Wave speed on a stretched string

What you will learn

Use v = √(T/μ); tension, linear mass density and instrument tuning.

Specification mapping: A1.5.2

Prerequisites: A1.5

Physical principles

For small-amplitude transverse waves on a flexible stretched string, v = √(T/μ). Here T means tension in newtons, not period, and μ is mass per unit length in kg m⁻¹. Determine μ from string mass divided by its length. The equation assumes approximately uniform tension and a flexible, uniform string.

Increasing tension increases wave speed, but with a square-root dependence: four times the tension doubles speed. Increasing linear density decreases speed; four times μ halves speed. At fixed vibrating length the fundamental frequency is proportional to speed, so tightening a string raises its pitch while using a heavier string lowers it.

Useful rearrangements are T = μv² and μ = T/v². A dimensional check gives T/μ in (kg m s⁻²)/(kg m⁻¹) = m² s⁻²; its square root has speed units. Total string mass is not a substitute for μ. In an experiment, control vibrating length when studying the effect of tension.

Scientific diagram and interpretation

Equations and relationships

v = √(T/μ)

Worked examples and applications

Common misconceptions

T in this equation is tension, while T in f = 1/T is period. Context and units distinguish them.

Examination technique

Square the speed when rearranging; retain extra calculator digits until the final answer.

For an explanation, connect the physical cause to its effect. For a calculation, state the model, rearrange before substitution, retain intermediate digits, and finish with units. For a data question, refer to the actual readings and distinguish a trend from proof of a cause.

Check your understanding

Five questions. Choose an option, then Check Answer. Read the feedback and use Try Again to reset. Use Tab between controls, arrow keys within radio groups, and Enter or Space on buttons. These local practice answers are not saved.