Sound Intensity and Pressure: How Sound Measurement Works in dB SPL

Introduction: This article explores the fundamentals of sound measurement, focusing on the relationship between sound intensity and sound pressure. We will illustrate the connections between intensity and pressure through the use of acoustic impedance, defining units of measurement such as the decibel. With a clear and concise approach, the article guides the reader in understanding the physical principles governing sound propagation, from the threshold of audibility to sound pressure levels that cause hearing damage.

In a previous post we introduced a measurement of the displacement of sound waves.

It is worth remembering the most important result we derived there, the formula

$$ y(t)=A\cdot\sin\left(2\pi f t+\phi\right)$$

being:

• $A$ the amplitude of the wave
• $f$ its frequency
• $t$ a given moment in time
• $\phi$ a displacement

for further details, refer to the previous post. Here we recall that sound is mechanical energy that propagates through a medium (such as air, water, or solids) in the form of pressure waves. These waves consist of compressions and rarefactions of the medium that carry mechanical energy, caused by the vibrations of a sound source.

Pressure can also be modeled with a function, called $p$ (for pressure), which depends on position in space. This function is:

$$p(x)=\Pi\sin⁡(2\pi fvx+\phi+\pi/2)$$

Where:

• $\Pi$ is the amplitude of the pressure, or the maximum pressure.
• $f$ is the frequency of the sound wave.
• $v$ is the speed of sound in the given medium.
• $\phi$ is the initial phase of the wave.

Now one may ask why the $\pi/2$ displacement. As always, when phases pop up, a bit of imagination is required. When we talk about sound waves, we describe both the movement of the particles of the medium (which move back and forth relative to their equilibrium position) and the pressure variations associated with the sound wave. These two quantities—the particle displacement and the sound pressure—are closely related but out of phase with each other in time.

In the specific mathematical model describing sound waves, the phase shift of π/2π/2 means that the maximum pressure occurs when the particle displacement is zero. Let’s see why this happens.

In a longitudinal sound wave, the particles of the medium oscillate back and forth in the same direction that the wave is propagating. As these particles move, they generate alternating zones of compression and rarefaction. When a particle moves forward, it pushes against the neighboring particles, creating a region of increased pressure called a compression. Conversely, as the particle moves away from its equilibrium position, it leaves behind a region of lower pressure, known as rarefaction. This process of alternating between compression and rarefaction allows the sound wave to carry mechanical energy through the medium.

The relationship between particle displacement and pressure in a sound wave reveals an interesting dynamic. When the particle displacement is at its maximum, the particles are at their furthest point from their equilibrium position. At this point, the particles are not actively compressing or rarefying the medium, which means that the pressure is relatively neutral. On the other hand, when the particle displacement is zero—meaning the particles are at their equilibrium position but are changing direction—this is when the pressure reaches its maximum or minimum values. Essentially, when particles are moving fastest through their equilibrium point, they create the most significant compressions and rarefactions, leading to maximum pressure variations.

The concept of a phase shift of $π/2$ helps to explain this relationship between pressure and displacement. The pressure variation is either ahead of or behind the particle displacement by a quarter of a cycle ($π/2$). This means that when the pressure reaches its maximum value (for instance, during maximum compression), the particle displacement is actually zero. In other words, the moments of highest pressure coincide with the moments when the particles are crossing their equilibrium position. This is a characteristic feature of sinusoidal waves, where the derivative of the wave function introduces a phase shift of $π/2$. In the context of sound waves, pressure is often related to the time derivative of displacement, which explains why there is this $π/2$ phase difference between the two quantities.

This phase relationship is fundamental to understanding how sound propagates through a medium. It shows that pressure and displacement do not peak simultaneously but instead are shifted in time by a quarter of a cycle, creating the characteristic behavior of sound waves as they move through air, water, or other materials.

In mathematical terms, the explanation is way simpler. The displacement of the particles in the medium describes how far and in which direction the particles move away from their equilibrium position as the wave propagates. On the other hand, the sound pressure represents the pressure variations that accompany the movement of the particles in the medium. It follows that the velocity of the particles is directly related to the time derivative of displacement. The pressure is highest when the particles are passing through their equilibrium position at maximum velocity, which is when the derivative of displacement (i.e., velocity) is at its maximum. The velocity of the particles, which is the time derivative of displacement, is given by:

$$v(t)=\frac{d}{dt}y(t)=A(2\pi f)\cos⁡(2\pi ft+\pi)$$

but for any real $x$, $\cos⁡(x)=\sin⁡(x+\pi/2)$ ...and this explains the further phase change!

Now it's worth taking a view of these formulas from a dimensional perspective. In these functions the values $A$, $\Pi$, $\phi$ and $v$ are all constants; the latter being the velocity of sound in a given medium (e.g. at room temperature, approximately 20°C, the speed of sound is approximately v=343 m/s). Observe that being constants doesn't mean being adimensional. $A$ is an amplitude, hence measured in meters. Being a velocity, $v$ is measured in m/sec.

So, let’s revisit $y(t)$. We know that $t$ is measured in seconds and the result of the sine function is a pure number, and that $f$ is measured in $sec^{−1}$. The argument of the sine function is a constant ($2\pi$) multiplied by the product of a frequency, which is measured in $sec^{−1}$, and time, which is measured in seconds. All of this is added to another pure number—so yes, the argument of the sine function is, coincidentally, dimensionless. However, being $A$ a spatial measurement, so is $y(t)$.

Similarly: $\Pi$ is the amplitude of the pressure, which represents the maximum pressure. It is measured in units of pressure, such as Pascal (Pa). $x$ represents position in space and is measured in meters (m). $λ$ is the wavelength and is also measured in meters (m). Here too, the argument of the sine function is the ratio of two lengths, multiplied and added to constants—another dimensionless value. However, the maximum pressure $\Pi$ is measured in Pascal (Pa), exactly like $p(x)$.

For those - like me - who are not too much into physics:

• Newton is the measurement unit for force. It is defined as
  $1N=1kg\cdot m\cdot s^{-2}$.
• Pascal is the measurement unit for pressure. It is defined as:
  $1Pa=1N\cdot m^{-2}$.
• The latter can be rewritten as $\frac{kg}{m\cdot s^2}$
• For instance: Standard atmospheric pressure: approximately 101,325 Pa or 101.3 kPa

The painful sound pressure for the human ear starts at around 20 Pa, while a grenade exploding near the point of detonation can easily generate pressures exceeding 100,000 Pa (or 100 kPa) at just a few meters away, which is immensely greater than those measured in micropascals.

It is interesting, as human beings, to consider some key references, such as:

1. Threshold of audibility: The weakest sound that the average human ear can perceive, the threshold of audibility, is typically 20 micropascals (µPa), or $2×10^{−5} Pa$. This pressure represents the lowest reference point for perceivable sound levels.
2. Threshold of pain: The threshold of pain is reached at much higher sound pressures, generally around 20 Pascals. This is the level at which sound becomes physically painful for most people.
3. Permanent hearing damage: Prolonged sounds above a pressure of approximately 0.63 Pa can cause long-term hearing damage, especially if the exposure is continuous. Brief exposures to sound pressure levels above 200 Pa can cause immediate and permanent hearing damage, even with a single exposure. Such levels can be generated, for example, by explosions or supersonic jets at close range.

Similar to what was observed in the previous post, pressure also attenuates with distance. In an open and homogeneous space, which we can consider isotropic, the intensity of pressure $\Pi$ (and therefore also $p(x)$) is inversely proportional to the distance $R$ from the sound source. Other factors that reduce pressure include sound absorption in the medium and dispersion. The sound is gradually absorbed by the propagation medium (such as air), especially at high frequencies, causing a reduction in the amplitude of the pressure waves. In enclosed or irregular environments, sound waves can reflect and scatter, further reducing the perceived pressure at a distance.

We can model all this as:

$$\Pi(R)=\frac{\Pi_0}{R}$$​​

This formula also includes a $\Pi_0$​, which is the pressure amplitude at a reference distance (often considered very close to the source, for example, $R=1m$).

This relationship is called the INVERSE SQUARE LAW because, in many physical phenomena, the intensity of a quantity decreases in proportion to the square of the distance from the source. However, in the specific case of pressure, the decay is inversely proportional to the distance and not to its square.

If we want to include other effects, such as the absorption of the sound wave in the medium (air, water, etc.), we can add an exponential term that models the attenuation due to the medium:

$$\Pi(R)=\frac{\Pi_0}{R} \exp⁡(−\alpha R)$$

Where:

• $$\alpha$$ is the acoustic absorption coefficient of the medium, which depends on the material properties (such as temperature and humidity in the air).
• $\exp⁡(−\alpha r)$ represents then the exponential attenuation due to sound absorption in the medium.This term represents the attenuation due to sound absorption, which increases with the distance $R$ from the point of origin. The value of $\alpha$ depends on the medium (for air, for example, it depends on the sound frequency and atmospheric conditions).
• $\Pi_0$​: This term follows the inverse square law, which indicates how pressure decreases with distance due to the dispersion of sound energy over increasingly larger surfaces.

If αα is small (as in air at low frequencies), the absorption effect will be less significant, and the pressure decrease will mainly follow the inverse square law. At very large distances or in highly absorbent media (such as water or under high humidity conditions), the exponential effect becomes more relevant.

Relationship between Pressure and Displacement

There is a close correlation between the decrease in $\Pi$ (the sound pressure amplitude) and the decrease in $A$ (the particle displacement amplitude) in the case of sound waves.

The sound pressure amplitude $\Pi$ is directly related to the displacement amplitude $A$ of the particles in the medium through a relationship that depends on the medium's density and the speed of sound. The approximate relation is:

$\Pi ∝\rho vfA$

Where:

• $\Pi$ is the pressure amplitude (Pascal, Pa).
• $\rho$ is the density of the medium (for air, about 1.2 kg/m³).
• $v$ is the speed of sound in the medium (about 343 m/s in air at room temperature).
• $f$ is the frequency of the sound wave.
• $A$ is the displacement amplitude (i.e., the maximum displacement of particles in the medium).

This equation tells us that pressure $\Pi$ and displacement $A$ are directly proportional to each other, given the same density, speed of sound, and frequency.

Decrease with Distance

Both $\Pi$ and $A$ follow the same principle of decreasing with distance from the sound source. This is because both depend on the energy carried by the sound wave, which disperses as it propagates through space.

Using the inverse square law:

$A(R)∝R^{-1}$

And since $\Pi∝A$, also $\Pi$ follows the same decrease:

$\Pi(r)∝R^{-1}$

Combined Effect

In a practical model, the decrease in both displacement amplitude $A$ and pressure amplitude $\Pi$ is proportional to the distance $R$ from the sound source. If we also consider the acoustic absorption of the medium, we can use the same form for both:

$A(r)=\frac{A_0}{R}exp⁡(−\alpha R)$

$\Pi(r)=\frac{\Pi_0}{R}exp⁡(−\alpha R)$

This means that as you move away from the source, both the displacement amplitude of the particles $A$ and the pressure amplitude $\Pi$ decrease following the same pattern: they decrease linearly with $R$ (due to dispersion) and exponentially with $R$ (due to sound absorption in the medium).

We had seen that AA decreases with time, not with space. When we talk about the exponential decrease of amplitude over time, we are referring to a phenomenon of damping. Sound can be subject to damping due to various factors, such as air resistance or internal friction in the material through which the wave propagates.

The most common model to describe the exponential decrease of amplitude over time is:

$$A(t)=A_0exp⁡(−\gamma t)$$

Where:

• $A(t)$ is the amplitude at time $t$.
• $A_0$​ is the initial amplitude (at time $t=0$).
• $\gamma$ is the damping coefficient, which depends on the properties of the medium and the wave frequency.
• $exp⁡(−\gamma t)$ describes the exponential decrease of the amplitude over time.

This exponential behavior describes the decay of the wave’s energy over time and manifests as a gradual reduction of the amplitude of the oscillations.

With distance (space), the amplitude decreases as $R^{-1}$​ due to the dispersion of sound energy. Over time, the amplitude may decrease exponentially as $exp⁡(−\gamma t)@ due to damping or friction.

So, when we say that $A$ decreases exponentially, we are referring to the decay of the wave's energy over time due to damping factors, while the spatial decrease follows the inverse square law. Both phenomena can coexist: a wave can decrease in both space (with distance) and time (with exponential decay).

Aren't you a physics enthusiast and hate math? No worries, for we've got your gluteus maximus! Here's the contents of the post for the rest of us!