Standing Waves, Resonance & Q (IPhO Waves)

Standing waves are what you get when a wave keeps reflecting and adding to itself in phase. Resonance is the driven version of the same idea: when the driving frequency matches a normal mode, energy piles up. The quality factor Q tells you how sharply tuned that pile-up is, and how slowly the oscillation dies away when you stop driving.

1. Definitions (Must Know)

• Standing wave: superposition of two equal-frequency waves traveling in opposite directions.
• Node / antinode: fixed zero-amplitude points and maximum-amplitude points of a standing wave.
• Normal mode: a standing-wave pattern that satisfies the boundary conditions; labeled by an integer mode number.
• Resonance: large steady-state response when a system is driven near a natural frequency.
• Bandwidth (half-power): the frequency width Δ f between the two points where power is half the peak value.
• Quality factor: Q = 2π (E/Δ E_cycle), and (for light damping) also Q = f₀/Δ f.
• Ring-down: exponential decay after the drive is removed; amplitude envelope A(t) falls exponentially.

2. Key Ideas (What Earns Marks)

• A standing-wave frequency is set by two things: wave speed v and a boundary-conditioned length scale L.
• Start from boundary conditions, not memorized formulas, to avoid mixing up fixed, free, open, and closed ends.
• For driven resonance, the high-scoring moves are: identify f₀, identify Δ f, then use Q = f₀/Δ f.
• For ring-down data, extract the exponential decay constant from amplitude ratios, then connect to Q.
• Always keep straight what is being measured: amplitude, intensity, power, or energy.

3. Detailed Explanations

3.1 Standing wave from two traveling waves

Add two waves of equal amplitude traveling in opposite directions:

The spatial factor cos(kx) gives stationary nodes and antinodes, while the time factor cos(ω t) makes them oscillate.

3.2 Quantization of k from boundary conditions

For a string of length L:

• Fixed-fixed: y(0,t) = 0 and y(L,t) = 0. A standing wave like y ∝ sin(kx) automatically satisfies y(0,t) = 0, and y(L,t) = 0 requires sin(kL) = 0:

  kₙ = nπ/L, fₙ = ωₙ/2π = nv/2L, n = 1,2,3,…

• Fixed-free: y(0,t) = 0 and ∂ y/∂ x = 0 at x = L. Using y ∝ sin(kx), the slope condition is cos(kL) = 0, giving

  kₙ = ((2n-1)π)/2L, fₙ = ((2n-1)v)/4L, n = 1,2,3,…

For air columns, “closed end” behaves like a displacement node (particle velocity is zero), and “open end” behaves like a pressure node. The mode structure is the same logic: pick the sinusoid that satisfies the boundary conditions.

3.3 Driven resonance and bandwidth

A standard model is a driven, damped oscillator:

The steady-state amplitude is

For light damping, the resonance is sharp and the half-power bandwidth in angular frequency is approximately

so

Since ω = 2π f, the same ratio works with ordinary frequency: Q = f₀/Δ f.

3.4 Ring-down and the time meaning of Q

When the drive is removed, the underdamped solution has an envelope

Energy scales like amplitude squared, so

Define the energy decay time constant τ_E by E(t) = E₀ e^(-t/τ_E), so τ_E = 1/(2β). Then

A useful interpretation is the per-cycle energy loss fraction:

4. Common Mistakes

• Using the fixed-fixed formula fₙ = nv/(2L) for open-closed pipes (only odd harmonics appear there).
• Confusing amplitude half-maximum with power half-maximum when reading a resonance curve.
• Mixing Δ f and Δω without the factor 2π.
• Extracting β from energy decay but plugging it into an amplitude decay formula (or vice versa).
• Forgetting that a mode number counts half-wavelengths along the length.

5. Exam Tips

• Sketch the mode shape and mark nodes at the boundaries; it forces the correct sine or cosine choice.
• If the question gives bandwidth, Q = f₀/Δ f is often the quickest mark.
• If the question gives ring-down, take logs of amplitude ratios to isolate β cleanly.
• If your computed resonance frequency is inconsistent with the boundary conditions (for example, a node where an antinode must be), restart from the boundary.

6. Worked Examples

7. Mind Stretchers

8. Practice