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) The partial differential equation for the small-amplitude vibrations of a string of length & is given byPA -T = 001² dx²where y(x, t) is the vibration amplitude, p is the material density of the string, A is the string cross sectionalarea, and T is the tension force in the string. Using only the parameters given, make this equation dimen-sionless. Hint 1: first find a combination of f and/or p and/or A and/or T to make z, y, and t dimensionless.(It is clear that a different combination will be needed to make t dimensionless than that for zand y.) Hint2: lete= What are the dimensions of e? (Note that e is known as the "wave speed.")

Given: Let us consider the given partial differential equation, ρA∂2y∂t2-T∂2y∂x2=0 c=TρA To find:…

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

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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)

The partial differential equation for the small-amplitude vibrations of a string of length & is given by
PA -T = 0
01² dx²
where y(x, t) is the vibration amplitude, p 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 dimen-
sionless. Hint 1: first find a combination of f and/or p and/or A and/or T to make z, y, and t dimensionless.
(It is clear that a different combination will be needed to make t dimensionless than that for zand y.) Hint
2: lete= What are the dimensions of e? (Note that e is known as the "wave speed.")

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