Honors Physics Unit 14 Review: Sound

Sound in air is a longitudinal mechanical wave. A vibrating source pushes on nearby particles and creates compressions (high pressure, particles crowded together) and rarefactions (low pressure, particles spread apart) that travel outward. Particles oscillate around their resting positions, while the disturbance carries energy through the medium. Sound cannot travel through a vacuum, though it can travel through gas or other matter in space. Solids can also carry transverse sound waves.

All waves follow $v = f\lambda$. The medium sets the speed and the source sets the frequency. Frequency is what you hear as pitch, and humans hear roughly 20-20,000 Hz.

Speed depends on stiffness and density. In a fluid, $v = \sqrt{B/\rho}$, where $B$ is the bulk modulus and $\rho$ is density. In solids, the formula depends on the wave type and the relevant elastic properties. Sound often travels faster in solids than in air because their greater stiffness outweighs their greater density. In air, warmer means faster, roughly $v \approx 331 + 0.6T$ m/s with $T$ in °C.

Worked example. A 440 Hz tuning fork rings in a 30°C room. First find the speed, $v = 331 + 0.6(30) = 349$ m/s. Then $\lambda = v/f = 349/440 \approx 0.79$ m. In 20°C air at 343 m/s, the same fork gives $\lambda \approx 0.78$ m. The temperature changed the wavelength, but the pitch stayed the same because the fork still vibrates 440 times per second.

Intensity is the power a wave carries per unit area, $I = P/A$, measured in W/m². It grows with the square of amplitude, so doubling the amplitude makes the intensity 4 times larger.

If a source spreads sound evenly in all directions and absorption and reflections are negligible, its power spreads over a sphere, so $I = \dfrac{P}{4\pi r^2}$. This is the inverse-square law. Double your distance and the intensity drops to 1/4. Triple it and it drops to 1/9.

Hearing spans a wide range of intensities, so physicists use a logarithmic scale. Sound level in decibels is $\beta = 10\log_{10}(I/I_0)$, with $I_0 = 10^{-12}$ W/m² as a reference intensity near the hearing threshold under favorable conditions. Actual hearing thresholds vary with frequency and listener. Each +10 dB means 10 times the intensity, and +3 dB roughly doubles it. A 60 dB conversation is a million times more intense than a 0 dB sound, not 60 times.

Worked example. A speaker puts out 0.50 W of sound power evenly in all directions. Neglect absorption and reflections, and find the sound level 2.0 m away. The intensity is $I = \dfrac{0.50}{4\pi(2.0)^2} = \dfrac{0.50}{50.3} \approx 1.0 \times 10^{-2}$ W/m². The level is $\beta = 10\log_{10}\left(\dfrac{10^{-2}}{10^{-12}}\right) = 10\log_{10}(10^{10}) = 100$ dB. At 4.0 m the intensity falls to 1/4, which is about 6 dB lower, so roughly 94 dB.

The Doppler effect is a change in the frequency you hear when the source or the listener is moving. The source keeps emitting the same frequency the whole time. A source moving toward you releases each crest from a position closer to you. The crests are closer together than they would be for a stationary source, so the wavelength shrinks and you hear a higher pitch. A source moving away stretches the waves and lowers the pitch. A moving listener changes how often they run into crests.

For motion along the line between source and observer, the equation is $f_o = f_s\dfrac{v \pm v_o}{v \mp v_s}$. Here $v$ is the sound speed, and $v_o$ and $v_s$ are the observer's and source's speeds relative to the medium. For subsonic motion, use the top sign for an object moving toward the other and the bottom sign for one moving away. As a sanity check, getting closer must give $f_o > f_s$.

Worked example. A 600 Hz siren moves at 30 m/s and the speed of sound is 343 m/s. Approaching a still listener, $f_o = 600 \times \dfrac{343}{343 - 30} = 600 \times \dfrac{343}{313} \approx 658$ Hz. After it passes, $f_o = 600 \times \dfrac{343}{343 + 30} \approx 552$ Hz. These idealized head-on values differ by about 106 Hz. For a siren passing at a finite distance, the observed frequency changes continuously as its direction relative to you changes.

When a source moves faster than sound, its waves pile up into a cone-shaped shock wave. The Mach number is $v_s/v$. You hear a sonic boom when the cone sweeps past you, and the aircraft produces it the entire time it flies supersonic.

When two sound waves overlap, their pressure changes add together. This is superposition. If compressions line up with compressions, you get constructive interference and a louder sound. If compressions meet rarefactions, destructive interference makes it quieter. For two speakers emitting the same frequency in phase, a path difference of a whole number of wavelengths gives constructive interference. A path difference of 0.5, 1.5, 2.5 wavelengths and so on gives destructive interference.

Two tones with close frequencies drift in and out of step, so the loudness pulses. These pulses are beats, and $f_{beat} = |f_1 - f_2|$. A 440 Hz fork and a 444 Hz string give 4 beats per second. Musicians tune by adjusting until the beats disappear.

Resonance happens when a system is driven at its natural frequency and its amplitude builds up. Air in a pipe resonates by forming standing waves, with displacement antinodes at open ends and a displacement node at a closed end. For ideal pipes with negligible end corrections, an open pipe has $f_n = n\dfrac{v}{2L}$ for positive integers $n$. A pipe closed at one end has $f_n = n\dfrac{v}{4L}$ for positive odd integers $n$ only.

Worked example. A 0.25 m tube is closed at one end, with $v = 343$ m/s. Its fundamental is $f_1 = \dfrac{343}{4(0.25)} = 343$ Hz, and the next resonance is the 3rd harmonic at 1029 Hz. Open both ends and $f_1 = \dfrac{343}{2(0.25)} = 686$ Hz, with the next resonance at 1372 Hz.