**Title: Verifying Solutions to the 1D Wave Equation****Objective:**To verify the validity of given functions as solutions for the one-dimensional (1D) wave equation.**Instructions:**Consider each function below and determine if it satisfies the standard form of the 1D wave equation.**Functions:**a) $ D(x, t) = A \ln(x + vt) $b) $ D(x, t) = A(x - vt)^4 $c) $ D(x, t) = A \sin(kx - \omega t) $**Explanation:**- Each function represents a potential solution to the wave equation, which models wave propagation on a string or through a medium.- The goal is to substitute each function into the wave equation and check if the equality holds, verifying if it's a valid solution.**Note:**- $ A $, $ k $, $ \omega $, and $ v $ are parameters which may represent amplitude, wave number, angular frequency, and wave speed, respectively.- The notation $ \ln $ refers to the natural logarithm, while $ \sin $ denotes the sine function, which is typical in sinusoidal wave descriptions. Use calculus and algebraic manipulation to perform your verification, ensuring each function meets the criteria of a solution to the wave equation.

Answered: WE-1 For each function shown below,…

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

See similar textbooks

Related questions

Concept explainers

Question

**Title: Verifying Solutions to the 1D Wave Equation**

**Objective:**
To verify the validity of given functions as solutions for the one-dimensional (1D) wave equation.

**Instructions:**
Consider each function below and determine if it satisfies the standard form of the 1D wave equation.

**Functions:**

a) $ D(x, t) = A \ln(x + vt) $

b) $ D(x, t) = A(x - vt)^4 $

c) $ D(x, t) = A \sin(kx - \omega t) $

**Explanation:**

- Each function represents a potential solution to the wave equation, which models wave propagation on a string or through a medium.

- The goal is to substitute each function into the wave equation and check if the equality holds, verifying if it's a valid solution.

**Note:**
- $ A $, $ k $, $ \omega $, and $ v $ are parameters which may represent amplitude, wave number, angular frequency, and wave speed, respectively.
- The notation $ \ln $ refers to the natural logarithm, while $ \sin $ denotes the sine function, which is typical in sinusoidal wave descriptions.

Use calculus and algebraic manipulation to perform your verification, ensuring each function meets the criteria of a solution to the wave equation.

Expert Solution

Step 1

The equation to describe the one dimensional wave function is given as

$\frac{\partial^{2} ψ}{\partial x^{2}} = \frac{1}{v} \frac{\partial^{2} ψ}{\partial t^{2}} . . . . . . . . . (1)$

where $ψ$ is the wave function and v is the velocity of the wave in the given medium which can be defined as $v = \frac{ω}{k}$ where $ω$ is the angular frequency and k is the wave vector.

Step by step

Solved in 4 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.

Similar questions