The sonar unit on a boat is designed to measure the depth of...

The sonar unit on a boat is designed to measure the depth of fresh water $\left(\rho=1.00 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}, B_{\mathrm{ad}}=2.20 \times 10^{9} \mathrm{~Pa}\right) .$ When the boat moves into seawater $\left(\rho=1025 \mathrm{~kg} / \mathrm{m}^{3}, B_{\mathrm{ad}}=2.37 \times 10^{9} \mathrm{~Pa}\right)$, the sonar unit is no longer calibrated properly. In seawater, the sonar unit indicates the water depth to be $10.0 \mathrm{~m}$. What is the actual depth of the water?

The sonar unit on a boat is designed to measure the depth of fresh water $\left(\rho=1.00 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}, B_{\mathrm{ad}}=2.20 \times 10^{9} \mathrm{~Pa}\right) .$ When the
boat moves into seawater $\left(\rho=1025 \mathrm{~kg} / \mathrm{m}^{3}, B_{\mathrm{ad}}=2.37 \times 10^{9} \mathrm{~Pa}\right)$, the
sonar unit is no longer calibrated properly. In seawater, the sonar unit indicates the water depth to be $10.0 \mathrm{~m}$. What is the actual depth of the water?

Key Concepts

Speed of Sound in Fluids

The speed of sound in any fluid is determined by its compressibility and density, and is given by the square root of the ratio between the fluid’s bulk modulus (a measure of its resistance to compression) and its density. This concept is central in understanding how sound waves propagate through different media, as variations in these properties directly influence the speed at which acoustic signals travel.

Sonar Operation Principles

Sonar operates by emitting acoustic pulses and measuring the time interval between sending the pulse and receiving its echo after reflection from an object or boundary. The depth or distance is then calculated using the speed of sound within the medium. Accurate knowledge of the medium's speed of sound is essential, as the measured travel time must be converted to distance with correct calibration.

Calibration Effects and Medium Property Mismatch

When a sonar system is calibrated for a specific medium with known acoustic properties, using it in a medium with different density and compressibility can lead to measurement errors. This discrepancy arises because the assumed speed of sound used in the calibration differs from the actual speed in the new medium, causing the converted distance (or depth) to be either underestimated or overestimated. Understanding this effect is crucial in adapting measurement techniques to various environmental conditions.

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