Study Notes: Sound Waves, Intensity, and the Doppler Effect

Sound Waves: Beats, Intensity, and Doppler Effect

Beats

When two sound sources emit waves of nearly the same frequency, an interference pattern called beats is produced. This phenomenon is important in acoustics and musical tuning.

• Definition: Beats are periodic variations in sound intensity resulting from the superposition of two waves of slightly different frequencies.

• Beat Frequency: The frequency of the beats is given by the absolute difference between the two source frequencies:

$f_b = |f_1 - f_2|$

• Example: If $f_1 = 50$ Hz and $f_2 = 48$ Hz, then $f_b = 2$ Hz.

Wave Intensity

Waves transfer energy through a medium, and the rate at which this energy is transferred is called power. The intensity of a wave quantifies how much energy passes through a unit area per unit time.

• Power: The rate of energy transfer by a wave:

$P = \frac{W}{t}$

• Units: Joules per second ($J/s$), also known as Watts (W).

• Intensity: Defined as the power per unit area perpendicular to the wave direction:

$I = \frac{P}{A}$

• Units: Watts per square meter ($W/m^2$).

• Example: If a wave delivers 2 W of power across an area of 4 $m^2$, then $I = 0.5$ $W/m^2$.

Spherical Waves

Many sources, such as light and sound, emit energy uniformly in all directions, forming spherical waves. The concept of wavefronts is used to describe surfaces of constant phase.

• Wavefronts: Surfaces over which the wave has the same phase (e.g., all points on a crest).

• Area of a sphere: $A_{sphere} = 4\pi r^2$

• Intensity for spherical waves:

$I = \frac{P}{4\pi r^2}$

• As distance $r$ from the source increases, intensity $I$ decreases.

• This relationship is known as the Inverse Square Law.

Sound Level

Sound intensity is often measured in decibels (dB), which is a logarithmic unit. The Sound Intensity Level quantifies how loud a sound is relative to a reference intensity.

• Formula:

$\beta = (10\,\mathrm{dB}) \log \left( \frac{I}{I_0} \right)$

• $I_0 = 10^{-12}\,W/m^2$ is the threshold of human hearing.

• Maximum tolerable intensity: $I \approx 10\,W/m^2$ ($\beta \approx 130$ dB, threshold of pain).

Sound Level Examples

dB	Intensity ($W/m^2$)	Source
0	$10^{-12}$	Threshold of human hearing
10	$10^{-11}$	Human breathing at 3 m
20	$10^{-10}$	Rustling leaves
50	$10^{-7}$	Quiet restaurant
90	$10^{-3}$	Loud factory
120			Rock concert
130		Threshold of pain
180	$10^6$	Rocket engine at 30 m

Logarithms in Sound Level Calculations

Logarithms are used to relate ratios of sound intensities to differences in sound levels. This is because logarithms convert multiplication/division into addition/subtraction.

• Properties:

$\log(A) - \log(B) = \log\left(\frac{A}{B}\right)$ $\log(A) + \log(B) = \log(AB)$ $10^{\log(A)} = A$

• A difference of 10 dB corresponds to a factor of 2 in perceived loudness.

• A difference of 3 dB corresponds to a factor of 2 in intensity ($W/m^2$).

Example Calculation

• At 2.0 m from a sound source, measured sound level is 75 dB.

• Find intensity, power emitted, and distance for loudness to drop to 65 dB.

• Answers: $3.2 \times 10^{-5}$ $W/m^2$, $1.6$ mW, $6.3$ m.

Doppler Effect

The Doppler Effect describes the change in frequency or wavelength of a wave in relation to an observer moving relative to the source. It is fundamental in acoustics, astronomy, and radar technology.

• Moving Source: If the source moves towards the observer, the observed frequency increases (higher pitch). If moving away, the frequency decreases (lower pitch).

• Moving Observer: If the observer moves towards a stationary source, the observed frequency increases; if moving away, it decreases.

• Observed Wavelength: The wavelength changes depending on relative motion.

• Speed of sound in air ($v$) is constant for a given temperature and pressure.

Equations for Doppler Effect

• Source moving towards stationary observer:

$f' = \frac{f}{1 - \frac{u_s}{v}}$

• Source moving away from stationary observer:

$f' = \frac{f}{1 + \frac{u_s}{v}}$

• Observer moving towards stationary source:

$f' = f \left(1 + \frac{u_d}{v}\right)$

• Observer moving away from stationary source:

$f' = f \left(1 - \frac{u_d}{v}\right)$

• General Doppler Shift Equation:

$f' = f \left( \frac{v \pm u_d}{v \mp u_s} \right)$

• $v$ = speed of wave, $u_d$ = speed of observer (detector), $u_s$ = speed of source

• Use $+$ when observer/source is approaching, $-$ when receding.

Applications of Doppler Effect

• Astronomy: Measuring red/blue shifts to determine stellar velocities.

• Police radar: Determining vehicle speeds using microwave Doppler shift.

• Meteorology: Doppler radar for tracking precipitation movement.

Sonic Boom

A sonic boom occurs when an object travels faster than the speed of sound, causing sound waves to accumulate and form a shock wave.

• Waves cannot precede the object, so they form a cone behind it.

• When the shock wave passes, all compressions and rarefactions are heard at once as a loud boom.

• Commonly produced by supersonic aircraft, meteors, and space shuttles.

• Example: F/A-18F Super Hornet producing a visible shock wave.

Doppler Effect: Example Problems

• Problem 1: Fire truck siren at rest: 1400 Hz; measured at 1600 Hz as it approaches. Speed of sound: 343 m/s. Answer: 43 m/s.

• Problem 2: Police car siren: 1000 Hz. Find observed frequencies for various motions (answers: 1088 Hz, 1081 Hz, 1084 Hz).

• Problem 3: Sound source: 5.0 kHz; plate moving towards source at 50 m/s. Speed of sound: 343 m/s. Answers: 5.7 kHz (at plate), 6.7 kHz (reflected wave).

Additional info: These notes cover topics from Chapter 14 (Wave Motion) and Chapter 13 (Oscillatory Motion), as well as applications relevant to acoustics and wave phenomena in college-level physics.