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 17a

Chapter 16, Problem 17a

What is the frequency of an electromagnetic wave that has the same wavelength as a 2.5 kHz sound wave in water?

Verified step by step guidance

Step 1: Understand the problem. The goal is to find the frequency of an electromagnetic wave that has the same wavelength as a sound wave in water. The relationship between wavelength, frequency, and speed is given by the formula:

v = f λ

, where

v

is the speed,

f

is the frequency, and

λ

is the wavelength.

Step 2: Calculate the wavelength of the sound wave in water. Use the formula

λ = v / f

, where the speed of sound in water is approximately 1500 m/s and the frequency of the sound wave is 2.5 kHz (or 2500 Hz). Substitute these values into the formula to find the wavelength.

Step 3: Recognize that the electromagnetic wave has the same wavelength as the sound wave. The speed of electromagnetic waves in a vacuum (and approximately in air) is the speed of light,

c = 3.0 × 10

⁸ m/s. Use the formula

f = c / λ

to calculate the frequency of the electromagnetic wave.

Step 4: Substitute the wavelength calculated in Step 2 into the formula for the electromagnetic wave frequency. Use

f = c / λ

, where

c

is the speed of light and

λ

is the wavelength.

Step 5: Simplify the expression to find the frequency of the electromagnetic wave. Ensure units are consistent throughout the calculation (e.g., meters for wavelength and seconds for time). The result will give the frequency of the electromagnetic wave in Hz.

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Key Concepts

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

Electromagnetic Waves

Electromagnetic waves are oscillations of electric and magnetic fields that propagate through space. They travel at the speed of light in a vacuum and encompass a wide range of frequencies and wavelengths, including radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays. The relationship between frequency and wavelength is fundamental to understanding their behavior.

Wave Equation

The wave equation relates the speed of a wave to its frequency and wavelength, expressed as v = fλ, where v is the wave speed, f is the frequency, and λ is the wavelength. This equation is crucial for calculating the frequency of an electromagnetic wave when its wavelength is known. For electromagnetic waves in a vacuum, the speed is approximately 3 x 10^8 m/s.

Sound Waves in Water

Sound waves are mechanical waves that require a medium to travel, such as air, water, or solids. In water, sound travels faster than in air due to the higher density and elasticity of the medium. The frequency of a sound wave, such as the 2.5 kHz mentioned, is related to its wavelength and speed in water, which is essential for comparing it to the frequency of an electromagnetic wave with the same wavelength.

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Standing Sound Waves

Related Practice

Textbook Question

Show that the displacement D(x,t) = ln(ax + bt), where a and b are constants, is a solution to the wave equation. Then find an expression in terms of a and b for the wave speed.

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Textbook Question

A hammer taps on the end of a 4.00-m-long metal bar at room temperature. A microphone at the other end of the bar picks up two pulses of sound, one that travels through the metal and one that travels through the air. The pulses are separated in time by 9.00 ms. What is the speed of sound in this metal?

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Textbook Question

What is the speed of sound in air (a) on a cold winter day in Minnesota when the temperature is -25°F, and (b) on a hot summer day in Death Valley when the temperature is 125°F?

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Textbook Question

FIGURE EX16.8 is a picture at t = 0 s of the particles in a medium as a longitudinal wave is passing through. The equilibrium spacing between the particles is 1.0 cm. Draw the snapshot graph D(x, t = 0 s) of this wave at t = 0 s.

Textbook Question

A 15-cm-long aluminum tank is filled with ethyl alcohol. A high-frequency ultrasound wave travels horizontally through one wall of the tank and then through the alcohol. There are 275 times more cycles of the wave in the alcohol than in the aluminum wall. How thick is the wall of the tank?

Textbook Question

Show that the displacement D(x,t) = cx² + dt², where c and d are constants, is a solution to the wave equation. Then find an expression in terms of c and d for the wave speed.

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