Question

A sound wave of intensity \(1.60 \mu \mathrm{W} / \mathrm{cm}^{2}\) passes through a surface of area \(4.70 \mathrm{~cm}^{2} .\) How much energy passes through the surface in \(1 \mathrm{~h}\) ?

Short answer

The amount of energy that passes through the surface in one hour is approximately \(27072 \, Joules\).

Step-by-step solution

Conversion to SI units

First, convert all the values to their respective SI units. 1 hour should be converted to seconds and also convert the intensity from microWatt to Watt. Here, we know \(1 \mathrm{~hour} = 3600 \mathrm{~s}\) and \(1 \mu \mathrm{W} = 10^{-6} \mathrm{W}\). So, the time becomes \(1 \mathrm{~hour} = 3600 \mathrm{~s}\) and intensity becomes \(1.60 \mu \mathrm{W} / \mathrm{cm}^{2} = 1.60 \times 10^{-6} \mathrm{W} / \mathrm{cm}^{2}\).

Calculate the total energy

The formula for the energy of wave is given by \(Energy = Intensity \times Area \times Time\). Now we can plug in the numbers and calculate the energy. \(Energy = 1.60 \times 10^{-6} \mathrm{W} / \mathrm{cm}^{2} \times 4.70 \mathrm{~cm}^{2} \times 3600 \mathrm{~s}\). When calculating this, make sure to keep your units straight, and remember cm^2 will cancel out.

Simplifying the Result

Simplify the equation and solve for the final energy. The answer will be expressed in Joules since the unit for energy in the International System of Units (SI) is Joules.

Key concepts

Energy Calculation

When trying to determine how much energy passes through a surface when a sound wave is present, a fundamental approach is the energy calculation. This type of problem typically requires calculating the energy transferred by the sound wave over a certain period of time.
The energy crossing a surface can be determined through the formula:\[\text{Energy} = \text{Intensity} \times \text{Area} \times \text{Time}\]

• Intensity: This measures the power per unit area and typically results in units such as watts per square meter (\(\text{W/m}^2\)). Intensity tells us how "spread out" the wave's energy is over an area.
• Area: This denotes the surface area through which the sound wave travels and is measured in square meters (\(\text{m}^2\)).
• Time: In most calculations, time is expressed in seconds (s), reflecting the duration over which the sound wave's energy is measured.

Each component of this equation impacts the total energy calculated, allowing us to compute the energy accurately when the intensity, area, and time are provided.

SI Units Conversion

Converting units to the International System of Units (SI) is a vital step in solving physics problems, including those involving sound wave intensity.
Using consistent units ensures that calculations based on physical formulas are accurate. Let's break down this process.
Firstly, the time must be converted. Since many physics problems require time in seconds, 1 hour is converted as follows: \[1 \text{ hour} = 3600 \text{ seconds (s)}\]Similarly, the sound wave intensity needs conversion. The initial given unit of microWatts per square centimeter (\(\mu \text{W/cm}^2\)) is not an SI unit. To convert it to watts per square meter (\(\text{W/m}^2\)), the relationship is:\[1 \mu \text{W} = 10^{-6} \text{ W} \]Thus, you adjust the given intensity accordingly:\[1.60 \mu \text{W/cm}^2 = 1.60 \times 10^{-6} \text{ W/cm}^2\]The area of interaction can be directly utilized in calculations with its original unit (\(\text{cm}^2\)) if you're consistently converting or canceling units as needed during calculations.

Wave Energy Formula

Understanding the wave energy formula is key in physics when dealing with sound waves. The formula incorporates different physical quantities which explain how energy is transferred through waves.
The core equation to calculate energy (E) involves:\[E = \text{Intensity} \times \text{Area} \times \text{Time}\]

• The intensity here signifies the concentration of sound energy at the surface and is essential to gauge how much energy a wave can transfer.
• The area (also represented as \(\Delta A\)) accounts for the size of the surface that the sound wave encounters. So, bigger the area, more energy passes through it.
• The time factor completes the equation by introducing the temporal element, illustrating how the energy transference occurs over a specific duration.

In essence, multiply these three terms to derive the energy transferred in Joules, the SI unit for energy.