Problem 6 - Applied Physics

Question

A sound wave is transmitted through water from one submarine, is reflected off another submarine \(15 \mathrm{~km}\) away, and returns to the sonar receiver on the first submarine. What is the round-trip transit time for the sound wave? Assume that the water temperature is \(25^{\circ} \mathrm{C}\).

Step-by-Step Solution

Answer

The round-trip transit time is approximately 20.04 seconds.

Step 1: Understand the Problem

The problem involves calculating the time it takes for a sound wave to travel from the first submarine, reflect off the second submarine located 15 km away, and return to the first submarine. This is a round-trip journey.

Step 2: Determine the Speed of Sound in Water

At a water temperature of 25\(^\circ\)C, the speed of sound in water is approximately 1497 meters per second (m/s). This value will be used to calculate the transit time.

Step 3: Calculate the Total Distance for the Round-Trip

The sound wave travels 15 km to the second submarine and 15 km back, making the total distance traveled 30 km. Convert this distance to meters: 30 km = 30,000 meters.

Step 4: Use the Formula to Calculate Time

Use the formula for time, which is distance divided by speed: \( \text{time} = \frac{\text{distance}}{\text{speed}} \). Substitute the values: \( \text{time} = \frac{30,000}{1497} \).

Step 5: Perform the Calculation

Calculate: \( \text{time} = \frac{30,000}{1497} \approx 20.04 \) seconds. This is the time it takes for the round trip.

Key Concepts

Speed of Sound in Water

Understanding the speed of sound in water is essential for solving many problems related to acoustic waves, such as determining transit times. Sound travels faster in water than in air due to its higher density and elasticity. This characteristic makes underwater communication and detection possible, using devices like sonar.
At a temperature of 25°C, the speed of sound in water is approximately 1497 meters per second. This speed varies based on several factors such as temperature, salinity, and pressure. In general, warmer water leads to faster sound speeds.

• Density: Water molecules are packed more tightly than air molecules, enhancing sound propagation.
• Elasticity: Water's elasticity allows sound waves to travel without losing much energy.

For the given problem, knowing the speed at the specific temperature of 25°C allows for accurate time calculations for sound waves traveling between submarines.

Round-Trip Calculation

The round-trip calculation is crucial to determine how long it takes for sound to travel from a source, reflect off a target, and return to the source. This process can be compared to a sonar system used in submarines to detect objects underwater.
Here's how the calculation is typically done:

• Identify the one-way distance. In this case, the sound travels to the second submarine 15 km away.
• Double the one-way distance to account for the return journey, making the total round-trip distance 30 km.
• Convert this distance into meters for consistency with speed units (1 km = 1000 meters), resulting in 30,000 meters for the round-trip.
• Use the time formula: \( \text{time} = \frac{\text{distance}}{\text{speed}} \).

By applying these methods, you can determine that the sound wave round-trip for this scenario takes approximately 20.04 seconds.

Temperature Effect on Sound Speed

Temperature significantly affects the speed of sound in water. As the temperature of water increases, the sound speed tends to rise as well. This relationship is important for various applications from oceanography to underwater acoustics.
Here's why temperature has such an impact:

• Chemical Reaction Rates: Higher temperatures increase the energy of water molecules, causing them to vibrate more and transmit sound faster.
• Speed Variation: At 0°C, sound travels at around 1402 m/s in water. At 25°C, it travels at approximately 1497 m/s—a noticeable difference.

By understanding the effect of temperature, one can better predict and measure sound waves' behaviors in differing aquatic environments. For the exercise problem, using the speed at 25°C is vital to achieving accurate results.