Q.57 An ultrasound signal of frequency 50 KHz is sent vertically down into a medium. The
signal gets reflected from a depth of 25 mm and returns to source 0.00005 seconds after it
is emitted. The wavelength of the ultrasound signal in that medium is ________ cm.
Problem Breakdown
An ultrasound signal travels to a 25 mm depth, reflects, and returns in 0.00005 seconds total. This round trip covers 50 mm (2 × 25 mm), allowing calculation of the speed in the medium. Frequency is given as 50 kHz, so wavelength follows from the wave relation v = f λ, where v is speed, f is frequency, and λ is wavelength.
Step-by-Step Solution
Convert depth to meters: 25 mm = 0.025 m. Total distance traveled: 2 × 0.025 = 0.05 m.
Time for round trip: 0.00005 s (or 5 × 10^{-5} s). Speed v = 0.05 / (5 × 10^{-5}) = 1000 m/s.
Frequency f = 50 kHz = 50,000 Hz. Wavelength λ = v/f = 1000/50000 = 0.02 m = 2 cm.
Key Physics Concepts
Ultrasound reflection measures depth via time-of-flight, assuming constant speed in the medium. Unlike soft tissue (1540 m/s average), this medium’s speed is 1000 m/s, slower like some fluids[web:5][web:14]. Wavelength decreases with higher frequency for fixed speed, aiding resolution in imaging.
Introduction: Ultrasound Wavelength Calculation in Reflection Medium
Ultrasound wavelength calculation is key for medical imaging and competitive exams like CSIR NET Life Sciences. This guide solves: An ultrasound signal of frequency 50 kHz sent vertically down reflects from 25 mm depth, returns in 0.00005 seconds—what’s the wavelength in cm? Master speed of sound from time-of-flight and λ = v/f formula.
Why Ultrasound Reflection Matters
Ultrasound uses high-frequency sound (>20 kHz) for non-invasive depth measurement via echo time. Round-trip time gives total path; divide by 2 for one-way depth. Here, no prior speed assumed—calculate from data unlike tissue’s 1540 m/s.
- Total distance: 2 × 25 mm = 50 mm = 0.05 m.
- Speed v = 0.05/0.00005 = 1000 m/s.
- Matches slower media, not soft tissue.
Detailed Wavelength Formula Derivation
Wave speed v = f λ, so λ = v/f[web:13]. Frequency 50 kHz = 5 × 10^4 Hz.
λ = 1000/50000 = 0.02 m = 2 cm.
Common Exam Traps
- Forgetting ×2 distance: wrongly halves speed to 500 m/s, λ=1 cm.
- Units mix (mm/s vs m/s): leads to 0.02 mm error.
- Tissue speed assumption (1540 m/s): ignores data, gives ~3 cm.
Exam Tips for CSIR NET, IIT JAM
Practice time-of-flight: depth d, time t → v = 2d/t. Verify with 13 μs/cm rule (soft tissue), but adapt here[web:9]. Units: convert mm to m, s consistent.
Parameters Summary Table
| Parameter | Value | Formula Role |
|---|---|---|
| Depth | 25 mm (0.025 m) | d = distance one-way |
| Time | 0.00005 s | t = round-trip |
| Speed | 1000 m/s | v = 2d/t |
| Frequency | 50 kHz | f in λ = v/f |
| Wavelength | 2 cm | Final answer |
