Chapter Review

Waves and Sound

Sound Waves and Speed · Stationary Waves · Doppler Effect

Progressive Waves and Sound

Sound is a mechanical longitudinal wave that propagates through a medium via alternating compressions (high pressure) and rarefactions (low pressure). It requires a medium and cannot travel through a vacuum.

Key Points

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Transverse waves: particle displacement perpendicular to wave travel (light, rope waves)
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Longitudinal waves: particle displacement parallel to wave travel (sound in air)
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Fluids support only longitudinal waves; solids support both types
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Human hearing range: 20 Hz – 20,000 Hz; below is infrasound, above is ultrasound
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The wave equation $v = f\lambda$ links speed, frequency, and wavelength
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When a wave crosses into a new medium, frequency stays constant while speed and wavelength change

Formula

$$v = f\lambda$$

Speed of Sound — Newton vs Laplace

Newton's isothermal model underestimates the speed of sound by ~16%. Laplace corrected this by recognizing that sound compressions are adiabatic, introducing the heat capacity ratio γ.

Key Points

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Newton's formula (isothermal): $v = \sqrt{P/\rho}$ gives ~280 m/s — too low
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Laplace's correction (adiabatic): $v = \sqrt{\gamma P/\rho}$ gives ~332 m/s — matches experiment
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γ = 1.40 for diatomic gases (air, O₂, N₂), 1.67 for monatomic (He, Ar), 1.29 for polyatomic (CO₂)
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Compressions occur too rapidly for heat exchange, making the process adiabatic rather than isothermal

Formula

$$v = \sqrt{\frac{\gamma P}{\rho}}$$

Factors Affecting Sound Speed

Sound speed depends on the medium's elasticity and density, and on temperature — but not on pressure alone, since density changes proportionally with pressure at constant temperature.

Key Points

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Pressure alone has no effect: if P doubles at constant T, ρ also doubles, so P/ρ is unchanged
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Lighter gases carry sound faster: $v \propto 1/\sqrt{\rho}$ (hydrogen ~4× faster than oxygen)
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Temperature: $v \propto \sqrt{T}$ (absolute); linear approximation $v_t \approx 332 + 0.61t$ m/s
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To double the speed, absolute temperature must quadruple
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General ranking: $v_{\text{solid}} > v_{\text{liquid}} > v_{\text{gas}}$ because elasticity increase outweighs density increase

Formula

$$\frac{v_t}{v_0} = \sqrt{\frac{T}{T_0}}$$

Superposition, Beats, and Echoes

When waves overlap, the resultant displacement is the algebraic sum of individual displacements. This principle produces interference, beats, and stationary waves.

Key Points

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Constructive interference: waves in phase → larger amplitude; destructive: out of phase → cancellation
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Beats: two slightly different frequencies produce periodic loudness fluctuations at $f_{\text{beat}} = |f_1 - f_2|$
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Beats > 10 Hz are indistinguishable to the ear; beats = 0 means perfect tuning
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Echo: sound reflection where round-trip distance gives $d = vt/2$
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Reverberation occurs when round-trip time < 0.1 s (reflected sound blends with original)

Formula

$$f_{\text{beat}} = |f_1 - f_2|$$

Stationary Wave Formation

Stationary (standing) waves form when two progressive waves of equal amplitude and frequency travel in opposite directions, creating fixed nodes and oscillating antinodes.

Key Points

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Nodes: points of permanent zero displacement, spaced λ/2 apart
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Antinodes: points of maximum amplitude (2A), midway between nodes, spaced λ/4 from nearest node
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Energy does not transfer along the wave — it oscillates between KE and PE within each loop
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All particles in one loop are in phase; adjacent loops are 180° out of phase
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Displacement equation: $y = [2A\sin(kx)]\cos(\omega t)$ — amplitude depends on position

Vibrations in Stretched Strings

A string fixed at both ends resonates at discrete harmonics where the length equals a whole number of half-wavelengths. The frequency depends on tension, length, and linear mass density.

Key Points

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Harmonic series: $f_n = nf_1$ where $f_1 = \frac{1}{2L}\sqrt{T/\mu}$
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The nth harmonic has n loops and wavelength $\lambda_n = 2L/n$
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$f \propto \sqrt{T}$: quadrupling tension doubles frequency
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$f \propto 1/L$: halving length doubles frequency (guitar fret principle)
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$f \propto 1/\sqrt{\mu}$: thicker/heavier strings vibrate at lower frequencies

Formula

$$f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}}$$

Resonance in Air Columns

Open pipes support all harmonics with antinodes at both ends, while closed pipes support only odd harmonics with a node at the closed end and antinode at the open end.

Key Points

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Open pipe: $f_n = nv/2L$ (n = 1, 2, 3…); fundamental wavelength $\lambda_1 = 2L$
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Closed pipe: $f_n = (2n-1)v/4L$ (n = 1, 2, 3… giving harmonics 1, 3, 5…); fundamental $\lambda_1 = 4L$
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For equal length, closed pipe fundamental is half the open pipe fundamental (one octave lower)
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Open pipes are richer in harmonics — they contain all integer multiples
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$f \propto 1/L$ for both types: longer pipes produce deeper notes

Formula

$$f_{\text{closed}} = \frac{(2n-1)v}{4L}$$

Doppler Effect — Moving Observer

When an observer moves relative to a stationary source, the observer encounters wave crests at a different rate, changing the perceived frequency without altering the physical wavelength in the medium.

Key Points

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Towards source: $f' = f(v + u_o)/v$ — higher frequency
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Away from source: $f' = f(v - u_o)/v$ — lower frequency
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Wavelength in the medium remains unchanged — only the encounter rate changes
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Fractional shift: $\Delta f/f = u_o/v$

Formula

$$f' = f\left(\frac{v \pm u_o}{v}\right)$$

Doppler Effect — Moving Source

A moving source physically compresses wavelengths ahead of it and stretches them behind, producing asymmetric frequency shifts that become extreme as the source speed approaches the speed of sound.

Key Points

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Towards observer: $f' = f \cdot v/(v - u_s)$ — compressed wavelength, higher frequency
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Away from observer: $f' = f \cdot v/(v + u_s)$ — stretched wavelength, lower frequency
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The approaching shift is always larger than the receding shift for equal speed
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As $u_s \to v$, frequency approaches infinity — shock wave / sonic boom
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Observer motion affects the numerator; source motion affects the denominator

Formula

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

Doppler Applications and Red/Blue Shift

The Doppler effect is used in radar, sonar, medical ultrasound, and astronomy to measure velocities. Red shift of distant galaxies provides key evidence for the expanding universe.

Key Points

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Red shift ($\Delta\lambda > 0$): source receding — wavelength increases (galaxies moving away)
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Blue shift ($\Delta\lambda < 0$): source approaching — wavelength decreases
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Radar speed traps: reflected microwaves undergo a double Doppler shift
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Medical ultrasound: frequency shifts of backscattered sound measure blood flow velocity
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General formula (both moving): $f' = f(v \pm u_o)/(v \mp u_s)$

Formula

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

Formulas

Wave Equation

Speed equals frequency times wavelength — universal for all wave types.

Formula

$v = f\lambda$

Laplace's Speed of Sound

Corrected formula using adiabatic compression with γ.

Formula

$v = \sqrt{\frac{\gamma P}{\rho}}$

Speed–Temperature Relation

Sound speed scales with the square root of absolute temperature.

Formula

$\frac{v_t}{v_0} = \sqrt{\frac{T}{T_0}}$

Beat Frequency

Beats per second equals the absolute difference of the two frequencies.

Formula

$f_{\text{beat}} = |f_1 - f_2|$

String Harmonics

Frequency of the nth harmonic for a string fixed at both ends.

Formula

$f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}}$

Closed Pipe Harmonics

Only odd harmonics for a pipe closed at one end.

Formula

$f_n = \frac{(2n-1)v}{4L}$

General Doppler Formula

Combined formula for both source and observer in motion.

Formula

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

Echo Distance

Divide round-trip time by 2 for one-way distance to reflector.

Formula

$d = \frac{v \times t}{2}$