Relationship Between Distance and Intensity (HSC SSCE Physics): Revision Notes

Relationship Between Distance and Intensity

What is sound intensity?

When a source produces sound waves, the energy spreads out into the surrounding air, covering an increasingly larger area. As this happens, the intensity of the sound decreases.

Intensity is the amount of sound energy passing through a unit area in one second. To understand this:

• Energy is measured in joules ($\text{J}$)
• Unit area means one square metre ($\text{m}^2$)
• Time is one second

This gives us the unit joules per second ($\text{J} \cdot \text{s}^{-1}$) per square metre. Since one joule per second equals one watt ($\text{W}$), we express intensity in watts per square metre ($\text{W} \cdot \text{m}^{-2}$).

The inverse square law

Sound intensity follows a pattern called the inverse square law. This describes how intensity changes as you move away from a sound source.

For this relationship to work accurately, we assume:

• The sound source is a point source (very small)
• There are no nearby objects to reflect or absorb the sound

The inverse square law states that intensity is inversely proportional to the square of the distance from the source:

$I \propto \frac{1}{d^2}$

where:

• $I$ is the intensity of sound
• $d$ is the distance from the sound source
• $\propto$ means "is proportional to"

What does this mean in practice?

If you double your distance from a sound source, the intensity becomes one-quarter of what it was. If you triple the distance, the intensity becomes one-ninth. The sound spreads out over a larger area, so the intensity at any point decreases rapidly.

Using the inverse square law for calculations

When comparing intensities at two different distances, we can use this mathematical relationship:

$\frac{I_1}{I_2} = \left(\frac{d_2}{d_1}\right)^2$

where:

• $I_1$ is the intensity at distance $d_1$
• $I_2$ is the intensity at distance $d_2$

This formula is very useful for solving problems about how sound intensity changes with distance.

Measuring intensity in decibels

Sound intensity is commonly measured using the decibel (dB) scale. This is a logarithmic scale, which means each step represents a multiplication rather than simple addition.

Why use a logarithmic scale? Because the range of sound intensities we can hear is enormous - from the faintest whisper to a jet engine represents a difference of over a trillion times!

The threshold of hearing

The quietest sound humans can typically hear is called the threshold of hearing. This has been assigned:

$I_0 = 10^{-12} \text{ W} \cdot \text{m}^{-2}$

This is defined as $0 \text{ dB}$.

How the decibel scale works

The decibel scale increases logarithmically:

• $10 \text{ dB}$ has 10 times the intensity of $0 \text{ dB}$, or $10^{-11} \text{ W} \cdot \text{m}^{-2}$
• $20 \text{ dB}$ has 10 times the intensity of $10 \text{ dB}$, or $10^{-10} \text{ W} \cdot \text{m}^{-2}$
• Each increase of $10 \text{ dB}$ represents a tenfold increase in intensity

Here's a table showing common sounds and their intensities:

Converting decibels to watts per square metre

Since most sound level meters and smartphone apps display readings in decibels, you often need to convert to $\text{W} \cdot \text{m}^{-2}$ for calculations. Use this formula:

$I = 10^{\left(\frac{\text{dB}}{10} - 12\right)}$

where:

• $I$ is the sound intensity in $\text{W} \cdot \text{m}^{-2}$
• $\text{dB}$ is the sound level in decibels

Investigation: Relationship between distance and intensity of sound

Aim

To investigate the relationship between the intensity of sound and the distance from the source.

Before conducting the investigation, write a hypothesis about how you predict distance from a sound source will affect the measured intensity.

Materials

• Constant sound source (e.g. signal generator with speaker, or electric bell)
• Decibel meter (or suitable smartphone app)
• Measuring tape

Method

1. Select a suitable quiet area outdoors, preferably away from walls and other buildings that might reflect sound. This helps ensure you're only measuring the direct sound from your source.

2. With the constant sound playing, measure the decibel reading at 10 different distances from the source.

3. Record your results in the table below.

Record your measurements in the first two columns of this table:

Analysis of results

1. Calculate the values for the third, fourth and fifth columns:

• For the third column (sound intensity), convert your decibel readings using: $I = 10^{\left(\frac{\text{dB}}{10} - 12\right)}$
• For the fourth column, square each distance value ($d^2$)
• For the fifth column, calculate the inverse of distance squared ($\frac{1}{d^2}$)

2. Plot a graph of intensity ($I$ in $\text{W} \cdot \text{m}^{-2}$) on the y-axis versus inverse distance squared ($\frac{1}{d^2}$) on the x-axis.

3. Alternatively, you can use spreadsheet software:

• Copy the fifth column (inverse distance squared) and third column (intensity) into the spreadsheet
• Important: The fifth column must be to the left of the third column, or your axes will be swapped
• Highlight both columns and insert a scatter graph
• Add appropriate labels and a title

4. Draw a line of best fit through your plotted data points. This line should pass as close to all data points as possible.

Discussion questions

Consider these questions when analysing your results:

1. Does your data support the inverse square law for sound? A straight line through the origin would confirm this relationship.

2. What other factors might have affected your measurements? Think about:

• Background noise
• Wind
• Reflections from nearby surfaces
• Accuracy of the decibel meter

3. How could you improve this investigation to get more reliable results?

Conclusion

Write a conclusion that:

• Refers to your original hypothesis
• Summarises what your data shows about the relationship between distance and intensity
• Links your findings to the inverse square law
• Notes any limitations or uncertainties in your results