17-4 Intensity and Sound Level

17-4 Intensity and Sound Level#

Prompts

  • Define sound intensity \(I\). What are its units? How does it relate to power \(P\) and area \(A\)?

  • How does intensity \(I\) depend on the displacement amplitude \(s_m\)? Why does doubling \(s_m\) quadruple \(I\)?

  • For an isotropic point source emitting power \(P_s\), write the intensity at distance \(r\). Why does \(I \propto 1/r^2\)?

  • Define sound level \(\beta\) in decibels. What is \(I_0\)? If intensity doubles, by how many dB does \(\beta\) change?

  • The sound level drops by 20 dB when you move away from a speaker. By what factor does the intensity change?

Lecture Notes#

Overview#

  • Intensity \(I\) is the power per unit area carried by a sound wave—a physical measure of how much energy crosses a surface per second.

  • Sound level \(\beta\) (in decibels) is a logarithmic scale for intensity, matching the wide range of human hearing (from barely audible to painful).

  • For a point source radiating equally in all directions, intensity falls off as \(1/r^2\)—energy spreads over an ever-larger spherical surface.

Intensity: definition and power#

The intensity of a sound wave at a surface is the average rate per unit area at which energy is transferred by the wave:

(126)#\[ I = \frac{P}{A} \]

  • \(P\): power (W)—time rate of energy transfer.

  • \(A\): area of the surface intercepting the sound (m²).

  • Units: W/m².

Intensity and displacement amplitude#

For a sinusoidal sound wave (section 17-2), intensity is related to the displacement amplitude \(s_m\):

(127)#\[ I = \frac{1}{2}\rho v \omega^2 s_m^2 \]

  • \(\rho\): density of the medium; \(v\): speed of sound; \(\omega\): angular frequency.

  • \(I \propto s_m^2\): doubling the amplitude quadruples the intensity.

Why \(I \propto s_m^2\)?

Energy in oscillatory motion scales as (amplitude)². The wave carries energy; the rate at which it delivers power per unit area is proportional to \(s_m^2\).

Isotropic point source#

An isotropic point source emits sound equally in all directions. The power \(P_s\) passes through every spherical surface centered on the source. At distance \(r\):

(128)#\[ I = \frac{P_s}{4\pi r^2} \]

  • \(I \propto 1/r^2\): doubling the distance halves the intensity (or reduces it by a factor of 4).

  • Same geometric idea as for light from a point source or Coulomb’s law for electric field.

Poll: Power through off-center sphere

A 10 kW radio source is inside a sphere. The sphere is not centered on the source. The total power through the sphere is:

  • (A) Less than 10 kW

  • (B) 10 kW

  • (C) More than 10 kW

  • (D) Depends on sphere size

Sound level in decibels#

Human hearing spans roughly 12 orders of magnitude in intensity. The sound level \(\beta\) uses a logarithmic scale:

(129)#\[ \beta = (10\;\text{dB})\,\log_{10}\left(\frac{I}{I_0}\right) \]

  • \(I_0 = 10^{-12}\) W/m²: reference intensity (near the threshold of hearing).

  • \(I = I_0\) → \(\beta = 0\) dB.

  • Factor of 10 in \(I\) → add 10 dB: \(\log(10) = 1\) → \(\Delta\beta = 10\) dB.

Change in intensity

Change in \(\beta\)

\(I \times 10\)

\(+10\) dB

\(I \times 100\)

\(+20\) dB

\(I \times 2\)

\(+3\) dB (since \(\log 2 \approx 0.30\))

Typical sound levels#

Environment

Sound level (dB)

Threshold of hearing

0

Rustle of leaves

~10

Conversation

~60

Rock concert

~110

Pain threshold

~120

Jet engine

~130

Intensity vs loudness

Intensity is a physical quantity (power per area). Loudness is subjective and depends on frequency—the ear is more sensitive to some frequencies than others.

Change in sound level#

When intensity changes from \(I_i\) to \(I_f\):

(130)#\[ \beta_f - \beta_i = (10\;\text{dB})\,\log_{10}\left(\frac{I_f}{I_i}\right) \]

  • \(\Delta\beta = -20\) dB → \(I_f/I_i = 10^{-2} = 0.01\) (intensity reduced by a factor of 100).

  • \(\Delta\beta = +3\) dB → \(I_f/I_i = 2\) (intensity doubled).

Poll: Sound level at doubled distance

At 50 m from a speaker, the sound level is 70 dB. Which is closest to the level at 100 m?

  • (A) 50 dB

  • (B) 56 dB

  • (C) 60 dB

  • (D) 64 dB

  • (E) 66 dB

Poll: Moving away

You stand 2 m from a speaker. You move to 6 m. By how many dB does the sound level change? (Assume \(I \propto 1/r^2\).)

  • (A) \(-\)3 dB

  • (B) \(-\)9.5 dB

  • (C) \(-\)20 dB

Summary#

  • \(I = P/A\)—intensity is power per unit area (W/m²).

  • \(I = \frac{1}{2}\rho v \omega^2 s_m^2\)—\(I \propto s_m^2\).

  • \(I = P_s/(4\pi r^2)\)—for isotropic point source \(I \propto 1/r^2\).

  • \(\beta = (10\,\text{dB})\,\log(I/I_0)\) with \(I_0 = 10^{-12}\) W/m².

  • Factor of 10 in \(I\) → add 10 dB; use \(\Delta\beta\) to relate changes in \(I\).