Answered: The partial differential equation for…

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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The partial di§erential equation for the small-amplitude vibrations of a string of length ` is given by A@2y@t2T@2y@x2 = 0 where y(x; t) is the vibration amplitude, is the material density of the string, A is the string cross sectional area, and T is the tension force in the string. Using only the parameters given, make this equation dimensionless. Hint 1: first find a combination of ` and/or and/or A and/or T to make x, y, and t dimensionless.
(It is clear that a different combination will be needed to make t dimensionless than that for x and y.) Hint 2: let c = q TA . What are the dimensions of c? (Note that c is known as the ìwave speed.î)

## Small-Amplitude Vibrations of a String
The partial differential equation for the small-amplitude vibrations of a string of length \( \ell \) is given by:

\[ \rho A \frac{\partial^2 y}{\partial t^2} - T \frac{\partial^2 y}{\partial x^2} = 0 \]

where \( y(x, t) \) is the vibration amplitude, \( \rho \) is the material density of the string, \( A \) is the string cross-sectional area, and \( T \) is the tension force in the string.

### Goal
Using only the parameters given, make this equation dimensionless.

#### Hint 1
First, find a combination of \( \ell \) and/or \( \rho \) and/or \( A \) and/or \( T \) to make \( x \), \( y \), and \( t \) dimensionless.
(It is clear that a different combination will be needed to make \( t \) dimensionless than that for \( x \) and \( y \).)

#### Hint 2
Let \( c = \sqrt{\frac{T}{\rho A}} \). What are the dimensions of \( c \)? (Note that \( c \) is known as the "wave speed.")

### Explanation
To make the variables dimensionless:
1. Identify the fundamental dimensions of \( \ell \), \( \rho \), \( A \), and \( T \).
2. Use dimensional analysis to form dimensionless groups.
3. Recognize that the wave speed \( c \) provides a natural scale for combining spatial and temporal dimensions.

This exercise will help you understand how physical parameters influence the behavior of the string's vibrations and how dimensionless variables simplify the analysis of such systems.

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