Problem 4 - University Physics with Modern Physics

Question

Ultrasound Imaging. Sound having frequencies above the range of human hearing (about \(20,000 \mathrm{Hz}\) ) 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 \(\mathrm{m} / \mathrm{s} .\) For a good, detailed image, the wavelength should be no more than 1.0 \(\mathrm{mm} .\) What frequency sound is required for a good scan?

Step-by-Step Solution

Answer

The required frequency for a good ultrasound scan is 1,500,000 Hz (or 1.5 MHz).

Step 1: Understand the Relationship between Speed, Frequency, and Wavelength

The relationship between speed, frequency, and wavelength of a wave is given by the formula \( v = f \times \lambda \), where \( v \) is the speed of the wave, \( f \) is the frequency, and \( \lambda \) is the wavelength.

Step 2: Convert the Wavelength to Meters

Given that the maximum wavelength for a detailed image is 1.0 mm, convert this to meters: \( 1.0 \text{ mm} = 0.001 \text{ m} \). This step is important because the speed of sound is given in meters per second.

Step 3: Substitute Values into the Wave Equation

Using the equation \( v = f \times \lambda \), substitute the given wave speed \( v = 1500 \text{ m/s} \) and the converted wavelength \( \lambda = 0.001 \text{ m} \) into the equation to solve for frequency \( f \):\[ 1500 = f \times 0.001 \]

Step 4: Solve for Frequency

Rearrange the equation to solve for the frequency: \( f = \frac{1500}{0.001} \). Now calculate:\[ f = 1500000 \text{ Hz} \]

Step 5: Confirm the Frequency is Above 20,000 Hz

Check that the calculated frequency \( f = 1500000 \text{ Hz} \) is indeed above the human hearing range of 20,000 Hz. Since it is above, this frequency is classified as ultrasound and is appropriate for imaging within the specified requirements.

Key Concepts

Wave Speed

Wave speed is a crucial concept in understanding ultrasound imaging. It refers to how fast a wave travels through a particular medium. In the context of ultrasound, sound waves travel through the body at a speed determined by the properties of the tissues. For medical imaging, we use the fact that ultrasound waves travel at approximately 1500 meters per second through body tissues.
This speed is critical because it determines how effectively sound waves can penetrate tissues and return useful images.

• Wave speed affects how sharp and detailed the ultrasound image is.
• It combines with frequency and wavelength to form a complete picture of wave behavior.

Understanding wave speed helps technicians adjust settings to optimize image quality in ultrasound diagnostics.

Frequency Calculation

Frequency calculation in ultrasound imaging helps determine how detailed the images will be. The frequency of a wave is how many cycles it completes in a second. In ultrasound, sound waves' frequency must be above 20,000 Hz, as these frequencies penetrate the body to provide clear images.
The frequency is calculated using the formula relating wave speed and wavelength:\[ f = \frac{v}{\lambda} \]
where:

• \( v \) is the wave speed (1500 m/s in tissue).
• \( \lambda \) is the wavelength (expressed in meters).

For a wavelength of 0.001 meters, the frequency necessary for a good scan is 1,500,000 Hz. This high frequency ensures that the waves can reflect off small structures, improving image resolution.

Wavelength Conversion

Wavelength is the distance between consecutive peaks of a wave. For ultrasound imaging, ensuring the correct wavelength is critical to acquiring clear images. Often, specifications use millimeters, but calculations typically use meters. Converting from millimeters to meters is straightforward and necessary:

• 1 millimeter = 0.001 meters

This conversion helps fit into the equation \( v = f \times \lambda \), where calculations are consistent in units. A typical scenario requires the wavelength to be no more than 1.0 mm (or 0.001 meters), ensuring that the ultrasound frequency is high enough to resolve fine details and provide clear, accurate images.

Acoustic Waves

Acoustic waves are fundamental to ultrasound imaging. These are sound waves that can travel through different media, like human tissues. For imaging, ultrasound uses frequencies higher than human hearing can detect, referred to as ultrasonic acoustic waves.
These acoustic waves interact with tissue by:

• Travelling through, being absorbed, or scattered by tissues.
• Reflecting back to a probe, which translates these reflections into images.

The quality of these images depends greatly on how well these waves penetrate tissues and the frequency and wavelength chosen. Understanding these interactions is critical for technicians and medical professionals to optimize and interpret ultrasound scans correctly.