Longitudinal wave from one thin rod to another

In summary, the problem involves a longitudinal wave passing from a thin rod of silver (E = 83 GPa, \rho = 10.5 g/cm3) into a thin rod of gold (E = 78 GPa, \rho = 19.3 g/cm3). Both rods have the same cross sectional area and an incident wave with amplitude 1. Using the equations for velocity (v = \sqrt{\frac{E}{\rho}}) and impedance (Z = \frac{AE}{v}), the amplitude of the transmitted and reflected waves can be calculated by comparing the impedances of silver and gold. The final step is to apply boundary conditions to determine the ratios B/A and C/A, which will
  • #1
Blueman52
1
0

Homework Statement


Suppose a thin rod of silver (E = 83 GPa, [itex]\rho[/itex] = 10.5 g/cm3) is joined to a thin rod of gold (E = 78 GPa, [itex]\rho[/itex] = 19.3 g/cm3).
Both rods have the same cross sectional area, and a longitudinal wave passes from the silver into the gold.

What amplitude will the transmitted and reflected waves be if the incident wave has amplitude 1?


Homework Equations


v = [itex]\sqrt{\frac{E}{\rho}}[/itex] (velocity)
Z = [itex]\frac{AE}{v}[/itex] (impedance)

The Attempt at a Solution


Esilver = 83 x 109 Pa.
[itex]\rho[/itex]silver = 1.05 x 104 Kgm-3.

Egold = 78 x 109 Pa.
[itex]\rho[/itex]gold = 1.93 x 104 Kgm-3.

vsilver.jpg

= 2.812 x 103 ms-1.

vgold.jpg

= 2.010 x 103 ms-1.

zsilver.jpg

= 2.952 x 107

zgold.jpg

= 3.881 x 107

Zsilver : Zgold
= 2.952 x 107 : 3.881 x 107
= 0.761 : 1


This is as far as I got.
I presume I have to multiply the amplitude with Zsilver : Zgold
= 1 x 0.761
= 0.761

but I don't know whether this is the amplitude of the reflected or transferred wave.

I would greatly appreciate any help at all.
Thank you!
 
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  • #2
I would approach the problem by setting up a coordinate system with x=0 as the boundary. To the left, the wave is Aeik1*x + B-ik1*x at t=0, where A is the initial wave and B is the reflection. To the right, it's Ceik2*x. You can calculate B/A and C/A by applying the appropriate boundary conditions.
 

Related to Longitudinal wave from one thin rod to another

1. What is a longitudinal wave?

A longitudinal wave is a type of mechanical wave that causes particles in a medium to vibrate in the same direction as the wave is traveling. This is in contrast to a transverse wave, where particles vibrate perpendicular to the direction of the wave.

2. How is a longitudinal wave created?

A longitudinal wave is created when a disturbance or energy is applied to a medium, causing particles in the medium to compress and expand in the same direction as the energy. For example, when you speak, the vibrations from your vocal cords create longitudinal waves in the air that travel to the listener's ear.

3. How does a longitudinal wave travel from one thin rod to another?

When a longitudinal wave reaches the end of a thin rod, some of the energy is reflected back towards the source, while the rest is transmitted to the next rod. The transmitted wave will then cause the particles in the second rod to vibrate in the same direction, continuing the wave's propagation.

4. What is the speed of a longitudinal wave?

The speed of a longitudinal wave depends on the properties of the medium it is traveling through. In general, the denser the medium, the faster the wave will travel. For example, sound waves travel faster through water than through air because water is more dense.

5. What are some applications of longitudinal waves?

Longitudinal waves have many practical applications, including in communication technology (such as radio waves and sound waves), medical imaging (such as ultrasound), and non-destructive testing (such as ultrasonic testing). They are also used in earthquake detection and monitoring, as well as in the study of geological structures in the Earth's interior.

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