Question

In an ultrasonic transducer with radius \( R \) and ultrasound wavelength \( \lambda \), the near field extends to a distance given by:

• A. \( \left(\frac{R^2}{\lambda}\right) - \left(\frac{\lambda}{4}\right) \)
• B. \( \left(\frac{R^2}{\lambda}\right) - \left(\frac{\lambda}{16}\right) \)
• C. \( \left(\frac{R}{\lambda}\right) - \left(\frac{\lambda}{2}\right) \)
• D. \( \left(\frac{R^2}{2\lambda}\right) - \left(\frac{\lambda}{4}\right) \)

Hint:

In clinical practice where the wavelength \( \lambda \) is extremely tiny compared to the transducer radius \( R \), the second term \( \frac{\lambda}{4} \) becomes so negligible that the formula is frequently abbreviated as simply \( x \approx \frac{R^2}{\lambda} \) or \( \frac{D^2}{4\lambda} \). However, the mathematically precise boundary expression includes the exact correction factor: \( \left(\frac{R^2}{\lambda}\right) - \left(\frac{\lambda}{4}\right) \).

Answer 1

The Correct Option is A

Concept: The sound beam emitted by a disc-shaped unfocused ultrasonic transducer is split structurally into two distinct spatial zones: the near field (also known as the Fresnel zone) and the far field (the Fraunhofer zone). The near field is characterized by a complex interference pattern of sound intensity and a converging beam shape.

Step 1: Deriving the boundary equation from wave interference.
Let us consider a circular transducer face of radius \( R \) (or diameter \( D = 2R \)) emitting continuous waves of wavelength \( \lambda \). The boundary or natural transition point of the near field occurs where the wave coming from the outermost edge of the transducer disc interferes with the wave coming from the absolute center point of the disc at a axial distance \( x \) along the beam axis. For constructive or destructive limit transitions, the path length difference between the edge path and the central axial path is set to a fraction of a wavelength, typically derived via the Pythagorean theorem. The distance from the center to the transition point is \( x \). The distance from the perimeter edge of the disc to that same axial point is given by: \[ \text{Path}_{\text{edge}} = \sqrt{R^2 + x^2} \] The central path distance is simply \( x \). The formal definition for the end of the near-field zone requires these paths to meet an exact phase offset condition related to the outermost maxima, which is a path difference of \( \frac{\lambda}{2} \): \[ \sqrt{R^2 + x^2} - x = \frac{\lambda}{2} \]

Step 2: Isolating the distance variable \( x \).
Move the term \( x \) to the right side of the equation: \[ \sqrt{R^2 + x^2} = x + \frac{\lambda}{2} \] Square both sides of the equation to eliminate the radical: \[ R^2 + x^2 = \left(x + \frac{\lambda}{2}\right)^2 \] Expanding the right hand side using the algebraic identity \( (a+b)^2 = a^2 + 2ab + b^2 \): \[ R^2 + x^2 = x^2 + 2 \cdot x \cdot \left(\frac{\lambda}{2}\right) + \left(\frac{\lambda}{2}\right)^2 \] Cancel out the common \( x^2 \) term from both sides: \[ R^2 = x\lambda + \frac{\lambda^2}{4} \] Isolate the term containing \( x \): \[ x\lambda = R^2 - \frac{\lambda^2}{4} \] Divide the entire expression by \( \lambda \) to solve explicitly for the near field distance \( x \): \[ x = \frac{R^2}{\lambda} - \frac{\lambda^2}{4\lambda} \] \[ x = \left(\frac{R^2}{\lambda}\right) - \left(\frac{\lambda}{4}\right) \] This algebraic derivation exactly yields option (A).