Extension of a rod segment dx due to a passing longitudinal wave

In summary, the conversation discusses a short segment of a rod with a length of dx and its movement due to a longitudinal wave. The left and right endpoints move in the direction of the x-axis, with the left moving for s and the right moving for s+ds. The speaker wants to calculate the elastic energy of the wave motion and needs to know the extension of dx in order to apply Hook's law. They are unsure if the extension is dx-ds or dx+ds, and are questioning the textbook's explanation that it is simply ds. They also ask about the length of the segment in its resting state and how it changes when the ends are displaced.
  • #1
bolzano95
89
7
Let us look at short segment of a rod with its length dx. Due to longitudinal wave, left endpoint moves for s in the direction of x-axis and the right endpoint moves in the same direction for s+ds.

Because I want to calculate the elastic energy of the wave motion, I need the extension of dx so I can apply Hook's law.

But I'm not sure what is my extension: dx-ds or dx+ds? In my textbook its written that the extension is simply ds, but I don't understand why.
 
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  • #2
What is the length of the segment in the resting state? How does this change when the ends are displaced as described?
 

1. What is the equation for calculating the extension of a rod segment due to a passing longitudinal wave?

The equation for calculating the extension of a rod segment (ΔL) due to a passing longitudinal wave is ΔL = −(ρ/A)∫(∂u/∂x)dx, where ρ is the density of the rod, A is the cross-sectional area of the rod, and u is the displacement of the wave.

2. How does the density of the rod affect the extension of a rod segment due to a passing longitudinal wave?

The density of the rod directly affects the amount of extension (ΔL) experienced due to a passing longitudinal wave. A higher density rod will experience a greater extension compared to a lower density rod with the same cross-sectional area and displacement.

3. Does the cross-sectional area of the rod play a role in the extension of a rod segment due to a passing longitudinal wave?

Yes, the cross-sectional area (A) of the rod is a crucial factor in determining the extension (ΔL) caused by a longitudinal wave. A larger cross-sectional area will result in a smaller extension compared to a rod with a smaller cross-sectional area experiencing the same displacement.

4. How does the displacement of the wave affect the extension of a rod segment?

The displacement (u) of the wave is directly proportional to the extension (ΔL) experienced by the rod segment. This means that a higher displacement will result in a greater extension, while a lower displacement will result in a smaller extension.

5. Are there any other factors that can influence the extension of a rod segment due to a passing longitudinal wave?

Aside from the density, cross-sectional area, and displacement, other factors that can influence the extension of a rod segment include the material properties of the rod (such as elasticity and stiffness), the frequency and amplitude of the wave, and the length of the rod segment.

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