Preview
Wave on a String
Explore transverse waves, reflection, and standing waves
Every string instrument ever built rests on a claim your hands will not believe at first: the speed of a wave on a string is none of your business. Shake the end faster — the wave does not go faster; the crests just bunch closer together. Shake it harder — taller crests, same speed. The wave's speed belongs to the STRING: v = √(T/μ), tension over linear density, square root of the medium's stiffness-to-heft ratio. This lab gives you an honest string to interrogate — one hundred coupled beads integrated with real mechanics, not a canned animation — and a panel that prices v, λ, and f live. You'll find the medium guards its speed jealously, and even when you grab the one knob that genuinely works — tension — the law charges you quadratically: DOUBLE the tension and the speed climbs by only √2, about 41%. Want double the speed? Pay four times the tension; the panel will show you 31.62 m/s at 50 N becoming exactly 63.25 at 200 N. And then there's the second act: hold both ends fixed and the reflected wave folds back over the incident one. At just the right frequencies — nv/(2L), the harmonic ladder — the two traveling waves lock into a standing pattern, with nodes pinned motionless not by any clamp but by perfect, permanent cancellation, while the energy panel shows kinetic and potential trading places twice a cycle without spilling a joule. A guitarist tuning a string is running this exact experiment: tightening the peg to raise v, because f₁ = v/(2L) and the ear wants a higher fundamental. Start with the question every tuner answers nightly: what exactly does doubling the tension buy you?
What you'll be able to do
- Assign wave speed to its true owner with the panel's numbers: at 80 N the v row reads 40.00 m/s and NOTHING the source does — frequency, amplitude — moves it; double the tension to 160 N and v climbs only to 56.57 m/s, a factor of √2, because v = √(T/μ) takes the square root of the medium's stiffness
- Read a standing wave as two traveling waves in disguise: on the n = 1 and n = 3 presets the amber node rings mark where incident and reflected waves permanently cancel — 'These positions are called nodes' — while the harmonic ladder f_n = nv/(2L) prices each allowed pattern (2.00 Hz for n = 1, 6.00 Hz for n = 3 at 80 N)
- Audit the energy ledger of a pure mode: the fundamental preset seeds an exact discrete normal mode, and the panel shows KE and PE trading places twice per cycle while Total holds to the digit — the machine conservation gate certifies the symplectic integrator's drift stays at float-dust level, and adding damping honestly breaks it
Formulas
Make a prediction
Fundamental preset: tension 80 N, and the v row reads 40.00 m/s. Now double the tension to 160 N (same string, same everything else). What does the wave speed become?
No grading here — commit to a guess, then scroll down and test it yourself.
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Check
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Your prediction
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Answer: About 56.57 m/s — only √2 times faster: v = √(T/μ), and the square root halves your winnings
The panel lands on 56.57 m/s — that's 40.00 × √2. Tension sits UNDER a square root: v = √(T/μ) (OpenStax Eq. 16.8, |v| = √(F_T/μ)), so doubling T multiplies v by √2 ≈ 1.414, not 2. The clean version of the law needs the full slider range: 50 N reads 31.62 m/s and 200 N — exactly four times the tension — reads exactly double, 63.25 m/s; this lab's machine gate certifies that ratio as 2.000000000 on the engine's own arithmetic. Option B is the natural linear guess the square root exists to correct. Option C smuggles in the deeper classic — the hand does not own the speed at all: 'The speed of a wave depends on the characteristics of the medium' (§16.3), and the drive frequency only repackages the fixed speed into shorter wavelengths, λ = v/f. Your hand chooses f; the string chooses v; the wavelength is the treaty between them.
Quiz (0/3)
A string (L=1m, v=10 m/s) — what are the first three harmonic frequencies?
How does doubling the string tension change wave speed?
Why do nodes not move in standing waves? What is happening physically?
You can now
- Assign wave speed to its true owner with the panel's numbers: at 80 N the v row reads 40.00 m/s and NOTHING the source does — frequency, amplitude — moves it; double the tension to 160 N and v climbs only to 56.57 m/s, a factor of √2, because v = √(T/μ) takes the square root of the medium's stiffness
- Read a standing wave as two traveling waves in disguise: on the n = 1 and n = 3 presets the amber node rings mark where incident and reflected waves permanently cancel — 'These positions are called nodes' — while the harmonic ladder f_n = nv/(2L) prices each allowed pattern (2.00 Hz for n = 1, 6.00 Hz for n = 3 at 80 N)
- Audit the energy ledger of a pure mode: the fundamental preset seeds an exact discrete normal mode, and the panel shows KE and PE trading places twice per cycle while Total holds to the digit — the machine conservation gate certifies the symplectic integrator's drift stays at float-dust level, and adding damping honestly breaks it