Physics · Waves · NEET
Faster. Higher tension means a stronger restoring force pulling each bit of string back to the middle, so a disturbance passes along quicker. In v = sqrt(T/mu), T is on top, so increasing T increases v. Loosen the string (lower T) and the wave slows down.
No. Because v is proportional to sqrt(T), doubling T only multiplies v by sqrt(2) (about 1.41 times), not 2. To actually double the speed you need 4 times the tension. This exact idea is the NEET 2022 question below, so do not skip it.
The formula comes from Newton's second law applied to a tiny curved element of string. The restoring force grows in proportion to T, but speed is a square-root of (force factor / mass factor), just like it is in v = sqrt(T/mu). Speed = sqrt(elastic factor / inertia factor) is a pattern you also see in sound: v = sqrt(B/rho). So the square root is built into how mechanical wave speeds work.
Yes. The string length L is fixed, so the wavelength of the fundamental is fixed at 2L. Since v = f x lambda and lambda is fixed, a higher v forces a higher f. That is why tightening a guitar or sonometer string raises the pitch. Here tension changes speed first, and frequency follows.
It is different at every point. Tension is largest where the rope must support more weight below it, so the wave speed is largest there and smallest at the free bottom end. The frequency stays the same everywhere, so the wavelength stretches where the speed is higher. This is exactly the NEET 2016 hanging-rope PYQ below.
Both matter. mu (linear mass density) is in the bottom of v = sqrt(T/mu). A thicker or heavier string has larger mu, so for the same tension the wave is slower. That is why the thick bass strings on a guitar give low notes and the thin strings give high notes.
If the tension in a stretched string is doubled, the ratio of the initial speed to the final speed of a transverse wave along the string is:
A uniform rope of length L and mass m1 hangs vertically from a rigid support, with a block of mass m2 attached to its free (lower) end. A transverse pulse of wavelength lambda1 is produced at the lower end of the rope; when it reaches the top its wavelength is lambda2. The ratio lambda2/lambda1 is:
Try the real previous-year questions from this chapter — each with the answer and a full solution.
v = sqrt(T / mu), where v is the transverse wave speed, T is the tension in newtons, and mu is the linear mass density (mass per unit length) in kg/m.
Wave speed increases with tension, but only as the square root: v is proportional to sqrt(T). Doubling the tension multiplies the speed by sqrt(2), and making it 4 times larger doubles the speed.
If the string is free-driven, no; frequency is set by the source. But for a string fixed at both ends (guitar, sonometer) the length fixes the wavelength, so raising tension raises the wave speed and therefore raises the frequency and pitch.
Fixing the ends fixes the fundamental wavelength at 2L. Tightening the string raises v = sqrt(T/mu), and since f = v / (2L), a higher v means a higher fundamental frequency, so a higher note.
No. For a given string and tension, v = sqrt(T/mu) is fixed by the medium. Changing how hard you pluck (amplitude) or how fast you shake it (frequency) does not change the wave speed; it only changes the wavelength through v = f x lambda.
