Ch 16: Traveling Waves

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 16: Traveling Waves

Problem 71

Chapter 16, Problem 71

A battery-powered siren emits 0.50 W of sound power at 1000 Hz. It is dropped from 100 m directly over your head on a 20°C day. 4.0 s after it is released, what are (a) the frequency and (b) the sound intensity level you hear?

Verified step by step guidance

Determine the speed of sound in air at 20°C using the formula:

v = 331 + 0.6 T

, where

T

is the temperature in Celsius. Substitute

T = 20

to find the speed of sound.

Calculate the distance the siren has fallen after 4.0 s using the kinematic equation:

y = h - \frac{1}{2} g t ²

, where

h

is the initial height (100 m),

g

is the acceleration due to gravity (9.8 m/s²), and

t

is the time (4.0 s).

Determine the relative velocity of the siren with respect to the observer using the formula:

v = g t

, where

g

is the acceleration due to gravity and

t

is the time (4.0 s).

Apply the Doppler effect formula to calculate the observed frequency:

f' = f \frac{v}{+}

_ov-v_s, where

f

is the source frequency (1000 Hz),

v

is the speed of sound,

v

_o is the observer's velocity (0 m/s), and

v

_s is the source velocity (calculated in step 3).

Calculate the sound intensity level using the formula:

β = 10 \log (\frac{I}{I}

_o), where

I

is the intensity of the sound at the observer's location and

I

_o is the reference intensity (

1 x 10 - 12

W/m²). To find

I

, use the formula:

I = \frac{P}{4}

, where

P

is the power of the source (0.50 W) and

r

is the distance from the source to the observer (calculated in step 2).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Doppler Effect

The Doppler Effect describes the change in frequency or wavelength of a wave in relation to an observer moving relative to the wave source. In this scenario, as the siren falls, the frequency of the sound waves it emits will appear to change due to its motion towards the observer, affecting the perceived pitch of the sound.

Sound Intensity and Intensity Level

Sound intensity is the power per unit area carried by a sound wave, typically measured in watts per square meter (W/m²). The sound intensity level, measured in decibels (dB), quantifies how loud a sound is perceived, using a logarithmic scale relative to a reference intensity. This concept is crucial for determining how the sound from the siren is perceived at a distance.

Free Fall Motion

Free fall motion refers to the motion of an object under the influence of gravity alone, without any air resistance. In this case, the siren is dropped from a height of 100 m, and understanding its acceleration due to gravity (approximately 9.81 m/s²) is essential for calculating the time it takes to reach the observer and the changes in sound frequency and intensity during its fall.

Recommended video:

Guided course

08:36

Vertical Motion & Free Fall

Related Practice

Textbook Question

An avant-garde composer wants to use the Doppler effect in his new opera. As the soprano sings, he wants a large bat to fly toward her from the back of the stage. The bat will be outfitted with a microphone to pick up the singer's voice and a loudspeaker to rebroadcast the sound toward the audience. The composer wants the sound the audience hears from the bat to be, in musical terms, one half-step higher in frequency than the note they are hearing from the singer. Two notes a half-step apart have a frequency ratio of 2^1/12 = 1.059. With what speed must the bat fly toward the singer?

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Textbook Question

A physics professor demonstrates the Doppler effect by tying a 600 Hz sound generator to a 1.0-m-long rope and whirling it around her head in a horizontal circle at 100 rpm. What are the highest and lowest frequencies heard by a student in the classroom?

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Textbook Question

A 1000 Hz sound wave traveling through 20°C air causes the pressure to oscillate around atmospheric pressure by ±0.050%. What is the maximum speed of an oscillating air molecule? Give your answer in mm/s.

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Textbook Question

A string that is under 50.0 N of tension has linear density 5.0 g/m. A sinusoidal wave with amplitude 3.0 cm and wavelength 2.0 m travels along the string. What is the maximum speed of a particle on the string?

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Textbook Question

LASIK eye surgery uses pulses of laser light to shave off tissue from the cornea, reshaping it. A typical LASIK laser emits a 1.0-mm-diameter laser beam with a wavelength of 193 nm. Each laser pulse lasts 15 ns and contains 1.0 mJ of light energy. During the very brief time of the pulse, what is the intensity of the light wave?

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Textbook Question

Some modern optical devices are made with glass whose index of refraction changes with distance from the front surface. FIGURE P16.72 shows the index of refraction as a function of the distance into a slab of glass of thickness L. The index of refraction increases linearly from n₁ at the front surface to n₂ at the rear surface. Evaluate your expression for a 1.0-cm-thick piece of glass for which n₁ = 1.50 and n₂ = 1.60.