Question

(II) Let two linear waves be represented by $D_{1}=f_{1}(x, t)$ and $D_{2}=f_{2}(x, t) .$ If both these waves satisfy the wave equation (Eq. $15-16$ ), show that any combination $D=C_{1} D_{1}+C_{2} D_{2}$ does as well, where $C_{1}$ and $C_{2}$ are constants.

Name: Problem 34, Chapter 15: Physics for Scientists and Engineers with Modern Physics
Uploaded: 2023-03-29T14:55:54-08:00
Channel: Lainey Roebuck
Description: (II) Let two linear waves be represented by D1=f1(x, t) and D2=f2(x, t) . If both these waves satisfy the wave equation (Eq. 15-16 ), show that any combination D=C1 D1+C2 D2 does as well, where C1 and C2 are constants.

   (II) Let two linear waves be represented by $D_{1}=f_{1}(x, t)$ and $D_{2}=f_{2}(x, t) .$ If both these waves satisfy the wave equation (Eq. $15-16$ ), show that any combination $D=C_{1} D_{1}+C_{2} D_{2}$ does as well, where $C_{1}$ and $C_{2}$ are constants.

Physics for Scientists and Engineers with Modern Physics

Douglas C. Giancoli 4th Edition

Chapter 15, Problem 34

Instant Answer

Solved by Expert Lainey Roebuck

03/29/2023

Step 1

This means that they satisfy the following equation: \[ \frac{\partial^2 f_{i}}{\partial t^2} = v^2 \frac{\partial^2 f_{i}}{\partial x^2} \quad \text{for} \quad i=1,2 \] Show more…

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(II) Let two linear waves be represented by $D_{1}=f_{1}(x, t)$ and $D_{2}=f_{2}(x, t) .$ If both these waves satisfy the wave equation (Eq. $15-16$ ), show that any combination $D=C_{1} D_{1}+C_{2} D_{2}$ does as well, where $C_{1}$ and $C_{2}$ are constants.

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Key Concepts

Wave Equation

The wave equation is a second?order partial differential equation that describes the propagation of waves, such as sound or light waves, through a medium. It typically involves second derivatives with respect to both space and time, capturing how disturbances evolve and spread over time.

Linearity of Differential Equations

Linearity implies that a differential operator applied to a sum of functions is equal to the sum of the operator applied to each function individually. This property is essential for understanding why individual solutions to the wave equation can be combined to form new solutions.

Superposition Principle

The superposition principle states that if two or more functions are solutions to a linear homogeneous differential equation, then any linear combination of those functions—with constant coefficients—will also be a solution. This is a fundamental property of linear systems such as those described by the wave equation.

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