A wave disturbance in a medium is described by y(x, t)=0.02 cos(50 πt+(π)/(2)) cos(10 πx) where

step 1 Dear students, in problem 172 172 of chapter 9 oscillation and waves we have asked that a wave d

A wave disturbance in a medium is described by y(x, t)=0.02...

A wave disturbance in a medium is described by $y(x, t)=0.02 \cos \left(50 \pi t+\frac{\pi}{2}\right) \cos (10 \pi x)$ where $x$ and $y$ are in meter and $t$ is in second. Then (A) First node occurs at $x=0.15 \mathrm{~m}$ (B) First anti-node occurs at $x=0.3 \mathrm{~m}$ (C) The speed of interfering waves is $5.0 \mathrm{~m} / \mathrm{s}$ (D) The wavelength is $0.2 \mathrm{~m}$

A wave disturbance in a medium is described by $y(x, t)=0.02 \cos \left(50 \pi t+\frac{\pi}{2}\right) \cos (10 \pi x)$
where $x$ and $y$ are in meter and $t$ is in second. Then
(A) First node occurs at $x=0.15 \mathrm{~m}$
(B) First anti-node occurs at $x=0.3 \mathrm{~m}$
(C) The speed of interfering waves is $5.0 \mathrm{~m} / \mathrm{s}$
(D) The wavelength is $0.2 \mathrm{~m}$

Key Concepts

Standing Waves

Standing waves occur when two waves of identical frequency and amplitude travel in opposite directions and interfere, forming a fixed vibration pattern characterized by alternating nodes and antinodes. This pattern can be mathematically represented as a product of a purely time-dependent function and a purely space-dependent function.

Nodes

Nodes are specific locations in a standing wave where the interference is completely destructive, causing the amplitude to remain zero at all times. They reflect the points of minimum energy and are determined by the spatial component of the wave’s mathematical representation.

Antinodes

Antinodes are points of maximum displacement in a standing wave, where the constructive interference between the two interfering waves is the strongest. These spots reach the highest amplitude during oscillation and are found through analysis of the spatial part of the wave equation.

Wavelength and Wave Number

The wavelength is the spatial period of the wave—the distance over which the wave's shape repeats—while the wave number (k) is defined as 2? divided by the wavelength. This relationship is crucial for determining the spatial characteristics of a wave.

Wave Speed and Angular Frequency

Wave speed is calculated as the ratio of the angular frequency (?) to the wave number (k), i.e., v = ?/k. Angular frequency describes how quickly the wave oscillates in time, and together with the wave number, it allows one to determine the propagation speed of the wave through the medium.

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