Which expression correctly represents the beam-divergence relation for a quartz crystal?

Which expression correctly represents the beam-divergence relation for a quartz crystal?

• A. sinΘ = diameter 1/2 x wavelength
• B. sinΘ diameter = frequency x wavelength
• C. sinΘ/2 = 1.22 x wavelength/diameter
• D. sinΘ = frequency x wavelength

Answer: (C)

Beam divergence from a circular transducer aperture is governed by diffraction. For a circular aperture, the first diffraction minimum occurs at an angle where sin θ = 1.22 λ / D, with D being the aperture diameter and λ the wavelength in the medium. If you define the full beam spread as Θ, the relevant half-angle is Θ/2, so sin(Θ/2) ≈ 1.22 λ / D. This is exactly the form shown: sin(Θ/2) = 1.22 × (wavelength / diameter). In a quartz ultrasonic transducer, the wavelength is λ = v/f, where v is the speed of sound in quartz and f is the drive frequency. Therefore the beam divergence decreases with a larger diameter and with higher frequency (shorter wavelength). Other forms don’t fit because they mix up the quantities involved in diffraction (dropping the 1.22 factor, replacing wavelength with frequency, or placing diameter on the wrong side), which would not match the diffraction-limited angular spread.

Beam divergence from a circular transducer aperture is governed by diffraction. For a circular aperture, the first diffraction minimum occurs at an angle where sin θ = 1.22 λ / D, with D being the aperture diameter and λ the wavelength in the medium. If you define the full beam spread as Θ, the relevant half-angle is Θ/2, so sin(Θ/2) ≈ 1.22 λ / D. This is exactly the form shown: sin(Θ/2) = 1.22 × (wavelength / diameter).

In a quartz ultrasonic transducer, the wavelength is λ = v/f, where v is the speed of sound in quartz and f is the drive frequency. Therefore the beam divergence decreases with a larger diameter and with higher frequency (shorter wavelength).

Other forms don’t fit because they mix up the quantities involved in diffraction (dropping the 1.22 factor, replacing wavelength with frequency, or placing diameter on the wrong side), which would not match the diffraction-limited angular spread.