Wave on a StringJEE Main

Concept

A transverse wave carries energy along the string while each particle merely oscillates up and down. Its shape is $y(x,t) = A\sin(kx - \omega t)$; the whole pattern moves at the wave speed $v = f\lambda$, set by the medium's tension and mass per length.

Key formula

Derivation

The wave number $k = 2\pi/\lambda$ measures phase per unit length and $\omega = 2\pi f$ phase per unit time. A point of constant phase satisfies $kx - \omega t = \text{const}$, so it moves at $x/t = \omega/k = f\lambda$.

For a string the speed is fixed by the medium: $v = \sqrt{T/\mu}$ where $T$ is the tension and $\mu$ the linear mass density.

Changing the source frequency changes $\lambda$ to keep $v$ constant — the medium, not the source, sets the speed.

Scenarios to explore

• Wave on a String — Travelling transverse wave: amplitude, wavelength, speed.

Real-world applications

• Musical strings and the pitch of stringed instruments.
• Seismic S-waves travelling through the Earth.
• Signal propagation along cables.

JEE exam tips

• On the same string $v$ is constant, so $\lambda \propto 1/f$.
• Particle velocity $v_p = -\omega A\cos(kx-\omega t)$ is different from wave velocity $v$.

Common mistakes

• Thinking the string particles travel along with the wave — they only oscillate transversely.
• Assuming a higher frequency means a higher wave speed on the same string.
• Mixing up $k = 2\pi/\lambda$ with $\omega = 2\pi f$.

Exam traps to avoid

• $y = A\sin(kx + \omega t)$ travels in the −x direction; watch the sign.
• Maximum particle speed $A\omega$ can exceed or be less than the wave speed $v$.