Ultrasound used in a medical imager has frequency \(4.8 \mathrm{MHz}\) and wavelength \(0.31 \mathrm{mm}\). Find (a) the angular frequency, (b) the wave number, and (c) the wave speed.

In the context of waves, specifically ultrasound waves used in medical imaging, a key concept to understand is angular frequency. Angular frequency, denoted by the Greek letter \( \omega \), measures how quickly the wave oscillates in radians per second. It is directly related to the regular frequency \( f \), which counts the number of oscillations per second measured in Hertz (\( Hz \) ). The relationship between them is simple: \( \omega = 2\pi f \) .

To visualize this, think of the wave as a circular motion, where each oscillation represents a rotation. Angular frequency then is how many radians of that circle the wave completes in one second. For an ultrasound with a frequency of \( 4.8 \mathrm{MHz} \), its angular frequency can be calculated as \($ \omega = 2\pi \times 4.8 \times 10^6 \mathrm{Hz} \$\), which brings out the rapid motion that ultrasound waves exhibit.

Another essential concept in the study of waves, such as those in medical ultrasounds, is the wave number. Represented by \( k \), the wave number is a spatial counterpart to angular frequency. It tells us the number of wave cycles in a unit distance, expressed typically in radians per meter \( \mathrm{rad/m} \). The wave number can be calculated using the formula: \( k = \frac{2\pi}{\lambda} \) where \( \lambda \), is the wavelength.

This measurement is reflective of the wave's spatial structure, describing how tightly packed the wave cycles are. In the example of an ultrasound with a wavelength of \( 0.31 \mathrm{mm} \), you can see that the wave number indicates how many times the wave repeats itself over the length of one meter. The smaller the wavelength, the higher the wave number, which implies a more compact wave structure.

Wave speed is a fundamental characteristic that describes how fast a wave travels through a medium. In the case of medical ultrasound, the wave speed \( v \) describes how quickly sound waves travel through body tissues. The wave speed is obtained by multiplying the wave's frequency \( f \) by its wavelength \( \lambda \): \( v = f \times \lambda \) .

This concept is critical in medical imaging because it helps to determine how the waves will interact with human tissue and how they can create an image of the internal structures of the body. For a wave with the given frequency and wavelength in the exercise, the wave speed can be calculated. If the speed of sound in the tissue is known, this value is instrumental in improving the accuracy and resolution of the resulting medical image.

The relationship between frequency and wavelength is at the heart of understanding wave behavior. Frequency \( f \) and wavelength \( \lambda \) are inversely related, as captured by the formula: \( v = f \times \lambda \) which we use to calculate wave speed. This formula tells us that if the wave's frequency increases, the wavelength must decrease for the wave speed to remain constant, and vice versa.

For medical ultrasounds, operating at various frequencies, different wavelengths are used to optimize image resolution and penetration depth. High-frequency waves provide better resolution due to shorter wavelengths but do not penetrate as deeply as lower-frequency waves. Therefore, understanding this relationship helps in selecting appropriate ultrasound settings for different diagnostic purposes.