A 1.20 m string fixed at both ends has mass 3.60 g and tension 48.0 N. Determine its wave speed, fundamental frequency and frequencies of the next two harmonics. At 300 Hz identify the harmonic and number of antinodes. If tension is varied experimentally, describe a graph that determines linear density and explain how end effects could shift the measured frequencies.

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μ = 3.00 × 10⁻³ kg m⁻¹, wave speed = 126 m s⁻¹, harmonics ≈ 52.7, 105 and 158 Hz. A 300 Hz drive is not exactly a normal mode; the nearest is n ≈ 5.69, so no ideal stationary resonance occurs. Linear density is μ = m/L = 0.00360/1.20 = 3.00 × 10⁻³ kg m⁻¹. Wave speed is v = √(T/μ) = √(48.0/0.00300) = 126 m s⁻¹. For two fixed ends, allowed wavelengths are λ_n = 2L/n and f_n = nv/(2L). Therefore f₁ = 52.7 Hz, f₂ = 105 Hz and f₃ = 158 Hz. The ratio 300/52.7 = 5.69 is not an integer. Therefore 300 Hz does not correspond to an ideal harmonic, so it is incorrect to assign an exact integer number of antinodes at the stated tension. For fixed L and chosen mode n, f² = [n²/(4L²μ)]T. A plot of f² against T should be straight; its gradient yields μ = n²/(4L² gradient). Real clamps can make the effective vibrating length differ from the ruler length, and string stiffness can alter boundary behaviour. Locate nodes experimentally, measure between effective nodes, repeat frequencies and use a best-fit gradient.

Standing waves and linear-density measurement

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