Sound Power

🎯 Learning Objectives:

In this topic we will look into sound power as our first power quantity, and we will find out how it is different from a field quantity like pressure. Sound power is a constant, non directional, calculated value and is quite significant in the field of acoustics.

This describes the applications of sound power as a tool for labeling, derive the formula for Sound Power Level (SWL) in decibels, and derive the formula for calculating sound pressure produced by a source of known power at any distance in an ideal space.
Previously we took a look at sound pressure. Sound pressure is a field quantity, changes based on the distance from the source, and can be easily measured using a microphone or some other pressure sensitive transducer. In this topic we will talk all about sound power and on all three points, sound power is completely different. It is a power or energy quantity, and not a field quantity. There is no concept of distance when talking about sound power, it is independent of directionality and is independent of distance from the observer or the source. It is a constant value no matter what the environmental conditions are. And sound power is a theoretically calculated value, which can't easily be measured by a single device.

Here is a comparison between sound pressure and sound power in the Figure below

So based on these starting points, we might think that they are totally unrelated concepts. But there is a cause and effect relationship between them.
In previous topic (sound pressure), we looked at a heater analogy, but this time, let us look at light example by watching the video lecture.

Video Lecture:

So, as we can see from the video lecture, the sound power or acoustic power is quite an important measure in the field of acoustics. Since sound power is a constant non directional value, it is possible to compare the sound output of different devices, without any knowledge of the environment in which they were tested or the distance at which measurements were taken. This makes sound power levels ideal for product labeling, for comparing the sound emission of different power tools as shown in the Figure below.

If an operator were to use a pneumatic drill for instance, if he knows the sound power generated from this tool, he would know the appropriate level of ear protection to use and the maximum time that he can spend on the equipment in one go to meet health and safety guidelines and to prevent long term hearing damage. We could theoretically compute the Sound pressure level at any distance from a source, just by knowing the sound power of the source.
Let us go a further with this topic, because this is the first time we are dealing with representing and quantifying a power quantity. So far, we have only looked at sound pressure, which is a field quantity. So let us dig a little deeper. Acoustic sound power is the total airborne acoustic energy emitted by a device over unit time. It is measured in Watts. Acoustic power levels of many devices are quite small. For example, a noisy coffee grinder may emit only about 5 milli Watts of acoustic power. On the other hand, a rocket at take off could produce 100 million Watts of sound power. The range of values, sounds power can take on is massive! So naturally, we want to represent these values in the logarithmic scale, or better yet, the decibel scale.
As we know that a Bel is said to be log of the ratio between two power quantities and a decibel is just one tenth of a Bel s we can see from the following Equation.

And since the decibel is a relative scale of values, we need to pick the reference power level against which all other power levels are measured. What do we pick for the reference value? But before that, we want the power level in decibels to be comparable to pressure level in decibel. Since they are dimensionless units, we want some means of comparing between the two. We do not want to have these values completely different from each other. To achieve this, we need to have similar conditions when choosing the reference value. When choosing the reference value for sound pressure, we chose the minimum pressure value that a good pair of ears could just about hear a 1 kHz sine tone in an anechoic chamber. This value was established to be 20 micro Pa. Now, the question is, what is the power required to generate this pressure value from the same measuring distance. The complicated answer to this question is $10^{−12}$ watts or 1 pico Watt, which is a tiny infinitesimal value.
So, here is the formula for sound power level in decibels, or SWL.

If you have read the lecture on sound pressure, we derived the sound pressure level or SPL, and it had the following formula.

Here we have some interesting things to note.
Why is one value multiplied by 10, and the other by 20? It’s a bit confusing? In fact if you look at a lot of other decibel units, they are all similar, but they have the same quirk as show below.

What do we use? 10 or 20? How do we keep track of all of them?

The simple answer is we have to identify if the unit we are looking at is a power quantity or a field quantity. Power quantities always have a multiplier of 10 and field quantities always have a multiplier of 20. That is because, in the real world, the power quantities are directly proportional to the square of the field quantities, as $power ∝ field^2$. So, sound power is directly proportional to square of the pressure. And electrical power is directly proportional to the square of the voltage or the square of the current. And the log of a squared quantity, is just 2 times the log of the quantity. Here we can see how it works.

As previously mentioned that by knowing the sound power of a source, we can "theoretically" calculate the sound pressure caused by it at any distance. I say theoretically, because the formula assumes an ideal space, where the source is suspended in mid air, with no primary or secondary reflections of any kind. In the real world, these sort of spaces are hard to come by, but engineers make clever modifications to the formula to fit the space being measured.
We will not get into that, however, let watch the following video lecture to understand the concept by animation.

📝 Key Takeaways

• Sound Pressure vs. Sound Power

  • Sound pressure depends on distance and environment, while sound power is constant, non-directional, and independent of distance.

• Measurement

  • Sound pressure is measured with microphones.
  • Sound power is calculated and used for labeling, product comparison, and regulatory compliance.

• Logarithmic Scale

  • Sound power and pressure cover wide ranges, so we use a logarithmic (decibel) scale for easier representation and comparison.

• Power vs. Field Quantities

  • Power quantities (like sound power) behave differently from field quantities (like pressure), which affects how we calculate decibels.

• Practical Relevance

  • Knowing sound power helps predict sound pressure levels at different distances and informs product labeling and safety guidelines.

• Real-World Application

  • Sound power is ideal for comparing devices because it is independent of the environment, while sound pressure is what we actually perceive or measure.

🧪 Interactive demos

🧠 Quick Quiz

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