Wave Speed On A String

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A guitarist stretches a string to a tension of 80 N, and the string has a linear mass density of 0.005 kg per metre. At what speed will a transverse pulse travel along this string?

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The speed of a transverse wave on a stretched string depends on the tension $T$ that provides the restoring force and the linear mass density $\mu$ that provides the inertia, through $v = \sqrt{T/\mu}$. A tighter or lighter string carries waves faster. Substituting $T = 80	ext{ N}$ and $\mu = 0.005	ext{ kg/m}$ gives $v = \sqrt{80/0.005} = \sqrt{16000} \approx 126	ext{ m/s}$. The value $16	ext{ m/s}$ takes the ratio $T/\mu$ but forgets the square root applied to the wrong quantity. The value $400	ext{ m/s}$ would arise from a tension of 800 N, an order-of-magnitude error. The value $63	ext{ m/s}$ halves the correct answer, likely from doubling $\mu$. This is the NCERT formula for wave speed on a string, derived from a force balance on a small curved element. As a plausibility check, dividing a large tension by a small mass density gives a large number under the root, so a speed greater than 100 m/s is expected, confirming that $126	ext{ m/s}$ is reasonable for a taut light string.

Wave Speed On A String

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