Connect string tension and mass density to wave speed

Give the wave a material model

The earlier traveling-wave lab prescribed a speed. Here, the speed follows from an ideal stretched string’s tension and mass per unit length. Tension supplies the transverse restoring effect when the string bends; the distributed mass provides inertia. Under small-displacement, flexible-string assumptions, the relation is c = √(F/μ).

Here c is wave speed in metres per second, F is tension force in newtons and μ, read “mu,” is linear mass density in kilograms per metre. F is a force, not a period. Linear density differs from volume density: a string with 0.020 kg distributed uniformly over 2 m has μ = 0.010 kg/m.

Check the units before calculating. A newton is kg·m/s², so F/μ has units m²/s². Taking its square root gives m/s. Using grams per metre without converting to kilograms per metre would give an incorrect numerical result.

Test square-root scaling

The browser fixes μ = 0.010 kg/m and vibrating length L = 2 m. Select mode 2 and compare two tensions:

• At 4 N, c = √(4/0.010) = √400 = 20 m/s.
• At 16 N, c = √(16/0.010) = √1600 = 40 m/s.

Four times the tension doubles the speed. At 32 N, speed is approximately 56.569 m/s: doubling 16 N multiplies speed by √2, not by two. Capture these comparisons and explain the ratio before inspecting the numerical ledger.

The graph is a mode shape at a displacement extreme, not a moving pulse. A faster calculated wave speed does not make this static shape travel across the screen. It changes the time scale of the mode’s oscillation.

Explain the unchanged wavelength

Both ends are fixed. Mode 2 contains two half-wavelength loops in a 2 m length, so its wavelength is 2 m. That geometric requirement stays the same when tension changes while length and mode number remain fixed. Frequency is c/λ: it changes from 10 Hz at 4 N to 20 Hz at 16 N.

This differs from the previous fixed-speed comparison, where changing frequency changed wavelength. State what is held fixed before using c = λf. Here the selected mode and string length fix wavelength, while tension changes speed and therefore frequency. A separately forced string driven at a specified frequency may not remain in the same natural mode after its tension changes.

Do not treat fixed parameters as experimental facts

The model assumes uniform density, constant tension, perfectly fixed endpoints, negligible bending stiffness and small transverse motion. It does not calculate how tightening a real string stretches it, changes its density or moves its supports. Nor does it predict a breaking load or recommend how hard to tension a real apparatus.

If you compare real strings, record the vibrating length, mass per length, tension estimate and support conditions. Changing material or thickness can change linear density as well as stiffness. A thicker string is not described by its visual thickness alone; its material and cross-sectional area matter. A controlled comparison needs these properties rather than an unsupported claim that every tighter string follows the exact ideal formula.

Transfer to a density change

An ideal string has tension 16 N and linear density 0.010 kg/m. A second string has the same length and tension but four times the linear density. Predict the wave speed and the mode-2 frequency for the second string without using a new browser control.

Check the string speed transfer

The second density is 0.040 kg/m. Wave speed is sqrt(16/0.040) = 20 m/s, half the original 40 m/s. With the same 2 m length and mode 2, wavelength remains 2 m and frequency is 10 Hz. These conclusions assume the same ideal flexible-string model; the browser keeps density fixed and does not simulate the second material.

OpenStax wave speed on a stretched string develops the tension and linear-density relationship. The numerical values and fixed-parameter comparisons here are synthetic teaching examples.