A-Level Physics May/June 2024 Q10(b)(ii): Determine the intensity reflection coefficient for ultrasound that is incident on a wat…

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  First, calculate the specific acoustic impedance ($Z$) for both water and glass using the formula $Z = \rho c$, where $\rho$ is the density and $c$ is the speed of sound.

  For water: $Z_{water} = \rho_{water} \times c_{water}$ $Z_{water} = 1000 \times 1420$ $Z_{water} = 1.42 \times 10^6 \, \text{kg m}^{-2} \text{s}^{-1}$

  For glass: $Z_{glass} = \rho_{glass} \times c_{glass}$ $Z_{glass} = 2500 \times 4560$ $Z_{glass} = 11.4 \times 10^6 \, \text{kg m}^{-2} \text{s}^{-1}$

  Next, use the formula for the intensity reflection coefficient, $\alpha$: $\alpha = \frac{(Z_{glass} - Z_{water})^2}{(Z_{glass} + Z_{water})^2}$

  Substitute the calculated values: $\alpha = \frac{(11.4 \times 10^6 - 1.42 \times 10^6)^2}{(11.4 \times 10^6 + 1.42 \times 10^6)^2}$

  The factor of $10^6$ cancels from the numerator and denominator: $\alpha = \frac{(11.4 - 1.42)^2}{(11.4 + 1.42)^2} = \frac{(9.98)^2}{(12.82)^2}$

  $\alpha = 0.6065...$

  $\alpha \approx 0.61$

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  How the marks are awarded

  • C1 — Awarded for the correct calculation of the specific acoustic impedance of water ($Z_{water} = 1.42 \times 10^6 \, \text{kg m}^{-2} \text{s}^{-1}$).
  • C1 — Awarded for the correct calculation of the specific acoustic impedance of glass ($Z_{glass} = 11.4 \times 10^6 \, \text{kg m}^{-2} \text{s}^{-1}$).
  • A1 — Awarded for the correct substitution of the two impedance values into the intensity reflection coefficient formula and calculating the final correct answer of 0.61.

  Common mistakes

  • Forgetting to square the numerator and/or denominator in the reflection coefficient formula, i.e. calculating (Z₂ - Z₁) / (Z₂ + Z₁).
  • Using the formula for the intensity transmission coefficient, 4Z₁Z₂ / (Z₂ + Z₁)², by mistake.
  • Making a calculation error when determining one of the specific acoustic impedances, but otherwise using the correct method.
  • Incorrectly handling the powers of ten, leading to an answer that is orders of magnitude wrong.

  Examiner tip: For multi-step calculations, always write down the formula you are using and show the substitution of values, as this can earn method marks even if the final answer is incorrect.

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