PHYS2250 Chap.12 Wave Motion and Superposition
Wave Motion and Superposition
Separate motion of the pattern from motion of the medium
A travelling wave carries energy through a medium while particles oscillate locally. Wave number and angular frequency encode wavelength and frequency, and the phase sign fixes direction. Superposition produces interference, standing modes and beats.
Fixed-end boundary conditions quantise allowed wavelengths and make visible node counts a direct route to harmonic number.
Reasoning route and exam method
A mechanical wave transports energy and momentum through a medium without carrying the medium bodily with it.
A sinusoidal travelling wave may be written y(x,t) = A sin(kx − ωt + φ), where k = 2π/λ, ω = 2πf and wave speed v = ω/k = fλ; this relationship belongs in the diagnostic checklist for the chapter. The sign linking x and t determines propagation direction; this is a useful checkpoint before any numerical substitution.
Particle velocity is the time derivative of displacement and is not the wave speed; confusing them mixes the motion of the medium with the motion of the pattern; the same result should survive the relevant limiting-case test.
When waves overlap, the resultant displacement is their algebraic sum; that distinction determines which quantities may be combined.
Equal-frequency coherent waves produce interference according to phase difference; this observation fixes the physical interpretation of the algebra. Oppositely travelling waves of equal amplitude and frequency form a standing wave with fixed nodes and antinodes; the statement supplies a direct check on the final result.
A string fixed at both ends admits wavelengths satisfying L = nλ/2, so its normal frequencies are integer multiples of the fundamental; this condition should remain visible beside the governing equation. Beats arise from nearby frequencies and have beat frequency equal to their absolute difference.
A string 1.20 m long is fixed at both ends and supports waves at 180 m s⁻¹.
The fundamental has λ₁ = 2L = 2.40 m, so f₁ = v/λ₁ = 75.0 Hz. The third harmonic has three half-wavelengths along the string, λ₃ = 2L/3 = 0.800 m and f₃ = 225 Hz. It contains two interior nodes and three antinodes. Counting visible loops gives the harmonic number directly.
TRAVELLING SIGN Direction check. Hold the phase constant.
kx − ωt = constant gives dx/dt = +ω/k; kx + ωt gives motion in the negative x direction.
The standing wave has fixed red nodes where displacement is always zero and antinodes of maximum amplitude between them. Adjacent loops oscillate in opposite phase while the pattern itself does not travel.
Always add the wave functions first.
Words such as constructive and destructive describe the result; they are not extra rules.
What this chapter covers
- 01
travelling-wave functions
- 02
wave speed and power
- 03
reflection and transmission
- 04
standing-wave modes
- 05
beats and superposition
Worked example · free
Find fixed-string harmonics
- 3Apply the fixed-end wavelength condition.
- 2Calculate the fundamental.
- 2Scale to the third harmonic and identify nodes.
Key terms
- Standing Wave
- A fixed node-and-antinode pattern from equal opposite travelling waves.
- Beat Frequency
- The absolute frequency difference between two nearby tones.
- Wave Number
- The spatial angular frequency equal to two pi divided by wavelength.
Wave Motion and Superposition FAQ
How do I read travelling direction from phase?
Hold phase constant. The form kx minus ωt travels toward positive x, while kx plus ωt travels toward negative x, independent of the instantaneous displacement sign.
Does material travel with a wave?
No. Medium particles oscillate locally while energy and the disturbance pattern propagate through the medium, so particle speed and wave speed are different physical quantities.
Why are string frequencies discrete?
Fixed endpoints must be nodes, so only wavelengths fitting an integer number of half-wavelengths along the string satisfy the boundary conditions and persist as normal modes.
Exam move
Translate among wave function, phase picture and node pattern. Practise direction signs, boundary conditions and superposition before calculating frequencies or power. Hold phase constant to derive the travel direction instead of relying on visual memory. At a fixed position, differentiate displacement to find particle velocity and compare that local motion with propagation speed.
Draw the first four fixed-string modes, count loops and interior nodes, and recover each wavelength from geometry. Add two equal opposite travelling waves algebraically to see why the nodes stay fixed. For beats, sketch the slow envelope separately from the rapid carrier and distinguish beat frequency from either source frequency.
Conclude with an energy question so amplitude, frequency and medium properties are connected to transport rather than treated as purely geometric labels. Check reflection at fixed and free boundaries to connect endpoint conditions with inversion, then relate that behaviour to the nodes or antinodes each boundary permits.
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