Click here to purchase the entire book in PDF format. Pressure If you listen to the radio in the mornings, they'll give you the news, the sports, the traffic and the weather. Part of the weather report is to tell you that the barometric pressure is something around 100 kilopascals (abbreviated kPa)4.1. What does this mean? Well, the air particles around you are all under pressure due to things like gravity and the weight of the air particles above them and other meteorological things that are outside the scope of this book. That pressure determines the amount of physical space between molecules in the air. When there's a higher barometric pressure, there's less space between the molecules than there is on a day with a lower barometric pressure. We call this the stasis pressure and abbreviate it $\wp _{o}$. When all of the particles in a gaseous medium (like air) in a given volume (like a room) are at normal pressure, then the gas is said to be at its volume density (also known as the constant equilibrium density), abbreviated $\rho_{o}$, and measured in kg/m$^3$. Remember that this is actually kilograms of air per cubic metre - if you were able to trap a cubic metre and weigh it, you'd find out that it's about 1.3 kg. These molecules like to stay at the same pressure all over, so if you bunch them up in one place in a room somehow, they'll move around to try and equalize the difference. This is kind of like when you pour a glass of water into a bucket, the water level of the entire bucket equalizes and therefore rises, rather than the water from the glass all bunching up in a little mound of water where you poured it in... Let's think of this as a practical example. We'll hang the piece of paper in front of a fan. If we turn on the fan, we're essentially increasing the pressure of the air particles in front of the blades. The fan does this by removing air particles from the space behind it, thus reducing the pressure of the particles behind the blades, and putting them in front. Since the pressure in front of the fan is greater than any other place in the room, we have a situation where there is a greater air pressure on one side of the piece of paper than the other. The obvious result is that the paper moves away from the fan. This is a large-scale example of how you hear sound. Let's say hypothetically for a moment, that you are sitting alone in a sealed room on a day when the barometric pressure is 100 kPa. Let's also say that you have a clarinet with you and that you play a concert A. What physically happens to convert air coming out of your mouth into a concert A coming in your ears? To begin with, let's pretend that a clarinet is just a tube with a hole in each end. One of the holes has a springy piece of wood next to it which, if you press on it, will close up the hole. When you blow into the hole, you bunch up the air particles and create a little area of high pressure inside the mouthpiece. Blowing into the hole with the reed on it also has the effect of pushing the reed against the hole and sealing it so that no more air can enter the clarinet. At that point the little high pressure area moves down the clarinet and leaves a low pressure behind it. Remember that the reed is springy, and it doesn't like being pushed up against the hole in the mouthpiece, so it bounces back and lets more air in. Now the cycle repeats and goes back to step 1 all over again. In the meantime, all of those high and low pressure areas move down the clarinet and radiate out the bell into the room like ripples on a lake when you throw in a rock. From there, they get to your ear and push your eardrum in and out (high pressure pushes in, low pressure pulls out) Those little fluctuations in the air pressure are small variations in the stasis pressure $\wp _{o}$. They're usually very small, never more than about $\pm1$ Pa (though we'll elaborate on that later...). At any given moment at a specific location, we can measure the the instantaneous pressure, $\wp$, which will be close to the stasis pressure, but slightly different because there's a sound source causing it to change. Once we know the stasis pressure and the instantaneous pressure, we can use these to figure out the instantaneous amplitude of the sound level, (also called the acoustic pressure or the excess pressure) abbreviated p, using Equation 3.1. $$p = \wp - \wp_{o}$$ (4.1) Figure 3.1: A graphic representation of the values $\wp _{o}$ (the stasis pressure), $p$ (instantaneous amplitude) , $\wp$ (instantaneous pressure), and $P$ (maximum peak pressure). To see an animation of what this looks like, check out www.gmi.edu/ drussell/Demos/waves/wavemotion.html. A sinusoidal oscillation of this pressure reaches a maximum peak pressure $P$ which is used to determine the sound pressure level or SPL. In air, this level is typically expressed in decibels as a logarithmic ratio of the effective pressure $P_e$ referenced to the threshold of hearing, the commonly-accepted lowest sound pressure level audible by humans at 1 kHz, 20 microPascals, using Equation 3.2 [Woram, 1989]. The intricacies of this equation have already been discussed in Section 2.2 on decibels. $$SPL = 20 \log_{10} \left[ \frac{P_e}{20 * 10^{-6} Pa} \right]$$ (4.2) Note that, for sinusoidal waveforms, the effective pressure can be calculated from the peak pressure using Equation 3.3. (If this doesn't sound familiar, it should - re-read Section 2.1.6 on RMS.) $$P_e = \frac{P}{\sqrt{2}}$$ (4.3)