Vibrations and Waves Problem Solving

Problems

Traveling Waves without Damping

Problem 1

There is a horizontal taut ideal string with a length of 8 meters, fixed at the left end ($x=0$) and attached to a massless ring free to slide on a frictionless rod on the right end at $x=8m$. The string has a mass of 0.5 kilograms and tension of 4 Newtons, assumed to be constant.

At $t=0$ the profiles of the transverse displacement $y$ and transverse velocity, ${\displaystyle \frac{\partial y}{\partial t}}$, of the string are as shown below.

On the graph below, sketch the shape of the string at $t=0.25$ sec, and at $t=1$ sec. Make sure you indicate the scale for $y$.

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This type of problem is done most easily by decomposing waveforms into superposition of progressive waves.

View the answer below:

 

Problem 2

Voltage, current waves propagating on a transmission line undergo reflection and transmission at a junction between lines with different characteristics. If one considers two transmission lines, joined at $x=0$, with inductance and capacitance (per unit length) of $L$ and $C$ for $x<0$, and $L$ and $C/4$ for $x>0$, then the characteristic phase velocity and impedance increase by a factor of 2 across the junction at $x=0$. Show that this results in transmission and reflected voltage amplitudes (relative to the incoming pulse) of

$$\begin{array}{rcl}T& =& \frac{4}{3}\\ R& =& \frac{1}{3}\end{array}$$

Also show that, in spite of the increased amplitude of transmission and the existence of a reflected pulse, energy is conserved in this system.

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Show that the sum of the power flowing in the transmitted and reflected pulses adds up, exactly, to the incident power.

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