Question

A wave propagates on a string in positive $x$ -direction with a speed of $40 \mathrm{~cm} / \mathrm{s}$. The shape of string at $t=2 \mathrm{~s}$ is $y=10 \cos \frac{x}{5}$, where $x$ and $y$ are in centimetre. The wave equation is: (a) $y=10 \cos \left(\frac{x}{5}-8 t\right)$ (b) $y=10 \sin \left(\frac{x}{5}-8 t\right)$ (c) $y=10 \cos \left(\frac{x}{5}-8 t+16\right)$ (d) $y=10 \sin \left(\frac{x}{5}-8 t+16\right)$

Name: Problem 4, Chapter 13: Arihant AIEEE Physics
Uploaded: 2021-09-09T13:58:05-08:00
Duration: 1 min 49 s
Channel: Ajay Singhal
Description: A wave propagates on a string in positive x -direction with a speed of 40 cm / s. The shape of string at t=2 s is y=10 cos(x)/(5), where x and y are in centimetre. The wave equation is: (a) y=10 cos((x)/(5)-8 t) (b) y=10 sin((x)/(5)-8 t) (c) y=10 cos((x)/(5)-8 t+16) (d) y=10 sin((x)/(5)-8 t+16)

   A wave propagates on a string in positive $x$ -direction with a speed of $40 \mathrm{~cm} / \mathrm{s}$. The shape of string at $t=2 \mathrm{~s}$ is $y=10 \cos \frac{x}{5}$, where $x$ and $y$ are in centimetre. The wave equation is:
(a) $y=10 \cos \left(\frac{x}{5}-8 t\right)$
(b) $y=10 \sin \left(\frac{x}{5}-8 t\right)$
(c) $y=10 \cos \left(\frac{x}{5}-8 t+16\right)$
(d) $y=10 \sin \left(\frac{x}{5}-8 t+16\right)$

Arihant AIEEE Physics

D.B. Singh 2006 Edition

Chapter 13, Problem 4

Instant Answer

Solved by Expert Ajay Singhal

09/09/2021

Step 1

Step 1: The wave equation at any time can be written as $y=10 \cos \left(\frac{x}{5}-\omega t + \phi\right)$, where $\omega$ is the angular frequency, $t$ is the time, and $\phi$ is the phase constant. Show more…

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A wave propagates on a string in positive $x$ -direction with a speed of $40 \mathrm{~cm} / \mathrm{s}$. The shape of string at $t=2 \mathrm{~s}$ is $y=10 \cos \frac{x}{5}$, where $x$ and $y$ are in centimetre. The wave equation is: (a) $y=10 \cos \left(\frac{x}{5}-8 t\right)$ (b) $y=10 \sin \left(\frac{x}{5}-8 t\right)$ (c) $y=10 \cos \left(\frac{x}{5}-8 t+16\right)$ (d) $y=10 \sin \left(\frac{x}{5}-8 t+16\right)$

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Key Concepts

Wave Speed, Wave Number, and Angular Frequency

These quantities are interrelated by the equation v = ?/k, where v is the wave speed, ? is the angular frequency, and k is the wave number. Knowing any two of these parameters allows the determination of the third, making this relationship key in understanding and formulating the wave equation.

Phase Shift

The phase shift (?) in the wave equation determines the initial displacement of the wave at a specific time and position. It is critical when the wave’s snapshot is provided at a particular moment, as it adjusts the equation so that the mathematical representation matches the given initial conditions.

Wave Equation Representation

The general form of a traveling wave is expressed as y(x, t) = A cos(kx - ?t + ?) or y(x, t) = A sin(kx - ?t + ?), which describes how the wave's amplitude, direction, and phase evolve in space and time. This representation is fundamental for analyzing wave behavior in various media.

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