DPST1021 Chap.14 Sound Waves, the Doppler Effect and Beats
Sound Waves, the Doppler Effect and Beats
Week 12 closes the waves block of Physics 1A with three lectures on sound, the last material assessed in Test 2. Lecture 30 treats sound as a longitudinal travelling wave in a gas. You write a harmonic equation for the displacement of air elements, connect it to the swing in pressure about the ambient value, and relate the average power of the wave to its intensity.
The formula sheet also gives the speed of sound in air as a function of temperature.
Lecture 31 turns to how loud a sound seems and how its pitch changes with motion. Loudness is measured on the decibel scale relative to a reference intensity near the threshold of hearing, and a point source spreads its power over an ever larger sphere.
The Doppler equation then links emitted and received frequencies when a source, a listener or the air itself is moving; anything faster than sound leaves a cone of fronts described by its Mach number.
Lecture 32 builds standing waves inside pipes.
Open and closed ends impose different boundary conditions, so the two kinds of pipe support different sets of harmonics, and a small end correction moves the real antinode just past an open end. Finally, two nearly equal frequencies combine to produce beats, a periodic rise and fall in loudness. The tube lab, Lab 8, runs that week, and homework quiz 10 covers all three lectures.
What this chapter covers
- 01
Longitudinal sound waves: describe how compressions and rarefactions form and write a harmonic displacement function for the air.
- 02
Pressure and displacement amplitudes: convert between them using air density, wave speed and angular frequency, noting their quarter-cycle offset.
- 03
Intensity and decibels: find intensity from power and distance for a point source, then express it as a sound level.
- 04
Doppler shifts: choose numerator and denominator signs for a moving listener and a moving source, including problems with wind.
- 05
Shock cones and Mach number: relate the cone half-angle behind a supersonic object to the ratio of the two speeds.
- 06
Resonant pipes and beats: list allowed harmonics for open and closed pipes with end corrections, and find beat rates from two frequencies.
Worked example · free
A cyclist, a buzzer and a steady breeze
- 1Work in the air frame: the buzzer recedes from the cyclist at 12 m/s and the cyclist approaches at 8.0 + 12 = 20 m/s.
- 1Signs: approaching listener, top line 343 + 20 = 363 m/s; receding source, bottom line 343 + 12 = 355 m/s.
- 1Substitute: f′ = 600 × 363/355 = 613.5 Hz.
- 1Ground check: downwind sound moves at 355 m/s, so λ = 355/600 = 0.5917 m; the cyclist meets fronts at 363 m/s, and 363/0.5917 = 613.5 Hz.
- 1Without wind: 600 × 351/343 = 614.0 Hz, about 0.5 Hz higher.
Key terms
- Longitudinal wave
- A wave whose medium oscillates parallel to the direction of travel, as air does in sound.
- Pressure amplitude
- The largest departure of local pressure from its undisturbed value as sound passes.
- Sound level
- A logarithmic measure of intensity in decibels, ten times the log of I over the reference intensity.
- Doppler effect
- The change in received frequency caused by source or observer motion relative to the medium.
- Mach number
- The ratio of an object's speed to the local speed of sound.
- End correction
- A length of roughly 0.6 of the pipe radius added at each open end, since the antinode sits just outside.
- Beat frequency
- The rate at which loudness swells and fades when two close tones overlap, equal to their frequency difference.
Sound Waves, the Doppler Effect and Beats FAQ
Why is the decibel scale logarithmic instead of linear?
Hearing responds roughly to ratios of intensity, and the range from a faint whisper to a painful noise covers a factor of about a trillion. A logarithm squeezes that span into roughly 0 to 120, and every tenfold rise adds the same 10 dB.
Does a moving listener give the same shift as a moving source at the same speed?
No. Source motion changes the wavelength laid down in the air, while a moving listener only changes how often existing fronts arrive. At low speeds the shifts nearly match, but they diverge as speed grows; a source at the speed of sound gives an unbounded result.
Why does a pipe closed at one end produce only odd harmonics?
The closed end forces a displacement node and the open end an antinode, so the pipe must hold an odd number of quarter wavelengths. Even multiples would need matching ends. The same pipe closed also sounds an octave below its open version.
How can I tell which of two tuning forks is higher when I only hear beats?
Load one prong with a little wax, which lowers its frequency. If the beats slow down, the loaded fork was the higher one and has moved closer to the other; if they speed up, it was already the lower fork and has moved further away.
What does temperature do to the notes from a wind instrument?
Warmer air carries sound faster, and the resonant frequencies of a pipe are proportional to that speed while the pipe length barely changes. A pipe tuned in a cold hall plays sharp once the room warms, by roughly 0.17% per degree near room temperature.
Exam move
Split revision by lecture. For the wave description, practise moving between displacement amplitude, pressure amplitude and intensity with the formula-sheet relations until the chain feels routine, and always compute the speed of sound from the stated temperature first. For levels, memorise a few ratios, such as doubling intensity adding 3 dB and doubling distance removing 6 dB,.
Doppler questions reward a fixed ritual: draw an arrow from source to listener, mark each velocity, convert to the air frame if there is wind, then set the signs before substituting. For pipes, sketch the displacement pattern for each harmonic before writing any formula, because the picture tells you which harmonics exist.
Finish with beat problems and the tenth homework quiz, entering answers with the significant figures common to the data.
Working through Sound Waves, the Doppler Effect and Beats in DPST1021? Sia is AskSia’s AI Physics tutor — ask any DPST1021 Sound Waves, the Doppler Effect and Beats question and get a clear, step-by-step explanation grounded in how DPST1021 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
