Q.What are longitudinal waves and transverse waves? Derive the formulas for the speed of a transverse wave on a stretched string, and the speed of a longitudinal wave in a medium. OR Defining the Doppler effect, match the following (Situation ↔ Observed frequency):

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/ …