Waves: Exam-Pattern Drill (8)

Question

A standing wave on a stretched string is described by $y(x, t) = 0.04 \sin(5\pi x)\cos(40\pi t)$ (SI units). Find (a) the wavelength, (b) the frequency, (c) the speed of the component travelling waves, and (d) the position of the third node from the origin. This is a JEE Main 2024 pattern question.

Solution — Step by Step

Why This Works

A standing wave $y = 2A\sin(kx)\cos(\omega t)$ is the superposition of two travelling waves of equal amplitude moving in opposite directions. The wavelength and frequency of the standing wave are the same as those of the component travelling waves — that’s why $v = f\lambda$ still applies.

The spatial part $\sin(kx)$ is fixed in space; it tells us where nodes (zeros) and antinodes (maxima) sit. The temporal part $\cos(\omega t)$ tells us how the standing pattern oscillates in time.

Alternative Method

Some textbooks include the node at the origin as “zeroth” rather than “first”. Read the question carefully — JEE typically counts $x = 0$ as the first node. If the question says “third node after the origin”, the answer is $x = 0.6$ m instead.