Understanding Sound Wave Calculations

This problem requires us to calculate the time taken by a sound wave to travel a certain distance. To do this, we first need to find the speed of the sound wave. The speed of any wave is related to its frequency and wavelength by the formula:

$\text{Speed (v)} = \text{Frequency (f)} \times \text{Wavelength (}\lambda\text{)}$

Once we know the speed, we can calculate the time taken to travel a given distance using the formula:

$\text{Time (t)} = \frac{\text{Distance (d)}}{\text{Speed (v)}}$

Given Sound Wave Information

We are provided with the following information about the sound wave:

• Frequency ($\text{f}$): 4 kHz
• Wavelength ($\lambda$): 40 cm
• Distance to travel ($\text{d}$): 3.2 km

Before we perform calculations, it's important to ensure all units are consistent. Let's convert the given values to standard SI units (Hertz for frequency, meters for wavelength and distance).

• Frequency: $4 \text{ kHz} = 4 \times 1000 \text{ Hz} = 4000 \text{ Hz}$
• Wavelength: $40 \text{ cm} = 40 \times 0.01 \text{ m} = 0.40 \text{ m}$
• Distance: $3.2 \text{ km} = 3.2 \times 1000 \text{ m} = 3200 \text{ m}$

Step-by-Step Calculation of Time

Step 1: Calculate the Speed of the Sound Wave

Using the formula $\text{v} = \text{f} \times \lambda$:

$\text{v} = 4000 \text{ Hz} \times 0.40 \text{ m}$

$\text{v} = 1600 \text{ m/s}$

So, the speed of the sound wave is 1600 meters per second.

Step 2: Calculate the Time Taken to Travel the Distance

Now, using the formula $\text{t} = \text{d} / \text{v}$:

$\text{t} = \frac{3200 \text{ m}}{1600 \text{ m/s}}$

$\text{t} = 2.0 \text{ s}$

The time taken by the sound wave to travel a distance of 3.2 km is 2.0 seconds.

Analyzing the Options

Let's compare our calculated time with the given options:

• Option 1: 0.5 s
• Option 2: 4.0 s
• Option 3: 2.0 s
• Option 4: 1.0 s

Our calculated time of 2.0 s matches Option 3.

Conclusion on Sound Wave Travel Time

Based on the frequency and wavelength, the speed of the sound wave was found to be 1600 m/s. Using this speed, the time required to cover a distance of 3.2 km is 2.0 seconds.

Revision Table: Sound Wave Formulas

Additional Information: Sound Wave Speed Factors

The speed of sound in a medium depends on the properties of the medium itself. Key factors include:

• Medium Type: Sound travels fastest in solids, slower in liquids, and slowest in gases. This is because particles are closer together in solids, allowing vibrations to transfer more quickly.
• Temperature: In gases, the speed of sound increases with increasing temperature. Higher temperature means gas molecules move faster and transmit vibrations more effectively. For air, the speed of sound increases by about 0.6 m/s for every 1°C increase in temperature.
• Humidity: In gases, humidity has a slight effect. Higher humidity makes the air less dense, slightly increasing the speed of sound.
• Pressure: For an ideal gas, the speed of sound is independent of pressure, assuming the temperature remains constant. However, real gases show a slight dependence.

In this specific problem, we were given the frequency and wavelength, which inherently determine the speed of the wave in that particular medium under those conditions.