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Chapter 15: Problem 4

Sound having frequencies above the range of human hearing (about 20,000 Hz) is called \(ultrasound\). Waves above this frequency can be used to penetrate the body and to produce images by reflecting from surfaces. In a typical ultrasound scan, the waves travel through body tissue with a speed of 1500 m/s. For a good, detailed image, the wavelength should be no more than 1.0 mm. What frequency sound is required for a good scan?

Short Answer

Expert verified
A frequency of 1.5 MHz is required for a good ultrasound scan.

Step by step solution

01

Understand the Relationship Between Speed, Frequency, and Wavelength

The speed of a wave, its frequency, and its wavelength are related by the equation \( v = f \lambda \), where \( v \) is the speed of the wave, \( f \) is the frequency, and \( \lambda \) is the wavelength.
02

Identify the Known Quantities

We know the speed of the ultrasound wave in the body tissue is 1500 m/s and the wavelength for a good image is 1.0 mm, which is 0.001 meters.
03

Rearrange the Formula to Solve for Frequency

We need to find the frequency, \( f \). Rearrange the wave equation to \( f = \frac{v}{\lambda} \).
04

Substitute the Values into the Equation

Substitute the known values into the rearranged equation: \( f = \frac{1500}{0.001} \).
05

Calculate the Frequency

Perform the calculation: \( f = \frac{1500}{0.001} = 1,500,000 \) Hz or 1.5 MHz.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Wave Speed
Wave speed is an essential concept in understanding how waves move through different mediums. It indicates how fast a wave travels from one point to another. Wave speed (\( v \)...), frequency (\( f \)...), and wavelength (\( \lambda \)...) are interconnected. The speed of a wave is calculated by multiplying the frequency by the wavelength: \[ v = f \times \lambda \].
  • For ultrasound waves traveling through the body, the speed is determined by the medium – in this case, body tissue, where it typically measures about 1500 m/s.
  • This speed is essential for accurately timing reflections during diagnostic imaging.
Understanding wave speed helps in calculating other important parameters crucial for ultrasound imaging and numerous other applications in science and technology.
Wavelength
Wavelength is the distance between consecutive points of the same phase on a wave, such as crest to crest or trough to trough. It is a crucial factor in determining the level of detail available in an image.
  • For ultrasound imaging, shorter wavelengths (such as 1.0 mm or 0.001 meters) are preferred as they produce higher resolution images.
  • These shorter wavelengths can penetrate deep into the body, providing detailed insights crucial for medical diagnoses.
By understanding wavelength, we can control the resolution of the images produced, leading to more accurate and detailed ultrasound scans.
Wave Frequency
Wave frequency refers to the number of wave cycles that pass through a given point per second. It is measured in hertz (Hz). In the field of ultrasound imaging, frequency is vitally important.
  • A higher frequency results in a shorter wavelength, which in turn, improves image resolution.
  • For example, in a typical ultrasound scan, to achieve a wavelength of 1.0 mm, a frequency of 1.5 MHz is necessary.
Using the formula \( f = \frac{v}{\lambda} \), we can calculate the frequency needed to achieve the desired image quality. Frequency control allows medical professionals to tailor diagnostic procedures to each patient's needs.
Ultrasound Imaging
Ultrasound imaging is a non-invasive diagnostic tool used widely in the medical field. It uses high-frequency sound waves beyond the range of human hearing to create images of the inside of the body.
  • Ultrasound waves are emitted by a probe and travel through body tissues, reflecting off structures and returning to the probe to form an image.
  • Image clarity and detail depend on controlling wave speed, frequency, and wavelength.
  • Typically, the wave speed in bodily tissues is around 1500 m/s, crucial for timing reflections correctly.
Understanding the principles behind ultrasound imaging helps in improving the diagnosis and treatment of various medical conditions, enhancing patient care.

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Most popular questions from this chapter

An ant with mass m is standing peacefully on top of a horizontal, stretched rope. The rope has mass per unit length \(\mu\) and is under tension \(F\). Without warning, Cousin Throckmorton starts a sinusoidal transverse wave of wavelength \(\lambda\) propagating along the rope. The motion of the rope is in a vertical plane. What minimum wave amplitude will make the ant become momentarily weightless? Assume that \(m\) is so small that the presence of the ant has no effect on the propagation of the wave.

A 1.50-m-long rope is stretched between two supports with a tension that makes the speed of transverse waves 62.0 m/s. What are the wavelength and frequency of (a) the fundamental; (b) the second overtone; (c) the fourth harmonic?

In your physics lab, an oscillator is attached to one end of a horizontal string. The other end of the string passes over a frictionless pulley. You suspend a mass \(M\) from the free end of the string, producing tension \(Mg\) in the string. The oscillator produces transverse waves of frequency \(f\) on the string. You don't vary this frequency during the experiment, but you try strings with three different linear mass densities \(\mu\). You also keep a fixed distance between the end of the string where the oscillator is attached and the point where the string is in contact with the pulley's rim. To produce standing waves on the string, you vary \(M\); then you measure the node-to-node distance \(d\) for each standingwave pattern and obtain the following data: (a) Explain why you obtain only certain values of \(d\). (b) Graph \(\mu d^2\) (in kg \(\cdot\) m) versus \(M\) (in kg). Explain why the data plotted this way should fall close to a straight line. (c) Use the slope of the best straight-line fit to the data to determine the frequency \(f\) of the waves produced on the string by the oscillator. Take \(g = 9.80 \, \mathrm{m/s}^2\). (d) For string A (\(\mu = 0.0260\) g/cm), what value of \(M\) (in grams) would be required to produce a standing wave with a node-to-node distance of 24.0 cm? Use the value of \(f\) that you calculated in part (c).

A fisherman notices that his boat is moving up and down periodically, owing to waves on the surface of the water. It takes 2.5 s for the boat to travel from its highest point to its lowest, a total distance of 0.53 m. The fisherman sees that the wave crests are spaced 4.8 m apart. (a) How fast are the waves traveling? (b) What is the amplitude of each wave? (c) If the total vertical distance traveled by the boat were 0.30 m but the other data remained the same, how would the answers to parts (a) and (b) change?

By measurement you determine that sound waves are spreading out equally in all directions from a point source and that the intensity is 0.026 W/m\(^2\) at a distance of 4.3 m from the source. (a) What is the intensity at a distance of 3.1 m from the source? (b) How much sound energy does the source emit in one hour if its power output remains constant?
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