Physics Study Guide/Sound

Sound is defined as mechanical sinosodial vibratory longitudinal impulse waves which oscillate the pressure of a transmitting medium by means of adiabatic compression and decompression consequently resulting in the increase in the angular momentum and hence rotational kinetic energy of the particles present within the transmitting medium producing frequencies audible within hearing range, that is between the threshold of audibility and the threshold of pain on a Fletchford Munson equal loudness contour diagram.

Intro

When two glasses collide, we hear a sound. When we pluck a guitar string, we hear a sound.

Different sounds are generated from different sources. Generally speaking, the collision of two objects results in a sound.

Sound does not exist in a vacuum; it travels through the materials of a medium. Sound is a longitudinal wave in which the mechanical vibration constituting the wave occurs along the direction of the wave's propagation.

The velocity of sound waves depends on the temperature and the pressure of the medium. For example, sound travels at different speeds in air and water. We can therefore define sound as a mechanical disturbance produced by the collision of two or more physical quantities from a state of equilibrium that propagates through an elastic material medium.

Sound

$decibel(\mathrm {dB} )=10\cdot \log \left({\frac {I_{1}}{I_{0}}}\right)$

Fig. 1: The Fletcher-Munson equal-loudness contours. Phons are labelled in blue.

The amplitude is the magnitude of sound pressure change within a sound wave. Sound amplitude can be measured in pascals (Pa), though its more common to refer to the sound (pressure) level as Sound intensity(dB,dBSPL,dB(SPL)), and the perceived sound level as Loudness(dBA, dB(A)). Sound intensity is flow of sound energy per unit time through a fixed area. It has units of watts per square meter. The reference Intensity is defined as the minimum Intensity that is audible to the human ear, it is equal to 10^-12 W/m², or one picowatt per square meter. When the intensity is quoted in decibels this reference value is used. Loudness is sound intensity altered according to the frequency response of the human ear and is measured in a unit called the A-weighted decibel (dB(A), also used to be called phon).

The Decibel

The decibel is not, as is commonly believed, the unit of sound. Sound is measured in terms of pressure. However, the decibel is used to express the pressure as very large variations of pressure are commonly encountered. The decibel is a dimensionless quantity and is used to express the ratio of one power quantity to another. The definition of the decibel is $10\cdot \log _{10}\left({\frac {x}{x_{0}}}\right)$, where x is a squared quantity, ie pressure squared, volts squared etc. The decibel is useful to define relative changes. For instance, the required sound decrease for new cars might be 3 dB, this means, compared to the old car the new car must be 3 dB quieter. The absolute level of the car, in this case, does not matter.

$I_{0}=10^{-12}{\mbox{ W}}/{\mbox{m}}^{2}$

Definition of terms

Intensity (I): the amount of energy transferred through 1 m2 each second. Units: watts per square meter

Lowest audible sound: I = 0 dB = 10-12 W/m2 (A sound with dB < 0 is inaudible to a human.) Threshold of pain: I = 120 dB = 10 W/m2

Sample equation: Change in sound intensity
Δβ = β₂ - β₁
= 10 log(I₂/I₀) - 10 log(I₁/I₀)
= 10 [log(I₂/I₀) - log(I₁/I₀)]
= 10 log[(I₂/I₀)/(I₁/I₀)]
= 10 log(I₂/I₁)
where log is the base-10 logarithm.

Doppler effect

$f'=f\,{\frac {v\pm v_{0}}{v\mp v_{s}}}$

f' is the observed frequency, f is the actual frequency, v is the speed of sound ($v=336+0.6T$), T is temperature in degrees Celsius $v_{0}$ is the speed of the observer, and $v_{s}$ is the speed of the source. If the observer is approaching the source, use the top operator (the +) in the numerator, and if the source is approaching the observer, use the top operator (the -) in the denominator. If the observer is moving away from the source, use the bottom operator (the -) in the numerator, and if the source is moving away from the observer, use the bottom operator (the +) in the denominator.

Example problems

A. An ambulance, which is emitting a 400 Hz siren, is moving at a speed of 30 m/s towards a stationary observer. The speed of sound in this case is 339 m/s.

$f'=400\,\mathrm {Hz} \left({\frac {339+0}{339-30}}\right)$

B. An M551 Sheridan, moving at 10 m/s is following a Renault FT-17 which is moving in the same direction at 5 m/s and emitting a 30 Hz tone. The speed of sound in this case is 342 m/s.

$f'=30\,\mathrm {Hz} \left({\frac {342+10}{342+5}}\right)$