Final answer:
To find the difference in echo times for tissues at varying depths, we calculate the time it takes for ultrasound to travel to the depth and back, and then compare these times. The period T of the ultrasound must be smaller than the echo time difference to resolve small details, which translates to a frequency that is within the normal range for diagnostic ultrasound.
Step-by-step explanation:
Understanding Echo Times in Ultrasound Imaging
The echo time of an ultrasound wave is the duration it takes for the sound wave to travel to a reflecting surface and return. Accurate measurement of echo times is crucial for resolving the distances to various tissues within the body. To find the difference in echo times for tissues located at 3.50 cm and 3.60 cm beneath the surface, we need to consider the speed of sound in human tissues, which is approximately 1540 m/s.
To calculate the difference in echo times, we must calculate the time taken for the sound to travel to each depth and back, then find the difference between them. The time for one depth is given by the formula:
Time = 2 * Depth / Speed of sound
For a depth of 3.50 cm (0.0350 m), the time is:
T1 = 2 * 0.0350 m / 1540 m/s ≈ 45.45 microseconds
Similarly, for a depth of 3.60 cm (0.0360 m), the time is:
T2 = 2 * 0.0360 m / 1540 m/s ≈ 46.75 microseconds
The difference in echo times is then:
ΔT = T2 - T1 ≈ 1.30 microseconds
For the ultrasound to resolve these two points separately, the period T of the ultrasound must be smaller than the minimum time resolution. The frequency is the inverse of the period (f = 1/T). If T must be less than 1.30 microseconds, the minimum frequency (fmin) must be:
fmin > 1 / 1.30 microseconds ≈ 769.23 kHz
Typically, the frequency range for diagnostic ultrasound is between 2 to 15 MHz, which is well above the minimum frequency calculated, thus falling within the normal operating range.