Q.Equation of a travelling wave on a stretched string of linear density 5 g/m is $y = 0.03\sin(450t - 9x)$, where distance and time are measured in SI units. The tension in the string is

The basic wave equation states that wave speed equals the product of its wavelength and its frequency: v = lambda f.

For any periodic wave, in the time of one full period T, the wave advances forward by exactly one wavelength lambda. So:

Speed v = distance / time = lambda / T

Since frequency f = 1/T, this becomes:

v = lambda x f

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The wave relation is v = f lambda, which rearranges to f = v/lambda - option (b).

The fundamental relation between the speed v, frequency f and wavelength lambda of a wave is

v = f lambda,

since a wave advances one wavelength in each time period T = 1/f. Rearranging gives

f = v/lambda. …

The speed of a transverse wave on a stretched string is $v=\sqrt{T/m}$.

For a string under tension $T$ with linear mass density (mass per unit length) $m$, applying Newton's second law to a small string element in a transverse disturbance gives the wave equation with wave speed:

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Wave speed on a stretched string is v = sqrt(T/mu); computing the linear mass density from the given length and mass, then substituting, gives v = 400 m/s.

Length of wire, L = 10 m. Mass, m = 5 g = 5 x 10^-3 kg.

Linear mass density, mu = m/L = (5 x 10^-3 kg) / (10 m) = 5 x 10^-4 kg/m.

Tension, T = 80 N.

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Frequency of a vibrating string increases with tension.

For a string of length $L$ and mass per unit length $\mu$ under tension $T$, the fundamental frequency of transverse vibration is $f=\dfrac{1}{2L}\sqrt{\dfrac{T}{\mu}}$. Since $f\propto\sqrt T$, increasing the tension incre …

The speed of a transverse wave on a stretched string is v = sqrt(T/m) (T = tension, m = linear mass density), matching option (b).

For a string under tension T with mass per unit length m (also written as mu), a transverse disturbance travels along it with speed:

v = sqrt(T/m)

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The speed of a transverse wave on a string is $v=\sqrt{T/\mu}$ — it increases with tension and decreases with a heavier string.

Step 1 — Physical basis.

A transverse wave on a stretched string is a restoring-force phenomenon: tension $T$ provides the restoring force that pulls a displaced element of the string back, while the string's inertia (its mass per unit length $\mu$) resists that acceleration. A wave equation derived from Newton's second law applied to a small string element under tension gives:

$v = \sqrt{\frac{T}{\mu}}$

Step 2 — Meaning of the symbols.

• $T$ = tension in the string (in newtons)
• $\mu = m/L$ = mass per unit length of the string (in kg/m) …

The wave velocity equals the angular frequency divided by the angular wave number: v = ω/k.

For a progressive wave y = A sin(kx - ωt), a point of constant phase (kx - ωt = constant) moves with the wave. Differentiating, k(dx/dt) - ω = 0, so dx/dt = ω/k. This dx/dt is exactly the wave (phase) velocity v. Also, since ω = 2πf (f = frequency) and k = 2π/λ (λ = wavelength), ω/k = (2πf)/(2π/λ) = fλ, which is the familiar relation v = fλ. So both forms — v = ω/ …