Pendulum period calculator, plus damping, resonance, waves and Doppler
Move sliders for a pendulum, a damped spring, resonance, interference, standing waves and the Doppler effect and see the chart and the number at once. The pendulum period is calculated both exactly and with the small-angle formula, and the calculation runs in your browser.
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- 6 topicsmechanics and waves
- T₀ = 2π√(L/g)and the exact period up to 170°
- 3 kinds of dampingunder-damped, critical, over-damped
- 2 Doppler formulassource and observer on one set of axes
Vibrations and waves: six topics
Vibrations and Waves is a free online tool from EduFacturing that calculates and plots six topics: the simple pendulum, a damped spring oscillator, resonance, two-wave interference, a standing wave on a string and in a pipe, and the Doppler effect.
Each tab has its own sliders and a chart that recalculates as you move them. The pendulum is compared with the exact period from the elliptic integral, and the three damping regimes share one chart.
The pendulum period and five more calculations
Pendulum period: exact versus small angle
For L from 0.1 to 3 m and an amplitude from 1° to 170° it shows T₀ = 2π√(L/g), the exact period and the difference in per cent. The animation integrates θ″ = −(g/L)·sin θ, without the sin θ ≈ θ approximation.
Damped spring and resonance
With damping c from zero up it gives ω₀, T₀, c_crit and the regime, and plots x(t). Resonance shows ω₀, ω_r, the quality factor Q and the curve A(ω).
Interference and standing waves
For two sources a distance d apart it calculates Δ = d·sin θ and the intensity I(θ). For a string, an open pipe and a closed pipe it gives λ, f, the nodes and antinodes and a table of the first six modes.
The Doppler effect
It calculates f′ for a moving source and for a moving observer at the same speed, approaching and receding, and plots both curves on one set of axes.
When you need a pendulum calculator
The formulas of these topics look alike; here the difference is visible.
A lesson on the small-angle formula
For L = 1 m, T₀ = 2.006 s, and the exact period is longer by 0.19% at 10°, 1.74% at 30° and 18.03% at 90°. The 1% mark is at about 22.8°.
Damping of a spring
With m = 0.1 kg and k = 10 N/m, c_crit is 2 N·s/m: at c = 0.3 the spring oscillates, at c = 2 it returns fastest, at a larger c more slowly.
Where the resonance peak is
At c = 0.3 N·s/m the peak is at ω_r = 9.77 rad/s while ω₀ is 10 rad/s. Above about c = 1.41 N·s/m there is no peak.
Pipe and string
At L = 0.5 m and v = 343 m/s an open pipe gives 343, 686 and 1029 Hz, a closed one 172, 515 and 858 Hz.
How the pendulum is calculated
Each tab calculates with closed formulas or its own numerical routine.
Pendulum
The integral K in the exact period is evaluated with Simpson's rule in 2000 steps, and the amplitude is capped at 179.5°.
Damping and resonance
From x₀ and v₀ = 0, m·ẍ + c·ẋ + k·x = 0 is solved for the three regimes. For forced oscillation with F₀ = 1 the amplitude A(ω) has a peak at ω_r.
Interference and standing waves
A maximum is at Δ = nλ and a minimum at Δ = (n + ½)λ, with a tolerance of 3% of λ. A closed pipe gives only odd harmonics.
Doppler
The source and the observer are calculated separately at the same speed. The formulas are not symmetric.
| Applies to | Formula | Meaning |
|---|---|---|
| Pendulum | T = 4√(L/g)·K(sin(θ₀/2)) | the exact period; T₀ = 2π√(L/g) is for small angles, g = 9.80665 m/s² |
| Spring | ω₀ = √(k/m); c_crit = 2√(mk) | under-damped for c < c_crit, critical for c = c_crit, otherwise over-damped |
| Resonance | A(ω) = F₀ / √((k − mω²)² + (cω)²); ω_r = √(ω₀² − 2γ²); Q = ω₀/(2γ) | γ = c/2m; there is no peak for γ² ≥ ω₀²/2 |
| Interference | Δ = d·sin θ; I = 4I₀·cos²(πΔ/λ) | I₀ is the intensity of one wave, the maximum is 4I₀ |
| Standing wave and Doppler | f = n·v/(2L); f = (2n − 1)·v/(4L); f′ = f·v/(v − v_s); f′ = f·(v + v_r)/v | string or open pipe; closed pipe; source; observer, when approaching |
What is different from a textbook
Three regimes on one chart
Under-damped (0.3 of c_crit), critical and over-damped (2.2 of c_crit) motion share the same m, k and x₀.
Two Doppler formulas
At 440 Hz and 100 m/s approaching, the source gives 621.1 Hz and the observer 568.3 Hz.
The pendulum is a simple pendulum with no air resistance. The spring starts from rest, resonance uses a force F₀ = 1, and interference is for two coherent sources at an angle from −60° to 60°.
Doppler is for sound at 343 m/s, with the source and the observer moving separately, up to 309 m/s. The charts are for learning, not for analysing machine vibration.
Questions about the pendulum, resonance and Doppler
How to calculate the period of a pendulum?
For small angles use T₀ = 2π√(L/g), with g = 9.80665 m/s². The exact period is T = 4√(L/g)·K(sin(θ₀/2)).
Does the period of a pendulum depend on the angle?
Yes, but only through the amplitude, not the length. The exact period is longer than T₀ = 2π√(L/g) by 0.19% at 10° and 18.03% at 90°.
Why does a heavily damped spring return more slowly?
Critical damping c_crit = 2√(mk) is the border between oscillating and not, and it returns fastest. Above it the resistance only slows the return.
Where is the resonance peak of a spring?
At ω_r = √(ω₀² − 2γ²), below the natural frequency ω₀ = √(k/m). The two coincide only with negligible damping. For γ² ≥ ω₀²/2 there is no peak.
How does the Doppler effect work?
A moving source shortens the wave ahead of it, while a moving observer crosses more wave fronts per second. The formulas are f′ = f·v/(v − v_s) for a source and f′ = f·(v + v_r)/v for an observer, when approaching, and they are not symmetric.
Why does a closed pipe have only odd harmonics?
There is a node at the closed end and an antinode at the open end, and f = (2n − 1)·v/(4L) leaves only the odd ones.
Do I need an account?
Not always. Without an account the tool opens 3 times a day. A free account removes the limit.
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Set the pendulum swinging
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