Understanding Sound Intensity: Formula, Hearing Range, and Differences Between some Animal Species

Definition 1 (isotropic material)

An isotropic material is a material that has the same physical properties in all directions. In other words, sound (or a wave in general) propagates in the same way in every direction within an isotropic material. Both air and water are considered good approximations of isotropic materials for sound propagation, at least on a macroscopic scale and under normal conditions.

Let's consider a sphere. The surface area of the sphere is given by $S=4\pir^2$, measured in square meters. Now, imagine a quantity of kinetic energy $\frac{d}{dt}$, where $E$ is the mechanical energy of the waves. Call this value $\Omega$, the sound power, which is measured in watts and represents this variation of energy over time. We define sound intensity $I(d)$ at a distance $d$ as:

$$I(d)=\frac{\Omega}{4\pi d^2}$$

This describes the dispersion of sound energy. Sound intensity $I$, which is the power per unit area, is proportional to the square of the displacement amplitude $A$ and the frequency $f$. It also depends on the medium's density $\rho$ and the speed of sound $v$. The relationship between $I$ and $A$ is:

$$I=\frac{1}{2}\rho v\left(2\pi fA\right)^2$$

This shows how sound intensity depends on both the square of the amplitude and the frequency.

Intensity $I$ is defined as power per unit area, related to the mechanical energy transported by the wave. The kinetic energy per unit volume is:

$$E_\text{kinetic}=\frac{1}{2}\rho v^2$$

where $v(t)$ is the velocity of a particle in oscillation, given by the derivative of displacement:

$$v(t)=A(2\pi f)\cos⁡(2\pi ft)$$

The maximum velocity is $v_\text{max}=A(2\pi f)$, and the kinetic energy becomes:

$$E_\text{kinetic}=\frac{1}{2}\rho \left(A(2\pi f)\right)^2$$

Definition 2 (Acoustic Impedance)

Acoustic Impedance measures the resistance a medium offers to the passage of a sound wave. If we denote it with $Z$, we can define it as:

$$Z=\rho v$$

being

• $\rho$ the density of the medium (e.g., 1.2 kg/m³ for air),
• $v$ the speed of sound (e.g., 343 m/s in air).

This brings that our definition of Sound Intensity can be rewritten as

$$I=\frac{1}{2}Z(2\pi fA)^2$$

Sound intensity is an objective measure of the energy transported by a sound wave per unit area and time, independent of whether the sound is audible. The human ear can perceive sounds between approximately 20 Hz and 20 kHz, but sound intensity applies even beyond this range.

It is noted that 20 µPa corresponds to $$10^{−12} W/m^2$$ (the threshold of hearing), while 20 Pa corresponds to around $$1000 W/m^2$$ (the threshold of pain). A simple division show that this dynamic range spans 15 orders of magnitude in intensity.

These observations offer an interesting comparison of the hearing ranges of different animals. Although cats and lions are both felids, their hearing capabilities differ greatly: cats can hear sounds up to 85 kHz, while lions are limited to 10 kHz. Similarly, dogs can hear up to 60 kHz, while wolves hear up to 40 kHz. Bats, with a hearing range of 1 kHz to 200 kHz, use echolocation to hunt.

• Lion: 50 Hz - 10 kHz
• Cat: 48 Hz - 85 kHz
• Dog: 40 Hz - 60 kHz
• Bat: 1 kHz - 200 kHz
• Wolf: 40 Hz - 40 kHz

Let’s estimate the ratio $I_\text{min}/​I$​, where $I_\text{min}=10^{−12} W/m^2$ is the minimum audible intensity. This ratio is dimensionless because both $I$ and $I_\text{min}$​ are measured in $W/m^2$. To simplify the range, we use logarithms:

$$L=10\log_⁡10(\frac{I}{I_text{min}})$$

The sound level in decibels (dB SPL) is given by:

$$L_\text{SPL}=20log⁡10(\frac{P}{P_0})$$

where $P_0$​ is the reference pressure. Then, the relationship between intensity and pressure is:

$$I=\frac{P^2}{2Z}$$

This concludes the introduction to the physics of sound. Surely, there are many more relevant concepts, and we will likely return to them in future posts. However, this introduction covers more than enough for us to get started.

The next post will focus on the auditory system of certain animals, to understand how sound, seen as a form of energy, is transformed into electrical impulses that can be processed by the brain.

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!