# 17.3 Sound intensity  (Page 2/14)

 Page 2 / 14
$\frac{dV}{V}=\underset{\text{Δ}x\to 0}{\text{lim}}\frac{A\left[s\left(x+\text{Δ}x,t\right)-s\left(x,t\right)\right]}{A\text{Δ}x}=\frac{\partial s\left(x,t\right)}{\partial x}.$

The fractional change in volume is related to the pressure fluctuation by the bulk modulus     $\beta =-\frac{\text{Δ}p\left(x,t\right)}{dV\text{/}V}.$ Recall that the minus sign is required because the volume is inversely related to the pressure. (We use lowercase p for pressure to distinguish it from power, denoted by P .) The change in pressure is therefore $\text{Δ}p\left(x,t\right)=\text{−}\beta \frac{dV}{V}=\text{−}\beta \frac{\partial s\left(x,t\right)}{\partial x}.$ If the sound wave is sinusoidal, then the displacement as shown in [link] is $s\left(x,t\right)={s}_{\text{max}}\text{cos}\left(kx\mp \omega t+\varphi \right)$ and the pressure is found to be

$\text{Δ}p\left(x,t\right)=\text{−}\beta \frac{dV}{V}=\text{−}\beta \frac{\partial s\left(x,t\right)}{\partial x}=\beta k{s}_{\text{max}}\text{sin}\left(kx-\omega t+\varphi \right)=\text{Δ}{p}_{\text{max}}\text{sin}\left(kx-\omega t+\varphi \right).$

The intensity of the sound wave is the power per unit area, and the power is the force times the velocity, $I=\frac{P}{A}=\frac{Fv}{A}=pv.$ Here, the velocity is the velocity of the oscillations of the medium, and not the velocity of the sound wave. The velocity of the medium is the time rate of change in the displacement:

$v\left(x,t\right)=\frac{\partial }{\partial y}s\left(x,t\right)=\frac{\partial }{\partial y}\left({s}_{\text{max}}\text{cos}\left(kx-\omega t+\varphi \right)\right)={s}_{\text{max}}\omega \phantom{\rule{0.2em}{0ex}}\text{sin}\left(kx-\omega t+\varphi \right).$

Thus, the intensity becomes

$\begin{array}{cc}\hfill I& =\text{Δ}p\left(x,t\right)v\left(x,t\right)\hfill \\ & =\beta k{s}_{\text{max}}\text{sin}\left(kx-\omega t+\varphi \right)\left[{s}_{\text{max}}\omega \phantom{\rule{0.2em}{0ex}}\text{sin}\left(kx-\omega t+\varphi \right)\right]\hfill \\ & =\beta k\omega {s}_{\text{max}}^{2}{\text{sin}}^{2}\left(kx-\omega t+\varphi \right).\hfill \end{array}$

To find the time-averaged intensity over one period $T=\frac{2\pi }{\omega }$ for a position x , we integrate over the period, $I=\frac{\beta k\omega {s}_{\text{max}}^{2}}{2}.$ Using $\text{Δ}{p}_{\text{max}}=\beta k{s}_{\text{max}},$ $v=\sqrt{\frac{\beta }{\rho }},$ and $v=\frac{\omega }{k},$ we obtain

$I=\frac{\beta k\omega {s}_{\text{max}}^{2}}{2}=\frac{{\beta }^{2}{k}^{2}\omega {s}_{\text{max}}^{2}}{2\beta k}=\frac{\omega {\left(\text{Δ}{p}_{\text{max}}\right)}^{2}}{2\left(\rho {v}^{2}\right)k}=\frac{v{\left(\text{Δ}{p}_{\text{max}}\right)}^{2}}{2\left(\rho {v}^{2}\right)}=\frac{{\left(\text{Δ}{p}_{\text{max}}\right)}^{2}}{2\rho v}.$

That is, the intensity of a sound wave is related to its amplitude squared by

$I=\frac{{\left(\text{Δ}{p}_{\text{max}}\right)}^{2}}{2\rho v}.$

Here, $\text{Δ}{p}_{\text{max}}$ is the pressure variation or pressure amplitude in units of pascals (Pa) or ${\text{N/m}}^{2}$ . The energy (as kinetic energy $\frac{1}{2}m{v}^{2}$ ) of an oscillating element of air due to a traveling sound wave is proportional to its amplitude squared. In this equation, $\rho$ is the density of the material in which the sound wave travels, in units of ${\text{kg/m}}^{3},$ and v is the speed of sound in the medium, in units of m/s. The pressure variation is proportional to the amplitude of the oscillation, so I varies as ${\left(\text{Δ}p\right)}^{2}.$ This relationship is consistent with the fact that the sound wave is produced by some vibration; the greater its pressure amplitude, the more the air is compressed in the sound it creates.

## Human hearing and sound intensity levels

As stated earlier in this chapter, hearing is the perception of sound. The hearing mechanism involves some interesting physics. The sound wave that impinges upon our ear is a pressure wave. The ear is a transducer    that converts sound waves into electrical nerve impulses in a manner much more sophisticated than, but analogous to, a microphone. [link] shows the anatomy of the ear.

The outer ear, or ear canal, carries sound to the recessed, protected eardrum. The air column in the ear canal resonates and is partially responsible for the sensitivity of the ear to sounds in the 2000–5000-Hz range. The middle ear converts sound into mechanical vibrations and applies these vibrations to the cochlea.

Watch this video for a more detailed discussion of the workings of the human ear.

The range of intensities that the human ear can hear depends on the frequency of the sound, but, in general, the range is quite large. The minimum threshold intensity that can be heard is ${I}_{0}={10}^{-12}\phantom{\rule{0.2em}{0ex}}{\text{W/m}}^{2}.$ Pain is experienced at intensities of ${I}_{\text{pain}}=1\phantom{\rule{0.2em}{0ex}}{\text{W/m}}^{2}.$ Measurements of sound intensity (in units of ${\text{W/m}}^{2}$ ) are very cumbersome due to this large range in values. For this reason, as well as for other reasons, the concept of sound intensity level was proposed.

A spring with 50g mass suspended from it,has its length extended by 7.8cm 1.1 determine the spring constant? 1.2 it is observed that the length of the spring decreases by 4.7cm,from its original length, when a toy is place on top of it. what is the mass of the toy?
In this first example why didn't we use P=P° + ¶hg where ¶ is density
Density = force applied x area p=fA =p = mga, then a=h therefore substitute =p =mgh
Hlehle
Hlehle
sorry I had a little typo in my question
Anita
Density = m/v (mass/volume) simple as that
Augustine
Hlehle vilakazi how density is equal to force * area and you also wrote p= mgh which is machenical potential energy ? how ?
Manorama
what is wave
who can state the third equation of motion
Alfred
wave is a distrubance that travelled in medium from one point to another with carry energy .
Manorama
wave is a periodic disturbance that carries energy from one medium to another..
Augustine
two particles rotate in a rigid body then acceleration will be ?
same acceleration for all particles because all prticles will be moving with same angular velocity.so at any time interval u find same acceleration of all the prticles
Zaheer
what is electromagnetism
It is the study of the electromagnetic force, one of the four fundamental forces of nature. ... It includes the electric force, which pushes all charged particles, and the magnetic force, which only pushes moving charges.
Energy
what is units?
units as in how
praise
What is th formular for force
F = m x a
Santos
State newton's second law of motion
can u tell me I cant remember
Indigo
force is equal to mass times acceleration
Santos
The acceleration of a system is directly proportional to the and in the same direction as the external force acting on the system and inversely proportional to its mass that is f=ma
David
The uniform seesaw shown below is balanced on a fulcrum located 3.0 m from the left end. The smaller boy on the right has a mass of 40 kg and the bigger boy on the left has a mass 80 kg. What is the mass of the board?
Consider a wave produced on a stretched spring by holding one end and shaking it up and down. Does the wavelength depend on the distance you move your hand up and down?
no, only the frequency and the material of the spring
Chun
Tech
beat line read important. line under line
Rahul
how can one calculate the value of a given quantity
means?
Manorama
To determine the exact value of a percent of a given quantity we need to express the given percent as fraction and multiply it by the given number.
AMIT
meaning
Winford
briefly discuss rocket in physics
ok let's discuss
Jay
What is physics
physics is the study of natural phenomena with concern with matter and energy and relationships between them
Ibrahim
a potential difference of 10.0v is connected across a 1.0AuF in an LC circuit. calculate the inductance of the inductor that should be connected to the capacitor for the circuit to oscillate at 1125Hza potential difference of 10.0v is connected across a 1.0AuF in an LC circuit. calculate the inducta
L= 0.002H
NNAEMEKA
how did you get it?
Favour
is the magnetic field of earth changing
what is thought to be the energy density of multiverse and is the space between universes really space
tibebeab
can you explain it
Guhan