Complex Harmonics · Optional Worked Solutions

These questions are optional enrichment. Try the original section first.

Two equal-phase contributions of the same standing mode have amplitudes 0.020 m and 0.030 m. What is their combined amplitude?

Answer: 0.050 m

1. Write y₁=A₁cos(ωt)sin(kx) and y₂=A₂cos(ωt)sin(kx). Why: The question specifies the same mode, frequency and phase, so both contributions have identical factors apart from amplitude.

2. Factor those common terms: y=(A₁+A₂)cos(ωt)sin(kx). Why: Linear superposition adds displacements; factoring is valid because the spatial and temporal factors match.

3. Add 0.020 m + 0.030 m to get 0.050 m. Why: The amplitudes have the same phase and sign here; their displacement units add rather than multiply.

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For a fixed-end string of length 10 m, does sin(πx/4) satisfy the endpoint at x = 10 m?

Answer: No: sin(10π/4) = 1.

1. Evaluate the spatial factor at the right endpoint: sin(π×10/4)=sin(5π/2)=1. Why: A fixed-end condition must be tested at the specified endpoint, rather than assumed for every sine function.

2. The displacement there is A cos(ωt), which is generally nonzero. Why: An occasional temporal zero does not make the endpoint fixed throughout the oscillation.

3. Choose k=nπ/10 for integer n to satisfy both fixed ends. Why: Then sin(kL)=sin(nπ)=0 at x=L and sin(0)=0 at x=0 for all times.

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Two different normal modes on the same ideal string are added. Which claim is justified?

Answer: Their different frequencies generally make the summed shape evolve.

1. Keep both mode terms, with their own k₁,k₂ and ω₁,ω₂. Why: Superposition adds whole solutions, so changing a mode frequency would change the physical solution.

2. Apply ωⱼ=vkⱼ on this uniform nondispersive string. Why: Both modes have the same wave speed set by the string tension and linear density, so distinct wave numbers imply distinct frequencies.

3. Their relative timing changes, giving an evolving summed shape. Why: Different frequencies do not generally reach extrema together; an arbitrary sum has no guaranteed traveling-wave form.

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