{°}\text{F}.$ Two observers see the flash and hear the bang. The first observer notes the time between the flash and the bang as 0.10 second. The second observer notes the difference as 0.15 seconds. The line of sight between the two observers meet at a right angle as shown below. What is the distance $\text{Δ}x$ between the two observers?

Suppose a bat uses sound echoes to locate its insect prey, 3.00 m away. (See Figure 17.6.) (a) Calculate the echo times for temperatures of $5.00\text{°}\text{C}$ and $35.0\text{°}\text{C}\text{.}$ (b) What percent uncertainty does this cause for the bat in locating the insect? (c) Discuss the significance of this uncertainty and whether it could cause difficulties for the bat. (In practice, the bat continues to use sound as it closes in, eliminating most of any difficulties imposed by this and other effects, such as motion of the prey.)

The warning tag on a lawn mower states that it produces noise at a level of 91.0 dB. What is the intensity of this sound in watts per meter squared?

What intensity level does the sound in the preceding problem correspond to?

What is the decibel level of a sound that is twice as intense as a 90.0-dB sound? (b) What is the decibel level of a sound that is one-fifth as intense as a 90.0-dB sound?

People with good hearing can perceive sounds as low as −8.00 dB at a frequency of 3000 Hz. What is the intensity of this sound in watts per meter squared?

Ten cars in a circle at a boom box competition produce a 120-dB sound intensity level at the center of the circle. What is the average sound intensity level produced there by each stereo, assuming interference effects can be neglected?

If a sound intensity level of 0 dB at 1000 Hz corresponds to a maximum gauge pressure (sound amplitude) of ${10}^{\mathrm{-9}}\text{atm}$, what is the maximum gauge pressure in a 60-dB sound? What is the maximum gauge pressure in a 120-dB sound?

Sound is more effectively transmitted into a stethoscope by direct contact rather than through the air, and it is further intensified by being concentrated on the smaller area of the eardrum. It is reasonable to assume that sound is transmitted into a stethoscope 100 times as effectively compared with transmission though the air. What, then, is the gain in decibels produced by a stethoscope that has a sound gathering area of $15.0{\text{cm}}^2$, and concentrates the sound onto two eardrums with a total area of $0.900{\text{cm}}^2$ with an efficiency of $40.0\text{\%}$?

The factor of 10^-12 in the range of intensities to which the ear can respond, from threshold to that causing damage after brief exposure, is truly remarkable. If you could measure distances over the same range with a single instrument and the smallest distance you could measure was 1 mm, what would the largest be?

Can you tell that your roommate turned up the sound on the TV if its average sound intensity level goes from 70 to 73 dB?

A person has a hearing threshold 10 dB above normal at 100 Hz and 50 dB above normal at 4000 Hz. How much more intense must a 100-Hz tone be than a 4000-Hz tone if they are both barely audible to this person?

What is the length of a tube that has a fundamental frequency of 176 Hz and a first overtone of 352 Hz if the speed of sound is 343 m/s?

Calculate the first overtone in an ear canal, which resonates like a 2.40-cm-long tube closed at one end, by taking air temperature to be $37.0\text{°}\text{C}$. Is the ear particularly sensitive to such a frequency? (The resonances of the ear canal are complicated by its nonuniform shape, which we shall ignore.)

A 4.0-m-long pipe, open at one end and closed at one end, is in a room where the temperature is $T=22\text{°}\text{C}\text{.}$ A speaker capable of producing variable frequencies is placed at the open end and is used to cause the tube to resonate. (a) What is the wavelength and the frequency of the fundamental frequency? (b) What is the frequency and wavelength of the first overtone?

A nylon guitar string is fixed between two lab posts 2.00 m apart. The string has a linear mass density of $\mu =7.20\text{g/m}$ and is placed under a tension of 160.00 N. The string is placed next to a tube, open at both ends, of length L. The string is plucked and resonates at the fundamental frequency while the tube resonates at the $n=3$ mode. The speed of sound is 343 m/s. What is the length of the tube?

Students in a physics lab are asked to find the length of an air column in a tube closed at one end that has a fundamental frequency of 256 Hz. They hold the tube vertically and fill it with water to the top, then lower the water while a 256-Hz tuning fork is rung and listen for the first resonance. (a) What is the air temperature if the resonance occurs for a length of 0.336 m? (b) At what length will they observe the second resonance (first overtone)?

What are the first three overtones of a bassoon that has a fundamental frequency of 90.0 Hz? It is open at both ends. (The overtones of a real bassoon are more complex than this example, because its double reed makes it act more like a tube closed at one end.)

What length should an oboe have to produce a fundamental frequency of 110 Hz on a day when the speed of sound is 343 m/s? It is open at both ends.

An organ pipe $\left(L=3.00\text{m}\right)$ is closed at both ends. Compute the wavelengths and frequencies of the first three modes of resonance. Assume the speed of sound is$v=343.00\text{m/s}\text{.}$

A sound wave of a frequency of 2.00 kHz is produced by a string oscillating in the $n=6$ mode. The linear mass density of the string is $\mu =0.0065\text{kg/m}$ and the length of the string is 1.50 m. What is the tension in the string?

A student holds an 80.00-cm lab pole one quarter of the length from the end of the pole. The lab pole is made of aluminum. The student strikes the lab pole with a hammer. The pole resonates at the lowest possible frequency. What is that frequency?

By what fraction will the frequencies produced by a wind instrument change when air temperature goes from $10.0\text{°}\text{C}$ to $30.0\text{°}\text{C}$? That is, find the ratio of the frequencies at those temperatures.

What beat frequencies result if a piano hammer hits three strings that emit frequencies of 127.8, 128.1, and 128.3 Hz?

Two identical strings, of identical lengths of 2.00 m and linear mass density of $\mu =0.0065\text{kg/m,}$ are fixed on both ends. String A is under a tension of 120.00 N. String B is under a tension of 130.00 N. They are each plucked and produce sound at the $n=10$ mode. What is the beat frequency?

A string with a linear mass density of $\mu =0.0062\text{kg/m}$ is stretched between two posts 1.30 m apart. The tension in the string is 150.00 N. The string oscillates and produces a sound wave. A 1024-Hz tuning fork is struck and the beat frequency between the two sources is 52.83 Hz. What are the possible frequency and wavelength of the wave on the string?

The middle C hammer of a piano hits two strings, producing beats of 1.50 Hz. One of the strings is tuned to 260.00 Hz. What frequencies could the other string have?

Twin jet engines on an airplane are producing an average sound frequency of 4100 Hz with a beat frequency of 0.500 Hz. What are their individual frequencies?

(a) What frequency is received by a person watching an oncoming ambulance moving at 110 km/h and emitting a steady 800-Hz sound from its siren? The speed of sound on this day is 345 m/s. (b) What frequency does she receive after the ambulance has passed?

What frequency is received by a mouse just before being dispatched by a hawk flying at it at 25.0 m/s and emitting a screech of frequency 3500 Hz? Take the speed of sound to be 331 m/s.

A commuter train blows its 200-Hz horn as it approaches a crossing. The speed of sound is 335 m/s. (a) An observer waiting at the crossing receives a frequency of 208 Hz. What is the speed of the train? (b) What frequency does the observer receive as the train moves away?

Two eagles fly directly toward one another, the first at 15.0 m/s and the second at 20.0 m/s. Both screech, the first one emitting a frequency of 3200 Hz and the second one emitting a frequency of 3800 Hz. What frequencies do they receive if the speed of sound is 330 m/s?

An ambulance with a siren $\left(f=1.00\text{kHz}\right)$ blaring is approaching an accident scene. The ambulance is moving at 70.00 mph. A nurse is approaching the scene from the opposite direction, running at $v_o=7.00\text{m/s}\text{.}$ What frequency does the nurse observe? Assume the speed of sound is $v=343.00\text{m/s}\text{.}$

What is the minimum speed at which a source must travel toward you for you to be able to hear that its frequency is Doppler shifted? That is, what speed produces a shift of $0.300\text{\%}$ on a day when the speed of sound is 331 m/s?

A jet flying at an altitude of 8.50 km has a speed of Mach 2.00, where the speed of sound is $v=340.00\text{m/s}\text{.}$ How long after the jet is directly overhead, will a stationary observer hear a sonic boom?

A plane is flying at Mach 1.2, and an observer on the ground hears the sonic boom 15.00 seconds after the plane is directly overhead. What is the altitude of the plane? Assume the speed of sound is $v_{\text{w}}=343.00\text{m/s}\text{.}$

A speaker is placed at the opening of a long horizontal tube. The speaker oscillates at a frequency of f, creating a sound wave that moves down the tube. The wave moves through the tube at a speed of $v=340.00\text{m/s}\text{.}$ The sound wave is modeled with the wave function $s\left(x,t\right)=s_{\text{max}}\text{cos}\left(kx-\omega t+\varphi \right)$. At time $t=0.00\text{s}$, an air molecule at $x=2.3\text{m}$ is at the maximum displacement of 6.34 nm. At the same time, another molecule at $x=2.7\text{m}$ has a displacement of 2.30 nm. What is the wave function of the sound wave, that is, find the wave number, angular frequency, and the initial phase shift?