The Timpson Lyceum · University Physics I (2025)

Sound Waves and the Doppler Effect- Lecture 16 (University Physics I)

Overview

This lecture treats sound as a longitudinal pressure wave — a chain of compressions and rarefactions traveling through a medium rather than the transverse string waves of the previous chapter. We build a pressure-wave equation from the displacement wave, derive the speed of sound from the bulk modulus and density of the medium, and define wave intensity and the logarithmic decibel scale used to measure loudness. We then look at standing sound waves (in front of a wall and inside wind instruments), constructive and destructive interference, and close with the Doppler effect — why an approaching siren sounds higher in pitch than a receding one — and the shock waves produced when a source outruns its own sound.

Key Concepts

Sound as a longitudinal wave

• A sound wave is a longitudinal disturbance: the medium oscillates parallel to the direction the energy travels. Audible sound spans roughly 20 Hz to 20,000 Hz.
• The basic wave relation always holds: $v = f\lambda$, where $f$ is frequency and $\lambda$ is wavelength.
• The same wave function $y(x,t)$ from before still applies; now $y$ is the molecule's displacement along $x$ from its equilibrium position.

Pressure wave vs. displacement wave

• It is often more useful to track the pressure variation $p(x,t)$ — how much higher or lower than atmospheric pressure — than the molecular displacement.
• Using the bulk modulus $B$, which relates pressure change to fractional volume change, $\dfrac{\Delta p}{} = -B\dfrac{\Delta V}{V}$, the displacement wave $y = A\cos(kx-\omega t)$ yields the pressure wave $p(x,t) = BkA\,\sin(kx-\omega t)$.
• Pressure amplitude: $p_{\max} = BkA$, where $k$ is the wave number, $B$ the bulk modulus, and $A$ the displacement amplitude.
• The pressure wave is a quarter-wavelength (90°) out of phase with the displacement wave: pressure extremes occur where molecular displacement is zero (where molecules are most crowded or most rarefied), and pressure is at equilibrium where displacement is maximal.

Speed of sound

• General form: speed = $\sqrt{\dfrac{\text{restoring property}}{\text{inertial property}}}$. For a fluid the restoring property is the bulk modulus and the inertia is the density: $v = \sqrt{\frac{B}{\rho}}$
• In a solid rod, replace $B$ with Young's modulus $Y$: $v = \sqrt{Y/\rho}$.
• For an ideal gas, $v = \sqrt{\dfrac{\gamma R T}{M}}$, where $\gamma$ is the ratio of heat capacities, $R = 8.314$, $M$ is the molar mass (kg/mol), and $T$ must be in kelvin ($T_K = T_C + 273.15$).
• The speed depends only on the medium, not on frequency. In air at 20 °C, $v \approx 344$ m/s; in water, $v \approx 1480$ m/s.

Intensity

• Intensity $I$ is power per unit area, $I = P/\text{area}$. Don't confuse it with power: a speaker's power is fixed, but the intensity reaching you falls off as the sound spreads over a larger area.
• For a sinusoidal sound wave, the time-averaged intensity is $I = \tfrac{1}{2}\,\omega B k A^2 = \frac{p_{\max}^2}{2\rho v}.$

Sound intensity level (decibels)

• The decibel level is $\beta = 10\log_{10}\!\left(\dfrac{I}{I_0}\right)$ dB, with reference intensity $I_0 = 10^{-12}\ \text{W/m}^2$ (the threshold of human hearing at 1000 Hz).
• The scale is logarithmic: every factor-of-10 increase in intensity adds 10 dB. So 0 dB is barely audible, and a millionfold ($10^6$) increase in intensity is only +60 dB.
• To invert: $I = I_0\,10^{\beta/10}$.

Standing waves and wind instruments

• A reflected sound wave interferes with itself to form a standing wave. Nodes never move; antinodes oscillate with maximum amplitude. A displacement node coincides with a pressure antinode, and vice versa.
• At a rigid wall, molecules cannot move: it is a displacement node and a pressure antinode.
• Open pipe (both ends open): $\lambda_n = \dfrac{2L}{n}$, $f_n = n f_1$ — all harmonics present, fundamental $f_1 = v/2L$.
• Closed pipe (one end closed): $\lambda_n = \dfrac{4L}{2n-1}$, $f_n = (2n-1)f_1$ — only odd harmonics, fundamental $f_1 = v/4L$.

Interference

• Two identical sources produce constructive interference (amplitudes add) where their path difference is a whole number of wavelengths, and destructive interference (cancellation) where it is an odd number of half-wavelengths.

Doppler effect

• Relative motion between source and observer shifts the perceived frequency. With $v$ the speed of sound, $v_L$ the observer (listener) speed, and $v_S$ the source speed: $f_L = f_S\,\frac{v + v_L}{v + v_S}$ using the sign convention that velocities are positive when the observer moves toward the source and the source moves away from the observer (signs flip for the opposite directions).
• Approaching shortens $\lambda$ ahead of the source (higher pitch); receding stretches it behind (lower pitch). The medium sets the wave speed, so a moving source is physically different from a moving observer.

Shock waves

• When a source moves faster than the speed of sound, the wavefronts pile up into a cone — a shock wave (e.g., a sonic boom, or the bow wave of a fast boat). An observer hears events arrive out of order because the source outruns its own sound.

Worked Example

Observer moving toward a stationary source. A car horn emits $f_S = 1000$ Hz. You drive toward it at $v_L = 30$ m/s on a day when the speed of sound is $v = 344$ m/s. What frequency do you hear?

With the source stationary ($v_S = 0$) and the observer approaching, the apparent wave speed relative to you is $v + v_L$, so:

The pitch rises by about 87 Hz because you sweep into the wavefronts faster than if you stood still. If instead you receded at the same speed, you would replace $+v_L$ with $-v_L$ and hear roughly 913 Hz.

Summary

• Sound is a longitudinal pressure wave; the pressure wave $p_{\max}=BkA$ lags the displacement wave by a quarter wavelength.
• The speed of sound depends only on the medium: $v=\sqrt{B/\rho}$ for fluids ($Y/\rho$ for solids), independent of frequency — about 344 m/s in air, 1480 m/s in water.
• Intensity is power per area, $I = p_{\max}^2/(2\rho v)$; loudness is reported on the logarithmic decibel scale $\beta = 10\log_{10}(I/I_0)$, where +10 dB means $10\times$ the intensity.
• Standing waves set instrument pitches: open pipes give all harmonics ($f_n=nf_1$), closed pipes only odd ones ($f_n=(2n-1)f_1$).
• The Doppler effect shifts perceived frequency with relative motion, $f_L = f_S\dfrac{v+v_L}{v+v_S}$ — higher when approaching, lower when receding.
• A source exceeding the speed of sound generates a shock-wave cone (sonic boom).