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In Section 16.1 we derived the wave equation for

Chapter 16, Problem 16.17

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QUESTION:

In Section 16.1 we derived the wave equation for transverse waves in a taut string. Here you will examine the possibility of longitudinal waves in the same string. Suppose that an element of string whose equilibrium position is x is displaced a short distance in the x direction to x u(x, t). (a) Consider a short piece of string of length 1 and use the definition (16.55) of Young's modulus YM to show that the tension is F = A YM au/ax, where A is the cross sectional area of the string. [If the string is already in tension in its equilibrium position, this F is the additional tension, that is, F = (actual equilibrium).] (b) Now consider the forces on a short section of string dx and show that u obeys the wave equation with wave speed c = ,/YM/Q where p is the density (mass/volume) of the string.

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QUESTION:

In Section 16.1 we derived the wave equation for transverse waves in a taut string. Here you will examine the possibility of longitudinal waves in the same string. Suppose that an element of string whose equilibrium position is x is displaced a short distance in the x direction to x u(x, t). (a) Consider a short piece of string of length 1 and use the definition (16.55) of Young's modulus YM to show that the tension is F = A YM au/ax, where A is the cross sectional area of the string. [If the string is already in tension in its equilibrium position, this F is the additional tension, that is, F = (actual equilibrium).] (b) Now consider the forces on a short section of string dx and show that u obeys the wave equation with wave speed c = ,/YM/Q where p is the density (mass/volume) of the string.

ANSWER:

Step 1 of 4

(a)

The increase in length of a segment of length \(l\) near position \(x\) is \(dl=u(x+l)-u(x) \approx l\partial u/\partial x\).

 

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