Problem 103 - Pearson Physics

Question

Medical ultrasound often uses a frequency of \(3.5 \mathrm{MHz}\) What is the wavelength of these ultrasound waves? Assume that the speed of sound waves in the human body is \(1,500 \mathrm{~m} / \mathrm{s}\), the same as the speed of sound in salt water.

Step-by-Step Solution

Answer

The wavelength of the ultrasound waves is approximately 0.429 mm.

Step 1: Understand the Formula

To find the wavelength of a wave, we use the formula \( \lambda = \frac{v}{f} \), where \( \lambda \) represents the wavelength, \( v \) is the speed of the wave, and \( f \) is the frequency.

Step 2: Identify the Given Values

We know that the frequency \( f = 3.5 \, \text{MHz} = 3.5 \times 10^6 \, \text{Hz} \) and the speed of the sound wave \( v = 1,500 \, \text{m/s} \).

Step 3: Substitute the Values into the Formula

Substitute the given speed and frequency into the formula: \[ \lambda = \frac{1,500}{3.5 \times 10^6} \].

Step 4: Calculate the Wavelength

Calculate the expression \( \lambda = \frac{1,500}{3.5 \times 10^6} \). Performing this calculation, \( \lambda \approx 4.29 \times 10^{-4} \text{ m} \).

Step 5: Express the Wavelength in Millimeters

Convert meters to millimeters: \( 4.29 \times 10^{-4} \text{ m} = 0.429 \text{ mm} \), since 1 millimeter is \( 10^{-3} \) meters.

Key Concepts

Wave Speed

Wave speed is an essential concept in understanding ultrasound physics and is a property of how quickly waves travel through a medium. In general, wave speed depends on the properties of the medium, such as its density and elasticity. For example, sound waves move differently through water compared to air, due to differences in these properties.
In the context of medical ultrasound, sound waves typically move through the human body at about 1,500 meters per second, similar to their speed in salt water. This speed is faster than in the air, due to the denser medium.
Key points to remember:

• Wave speed is impacted by the medium through which the wave travels.
• In the human body, the speed of sound waves is approximately 1,500 m/s.
• Sound waves travel faster in liquids and solids compared to gases.

Understanding wave speed is crucial for calculating other properties of sound waves, such as frequency and wavelength.

Frequency

Frequency is the number of wave cycles that pass a point per second, measured in Hertz (Hz). In medical ultrasound, frequency determines how detailed the imaging will be. High-frequency waves provide more precise images but don't penetrate as deeply into the tissues, while low-frequency waves travel deeper but offer less detail.
In our exercise, the frequency is given as 3.5 MHz, which is equivalent to 3.5 million cycles per second. This frequency is typical for medical ultrasounds as it balances penetration depth and image resolution.
Remember, frequency is crucial because:

• It affects the quality and depth of the ultrasound image.
• Higher frequencies offer better resolution.
• Megahertz (MHz) is a common unit used in medical ultrasound imaging.

Understanding frequency helps in choosing the right settings for different ultrasound applications.

Wavelength

Wavelength is the distance between two corresponding points on successive waves, such as crest to crest. It is inversely related to both wave speed and frequency. Using the formula

Calculating Wavelength

You can find the wavelength (\( \lambda \)) using the formula:\[ \lambda = \frac{v}{f} \]where \( v \) is the wave speed and \( f \) is the frequency. With ultrasound, knowing the wavelength helps in understanding the resolving power of the imaging.

Practical Example

Given our exercise: the speed of sound \( v \) is 1,500 m/s and frequency \( f \) is 3.5 MHz, the wavelength \( \lambda \) can be calculated as:\[ \lambda = \frac{1,500}{3.5 \times 10^6} \approx 4.29 \times 10^{-4} \text{ m} \]or about 0.429 mm. This wavelength is well-suited for capturing detailed medical images.
Key takeaways about wavelength:

• It is inversely proportional to frequency; as frequency increases, wavelength decreases.
• Shorter wavelengths provide better resolution.
• In ultrasounds, wavelengths are typically in the range of millimeters or less.

Wavelength helps determine the level of detail an ultrasound can achieve.