Waves · Lesson 8 of 8
Chapter Summary and Practice
“Connect wave propagation, speed, interference, normal modes and beats, then apply the chapter in mixed problems.”
• Choose an appropriate wave model from its physical mechanism and boundaries. • Use the chapter’s formula relations with their symbols, units and assumptions. • Distinguish progressive waves, standing modes and beats using their spatial and time structure. • Solve mixed problems involving string tension, boundary-selected frequencies, phase and sound travel.
One chain of reasoning connects the chapter
Begin with a medium whose neighbouring pieces interact. Elastic restoring forces and inertia pass a disturbance from place to place. Choose the displacement variable and compare its direction with propagation to identify transverse or longitudinal motion. A uniform harmonic progressive wave gives every element the same oscillation frequency and amplitude but a position-dependent phase. Following constant phase gives the pattern speed. The source sets the driving rhythm, while the prepared medium and type of disturbance set the speed in the ideal nondispersive model.
Next consider what can be added and what the boundaries permit. Superposition predicts reinforcement and cancellation. Reflection reverses propagation, and its displacement sign is set by whether the boundary is fixed or free. Equal opposite travelling waves can make a standing pattern. Applying both endpoints then selects normal modes. Finally, two close but unequal frequencies produce a changing relative phase and a slow beat envelope. These are different outcomes of the same addition principle, not unrelated formulas to memorise.
Quantities, units and conditions
| Quantity | Symbol and SI unit | Meaning / dimension |
|---|---|---|
| Displacement amplitude | a, m | Maximum local displacement; L |
| Wavelength | λ, m | Full spatial phase repeat; L |
| Angular wave number | k, rad m⁻¹ | 2π/λ; L⁻¹ |
| Period | T, s | Full local time repeat; 𝕋 |
| Frequency | f or ν, Hz | Cycles per second; 𝕋⁻¹ |
| Angular frequency | ω, rad s⁻¹ | 2πf; 𝕋⁻¹ |
| Wave speed | v, m s⁻¹ | Pattern propagation speed; L𝕋⁻¹ |
| Beat frequency | fᵦ, Hz | Magnitude of nearby frequency difference; 𝕋⁻¹ |
| String tension | F, N | Restoring force in the stretched-string model; ML𝕋⁻² |
| Linear mass density | μ, kg m⁻¹ | Mass per vibrating length; ML⁻¹ |
| Volume density | ρ, kg m⁻³ | Mass per volume; ML⁻³ |
| Elastic modulus | B or Y, Pa | Relevant stiffness; ML⁻¹𝕋⁻² |
Comparisons that prevent common mistakes
| Question | Correct distinction | Useful test |
|---|---|---|
| Particle or wave speed? | Particle velocity varies locally; the phase pattern can move at constant speed | Hold x fixed for particle velocity; hold phase fixed for wave velocity |
| Same displacement or same phase? | Equal height can occur at different stages of oscillation | Require a full phase repeat, including local motion |
| Node or momentary zero? | A node has zero amplitude for all times | Check whether the point stays motionless during the whole cycle |
| Standing-wave phase? | Within a loop, same phase; adjacent loops, opposite phase | Check the sign of the spatial amplitude factor |
| Open end or closed end? | Open: pressure node; closed: displacement node | Specify which physical variable is being graphed |
| First overtone or first harmonic? | First harmonic is the fundamental; first overtone is the next allowed mode | Closed–open first overtone is the third harmonic |
| Envelope frequency or beat rate? | Signed envelope repeats after two loud maxima | Count maxima of amplitude magnitude or intensity |
| Constant speed or dispersion? | Frequency-independent speed preserves a common pulse profile; dispersion changes relative component phases | Ask whether the phase speed depends on frequency |
Carry all units through calculations. If k is quoted in rad cm⁻¹, either use x in cm or convert k to rad m⁻¹. Put a factor of 2π outside a bracket into both the time and position coefficients before identifying ω and k. Use radians for trigonometric evaluation. Use kelvin for gas-temperature ratios. Preserve unrounded values until the final result. A correct formula with mismatched units can be wrong by factors of 100 or a million.
Worked synthesis before independent assessment
Problem
A uniform 0.80 m string has mass 0.016 kg and tension 128 N. Its fundamental drives a pipe containing air at 340 m s⁻¹. Find the string speed and frequency, then the shortest ideal resonant pipe lengths for open–open and closed–open ends.
- 1.The string has μ = 0.016/0.80 = 0.020 kg m⁻¹, so its transverse speed is √(128/0.020) = 80 m s⁻¹.
- 2.Fixed string ends give fundamental f = 80/[2(0.80)] = 50 Hz. The air is driven at 50 Hz too, but its sound speed is 340 m s⁻¹, not 80 m s⁻¹.
- 3.The wavelength in air is 340/50 = 6.8 m. An open–open fundamental needs half a wavelength: L = 3.4 m.
- 4.A closed–open fundamental needs a quarter wavelength: L = 1.7 m. The boundary change halves the required fundamental length for the same sound frequency.
Problem
Case A: two waves have equal amplitude, direction, k and ω with phase difference π/3. Case B: equal waves have opposite directions. Case C: equal-amplitude sounds have frequencies 300 and 304 Hz. Explain the characteristic result in each case.
- 1.A: the phase difference is fixed, so the sum is one travelling harmonic wave. Its amplitude is 2a cos(π/6) = √3 a, with unchanged wavelength and frequency.
- 2.B: opposite travelling waves can form a standing pattern. With a node chosen at the origin, y = 2a sin(kx) cos(ωt), with fixed nodes separated by λ/2 and adjacent loops in opposite phase.
- 3.C: the frequency difference makes relative phase drift. The average fast oscillation is 302 Hz and loudness beats occur at |304 − 300| = 4 Hz, every 0.25 s.
- 4.The source and medium determine which combination is present. A varying amplitude in time is not by itself a spatial standing-wave node pattern.
Problem
An echo returns after 0.12 s in air of speed 340 m s⁻¹. Separately, a closed–open pipe has successive resonant lengths separated by 0.25 m at a fixed source frequency. Find reflector distance and source frequency.
- 1.Echo time includes the outward and return paths, so the reflector is d = 340(0.12)/2 = 20.4 m away.
- 2.The resonance-length spacing is λ/2, giving λ = 0.50 m. Thus f = v/λ = 340/0.50 = 680 Hz.
- 3.The factor of two has a different origin in each calculation: two-way propagation for the echo, versus an extra half-wavelength between successive closed–open modes.
Using the whole chapter as a problem-solving method
Before substituting numbers, state the physical model: freely travelling pulse, harmonic progressive wave, standing normal mode, beat superposition or sound echo. Identify the medium and choose the relevant speed relation. Apply the boundary conditions only when the problem has boundaries. Decide whether a requested amplitude belongs to an individual travelling wave, a standing antinode or a selected position. Finally check limiting cases: zero phase difference should reinforce equal waves; opposite phase should cancel equal co-propagating waves; raising string tension should raise both speed and natural frequencies. If measured data cannot satisfy the stated ideal assumptions, explain that instead of hiding the inconsistency.
Quick Quiz
Quiz
A 1.0 m fixed-end string carries waves at 100 m s⁻¹. Its fundamental frequency is
The fundamental wavelength is 2L = 2 m, so f = 100/2 = 50 Hz.
For air at 340 m s⁻¹ in a 0.50 m closed–open pipe, the first overtone is
f₁ = 170 Hz, and the first overtone is the third harmonic at 510 Hz.
Two 5 mm equal co-propagating waves differ by 2π/3. Their resultant amplitude is
2a cos(π/3) = a = 5 mm.
An echo takes 0.20 s at sound speed 340 m s⁻¹. The reflector is
The two-way path is 68 m, so one-way distance is 34 m.
Equal-amplitude 250 and 254 Hz sounds give
The beat rate is the difference and the fast oscillation uses the average.
Practice Problems
- A 0.80 m fixed-end string of mass 0.016 kg is under 128 N tension. Find wave speed and its first three harmonic frequencies. State how each changes if the tension becomes 32 N.
- For y = 0.010 sin(8πx + 40πt − π/3), with SI units, find amplitude, wavelength, frequency, direction and wave speed. Calculate the signed phase difference between points 0.125 m apart at one time.
- Equal opposite travelling waves each have amplitude 4 mm and wavelength 0.80 m. Put a node at x = 0, locate the next three nodes and the first three antinodes, and calculate amplitude at x = 0.10 m.
- A 0.40 m pipe contains sound at 320 m s⁻¹. List the first three modes when both ends are open and when one end is closed. Identify the first overtone in each case and test resonance with an 800 Hz source.
- A reference string at 400 Hz makes 5 Hz beats with a second string. Raising the second string’s tension slightly increases the beat rate. Determine its original frequency under a small-adjustment assumption.
- A stone is dropped from rest through 80 m. Find when its splash is heard at the release point using g = 10 m s⁻² and sound speed 320 m s⁻¹. Explain why an echo formula alone cannot solve it.
- Explain pressure and displacement boundary conditions at a rigid end and an open end. Sketch a closed–open fundamental with both amplitude patterns and label the different node positions.
- A pulse contains several sinusoidal components. Explain why it can keep its shape when their propagation speeds match and why it generally distorts in a dispersive medium. Relate this to the distinction between a pure harmonic wave and a pulse.
- Design a consistent method to estimate sound speed from two consecutive closed–open resonance lengths. State what subtraction removes and how you would recognise a pair that conflicts with a zero-end-correction idealisation.
- Explain, using examples from this chapter, how waves transmit information without carrying the medium bodily from source to receiver, and how the same nominal pitch can have different timbres.
Key Takeaways
• Choose the medium, variable and boundary conditions before choosing a formula. • Phase repetition links λ, k, f, T and speed; restoring stiffness and inertia set the speed in the chosen model. • Superposition produces travelling interference, stationary modes or beats depending on directions and frequencies. • Always check units, phase conventions and the limits of the idealisation; inconsistent measurements should be explained rather than forced into an answer.
Previous · Lesson 7
Beats and Musical Sound
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