3.2 Sound pressure and sound pressure level

Sound Pressure Fundamentals

Sound pressure describes the small, rapid fluctuations in air pressure caused by a sound wave. These fluctuations ride on top of the existing atmospheric pressure, and they're what your eardrum physically responds to. Understanding sound pressure and how we measure it with decibels is central to everything else in acoustics.

Because the range of pressures our ears can detect is enormous (spanning roughly six orders of magnitude), we use a logarithmic scale called the sound pressure level (SPL) to express measurements in decibels. This scale compresses that huge range into manageable numbers and also reflects how human hearing actually perceives loudness.

Sound Pressure and Wave Amplitude

Sound pressure is measured in pascals (Pa). As a sound wave travels through air, it creates alternating regions of slightly higher pressure (compressions) and slightly lower pressure (rarefactions) relative to the ambient atmospheric pressure.

• Sound pressure is directly proportional to the amplitude of the wave, which represents the maximum displacement of air particles from their resting position.
• Larger amplitude means greater pressure fluctuation, which you perceive as a louder sound.
• The frequency of these oscillations determines pitch: rapid oscillations produce high-pitched sounds, while slow oscillations produce low-pitched sounds. Frequency affects how fast the pressure cycles, but amplitude determines how much the pressure swings.

Sound Pressure Level in Decibels

Sound pressure level (SPL) expresses how loud a sound is by comparing its pressure to a fixed reference value, using a logarithmic ratio. The result is given in decibels (dB).

The SPL formula:

$SPL = 20 \log_{10}\left(\frac{p}{p_{\text{ref}}}\right) \text{ dB}$

where $p$ is the measured sound pressure and $p_{\text{ref}}$ is the reference pressure.

Why logarithmic? Two reasons:

• The range of audible pressures is massive. The quietest sound you can hear is about $2 \times 10^{-5}$ Pa, while the threshold of pain is around 20 Pa. That's a ratio of 1,000,000:1. A linear scale would be impractical.
• Human hearing perceives loudness roughly logarithmically. Doubling the perceived loudness doesn't require doubling the pressure; it requires roughly a 10 dB increase.

Calculations and Reference Pressure

Calculations for Sound Measurements

Instantaneous sound pressure for a pure tone is described by:

$p(t) = p_0 \sin(2\pi f t)$

where $p_0$ is the pressure amplitude, $f$ is frequency in Hz, and $t$ is time in seconds. This gives you the pressure at any specific moment.

For SPL calculations, you don't use the peak pressure directly. Instead, you use the root mean square (RMS) pressure, which represents the effective average pressure of the wave:

$SPL = 20 \log_{10}\left(\frac{p_{\text{rms}}}{p_{\text{ref}}}\right) \text{ dB}$

For a pure sine wave, $p_{\text{rms}} = \frac{p_0}{\sqrt{2}}$.

To convert back from dB to pascals, rearrange the formula:

$p = p_{\text{ref}} \times 10^{SPL/20}$

Quick example: What's the RMS pressure of a 94 dB SPL sound?

1. Start with $p = p_{\text{ref}} \times 10^{SPL/20}$
2. Plug in values: $p = 2 \times 10^{-5} \times 10^{94/20}$
3. Calculate the exponent: $10^{4.7} \approx 50119$
4. Multiply: $p \approx 2 \times 10^{-5} \times 50119 \approx 1 \text{ Pa}$

So 94 dB SPL corresponds to an RMS pressure of about 1 Pa. This is a handy benchmark to remember.

Reference Pressure in Acoustics

The standard reference pressure for airborne sound is $p_{\text{ref}} = 2 \times 10^{-5}$ Pa (20 µPa). This value approximates the threshold of human hearing at 1 kHz for a young, healthy listener.

• Using a fixed reference ensures that SPL values are consistent and comparable across different measurements, environments, and instruments.
• This reference is internationally standardized by organizations including ISO, ANSI, and IEC.

The reference pressure changes depending on the medium:

Water uses a lower reference pressure because of its much higher density and sound speed. This means you cannot directly compare a dB value measured in air with one measured in water. A sound that's 120 dB in water is not the same intensity as 120 dB in air.