Waves/Acoustics

The sound from a trumpet radiates uniformly in all directions in 20°C air. At a distance of 5.00 m from the trumpet the sound intensity level is 52.0 dB. The frequency is 587 Hz. (a) What is the pressure amplitude at this distance? (b) What is the displacement amplitude? (c) At what distance is the sound intensity level 30.0 dB?

A person is playing a small flute $\mathbf{10}\mathbf{.}\mathbf{75}\mathbf{ }\mathbf{cm}$  long, open at one end and closed at the other, near a taut string having a fundamental frequency of  $\mathbf{600}\mathbf{.}\mathbf{0}\mathbf{Hz}$ . If the speed of sound is $\mathbf{344}\mathbf{.}\mathbf{0}\mathbf{m}/\mathbf{s}$ , for which harmonics of the flute will the string resonate? In each case, which harmonic of the string is in resonance?

A uniform 165-Nbar is supported horizontally by two identical wires A and B (Fig. P16.62). A small 185-N cube of lead is placed three fourths of the way from A to B. The wires are each 75.0 cm long and have a mass of 5.50 g. If both of them are simultaneously plucked at the centre, what is the frequency of the beats that they will produce when vibrating in their fundamental?

An organ pipe has two successive harmonics with frequencies 1372 and 1764 Hz. (a) Is this an open or a stopped pipe? Explain. (b) What two harmonics are these? (c) What is the length of the pipe?

The frequency of the note F4 is 349 Hz. (a) If an organ pipe is open at one end and closed at the other, what length must it have for its fundamental mode to produce this note at 20.0°C? (b) At what air temperature will the frequency be 370 Hz, corresponding to a rise in pitch from F to F-sharp? (Ignore the change in length of the pipe due to the temperature change.)

Two identical loudspeakers are located at points A and B, 2.00 m apart. The loudspeakers are driven by the same amplifier and produce sound waves with a frequency of 784 Hz. Take the speed of sound in air to be 344 m/s. A small microphone is moved out from point B along a line perpendicular to the line connecting A and B (line BC in Fig. P16.65). (a) At what distances from B will there be destructive interference? (b) At what distances from B will there be constructive interference? (c) If the frequency is made low enough, there will be no positions along the line BC at which destructive interference occurs. How low must the frequency be for this to be the case?

A bat flies toward a wall, emitting a steady sound of frequency 1.70 kHz. This bat hears its own sound plus the sound reflected by the wall. How fast should the bat fly in order to hear a beat frequency of 8.00 Hz?

The sound source of a ship’s sonar system operates at a frequency of 18.0 kHz. The speed of sound in water (assumed to be at a uniform 20°C) is 1482 m>s. (a) What is the wavelength of the waves emitted by the source? (b) What is the difference in frequency between the directly radiated waves and the waves reflected from a whale traveling directly toward the ship at 4.95 m/s? The ship is at rest in the water.

BIO Ultrasound in Medicine. A 2.00-MHz soundwave travels through a pregnant woman’s abdomen and is reflected from the fatal heart wall of her unborn baby. The heart wall is moving toward the sound receiver as the heart beats. The reflected sound is then mixed with the transmitted sound, and 72 beats per second are detected. The speed of sound in body tissue is 1500 m/s. Calculate the speed of the fatal heart wall at the instant this measurement is made.

BIO Horseshoe bats (genus Rhinolophus) emit sounds from their nostrils and then listen to the frequency of the sound reflected from their prey to determine the prey’s speed. (The “horseshoe” that gives the bat its name is a depression around the nostrils that acts like a focusing mirror, so that the bat emits sound in a narrow beam like a flashlight.) A Rhinolophus flying at speed v_bat  emits sound of frequency f_bat ; the sound it hears reflected from an insect flying toward it has a higher frequency f_refl .(a) Show that the speed of the insect is

${\mathrm{v}}_{\mathrm{in} \mathrm{sect}}=\frac{\mathrm{v}\left({\mathrm{f}}_{\mathrm{refl}}\left(\mathrm{v}-{\mathrm{v}}_{\mathrm{bat}}\right)-{\mathrm{f}}_{\mathrm{bat}}\left(\mathrm{v}+{\mathrm{v}}_{\mathrm{bat}}\right)\right)}{{\mathrm{f}}_{\mathrm{refl}}\left(\mathrm{v}-{\mathrm{v}}_{\mathrm{bat}}\right)+{\mathrm{f}}_{\mathrm{bat}}\left(\mathrm{v}+{\mathrm{v}}_{\mathrm{bat}}\right)}$$cv$

where v is the speed of sound. (b) If f_bat = 80.7 kHz,  f_refl =83.5 kHz, and  v_bat = 3.9 m>s, calculate the speed of the insect.

CP A police siren of frequency ${\mathrm{f}}_{\mathrm{siren}}$ is attached to a vibrating platform. The platform and siren oscillate up and down in simple harmonic motion with amplitude ${\mathbf{A}}_{\mathbf{P}}$and frequency ${\mathbf{f}}_{\mathbf{p}}$.(a) Find the maximum and minimum sound frequencies that you would hear at a position directly above the siren. (b) At what point in the motion of the platform is the maximum frequency heard? The minimum frequency? Explain.

A turntable 1.50 m in diameter rotates at 75 rpm. Two speakers, each giving off sound of wavelength 31.3 cm, are attached to the rim of the table at opposite ends of a diameter. A listener stands in front of the turntable. (a) What is the greatest beat frequency the listener will receive from this system? (b) Will the listener be able to distinguish individual beats?

A long, closed cylindrical tank contains a diatomic gas that is maintained at a uniform temperature that can be varied. When you measure the speed of sound v in the gas as a function of the temperature T of the gas, you obtain these results:   

(a) Explain how you can plot these results so that the graph will be well fit by a straight line. Construct this graph and verify that the plotted points do lie close to a straight line. (b) Because the gas is diatomic, g = 1.40. Use the slope of the line in part (a) to calculate M, the molar mass of the gas. Express M in grams/mole. What type of gas is in the tank?

A long tube contains air at a pressure of 1.00 atm and a temperature of 77.0°C. The tube is open at one end and closed at the other by a movable piston. A tuning fork that vibrates with a frequency of 500 Hz is placed near the open end. Resonance is produced when the piston is at distances 18.0 cm, 55.5 cm, and 93.0 cm from the open end. (a) From these values, what is the speed of sound in air at 77.0°C? (b) From the result of part (a), what is the value of g? (c) These results show that a displacement antinode is slightly outside the open end of the tube. How far outside is it?

DATA Supernova! (a) Equation (16.30) can be written as ${\mathbf{f}}_{\mathbf{r}}\mathbf{=}{\mathbf{f}}_{\mathbf{s}}{\left(1‐\frac{v}{c}\right)}^{\frac{\mathbf{1}}{\mathbf{2}}}{\left(1+\frac{v}{c}\right)}^{\frac{\mathbf{1}}{\mathbf{2}}}$

where c is the speed of light in vacuum $\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{\times }{\mathbf{10}}^{\mathbf{8}}\mathbf{m}\mathbf{/}\mathbf{s}$. Most objects move much slower than this (v/c is very small), so calculations made with Eq. (16.30) must be done carefully to avoid rounding errors. Use the binomial theorem to show that if v, Eq. (16.30) approximately reduces to ${\mathbf{f}}_{\mathbf{r}}\mathbf{=}{\mathbf{f}}_{\mathbf{s}}\left(1-\left(\frac{v}{c}\right)\right)$. (b) The gas cloud known as the Crab Nebula can be seen with even a small telescope. It is the remnant of a supernova, a cataclysmic explosion of a star. (The explosion was seen on the earth on July 4,1054 C.E.) Its streamers glow with the characteristic red colour of heated hydrogen gas. In a laboratory on the earth, heated hydrogen produces red light with frequency  $\mathbf{4}\mathbf{.}\mathbf{568}\mathbf{\times }{\mathbf{10}}^{\mathbf{14}}\mathbf{Hz}$; the red light received from streamers in the Crab Nebula that are pointed toward the earth has frequency $\mathbf{4}\mathbf{.}\mathbf{568}\mathbf{\times }{\mathbf{10}}^{\mathbf{14}}\mathbf{Hz}$ Estimate the speed with which the outer edges of the Crab Nebula are expanding. Assume that the speed of the centre of the nebula relative to the earth is negligible. (c) Assuming that the expansion speed of the Crab Nebula has been constant since the supernova that produced it, estimate the diameter of the Crab Nebula. Give your answer in meters and in light-years. (d) The angular diameter of the Crab Nebula as seen from the earth is about 5 arc-minutes $\mathbf{(}\mathbf{1}\mathbf{arc}\mathbf{‐}\mathbf{min}\frac{\mathbf{1}}{\mathbf{60}}\mathbf{degree}\mathbf{)}$ Estimate the distance (in light-years) to the Crab Nebula, and estimate the year in which the supernova actually, took place.

Figure P16.75 shows the pressure fluctuation p of a non-sinusoidal sound wave as a function of x for t = 0. The wave is traveling in the +x-direction. (a) Graph the pressure fluctuation p as a function of t for x = 0. Show at least two cycles of oscillation. (b) Graph the displacement y in this sound wave as a function of x at t = 0. At x = 0, the displacement at t = 0 is zero. Show at least two wavelengths of the wave. (c) Graph the displacement y as a function of t for x = 0. Show at least two cycles of oscillation. (d) Calculate the maximum velocity and the maximum acceleration of an element of the air through which this sound wave is traveling. (e) Describe how the cone of a loudspeaker must move as a function of time to produce the sound wave in this problem.

Longitudinal Waves on a Spring. A long spring such as a ${\mathbf{Slinky}}^{\mathbf{TM}}$ is often used to demonstrate longitudinal waves. (a) Show that if a spring that obeys Hooke’s law has mass m, length L, and force constant k′, the speed of longitudinal waves on the spring is  $\mathbf{v}\mathbf{=}\mathbf{L}\sqrt{\mathbf{k}\mathbf{\text{'}}}\mathbf{m}$

 (see Section 16.2). (b) Evaluate v for a spring with m = 0.250 kg, L = 2.00 m, and k′ = 1.50 N/m.

If the deepest structure you wish to image is $\mathbf{10}\mathbf{.}\mathbf{0}\mathbf{cm}$ from the transducer, what is the maximum number of pulses per second that can be emitted? (a) $\mathbf{3850}$ ; (b) $\mathbf{7700}$ ; (c) $\mathbf{15400}$ ; (d) $\mathbf{1}\mathbf{,}\mathbf{000}\mathbf{,}\mathbf{000}$ .

After a beam passes through $\mathbf{10}\mathbf{ }\mathbf{cm}$ of tissue, what is the beam’s intensity as a fraction of its initial intensity from the transducer?

(a) $\mathbf{1}\mathbf{\times }{\mathbf{10}}^{-11}$; (b)$\mathbf{0}\mathbf{.}\mathbf{001}$ ; (c) $\mathbf{0}\mathbf{.}\mathbf{01}$; (d)   $\mathbf{0}\mathbf{.}\mathbf{1}$

 Because the speed of ultrasound in bone is about twice the speed in soft tissue, the distance to a structure that lies beyond a bone can be measured incorrectly. If a beam passes through $\mathbf{4}\mathbf{ }\mathit{c}\mathit{m}$ of tissue, then $\mathbf{2}\mathbf{ }\mathit{c}\mathit{m}$ of bone, and then another $\mathbf{1}\mathbf{ }\mathit{c}\mathit{m}$ of tissue before echoing off a cyst and returning to the transducer, what is the difference between the true distance to the cyst and the distance that is measured by assuming the speed is always $\mathbf{1540}\mathbf{ }\mathit{m}\mathbf{/}\mathit{s}$? Compared with the measured distance, the structure is actually

(a) $\mathbf{1}\mathit{c}\mathit{m}$ farther

(b) $\mathbf{2}\mathbf{ }\mathit{c}\mathit{m}$ farther

(c) $\mathbf{1}\mathbf{ }\mathit{c}\mathit{m}$closer

(d) $\mathbf{2}\mathbf{ }\mathit{c}\mathit{m}$closer

In some applications of ultrasound, such as its use on cranial tissues, large reflections from the surrounding bones can produce standing waves. This is of concern because the large pressure amplitude in an antinode can damage tissues. For a frequency of $\mathbf{1}\mathbf{.}\mathbf{00}\mathbf{MHz}$ , what is the distance between antinodes in tissue?

For cranial ultrasound, why is it advantageous to use frequencies in the $\mathbf{kHz}$  range rather than the $\mathbf{MHz}$ range?