P2: Energy and Power for a Traveling Pulse

(P2a) Energies and Energy Density

The total energy U for a wave y(x,t) on a string is the integral of the energy density u(x) over the string, where

the first term under the integral is the potential energy density of the spring, and the second term is the kinetic energy density. One can estimate the energy in a boring, simple, structureless, up-and-down pulse like the one studied here, simply from dimensional analysis.

Make a table with four columns and the following three rows:

potential energy density
kinetic energy density
energy density

If you use the amplitude A for all the dy's in the formula, the pulse length in space for all the dx's, and the pulse length in time for the dt's, you can write an algebraic formula for each of these quantities in terms of A,

, and the string parameters tau, Mu1, and L. Put these formulas in the first column of the table.

Evaluate the formulas using the parameters of our problem, and enter these in the second column. For , you can either measure the FWHM, or you can calculate it from , tau, and mu.

By selecting the relevant quantity to measure on the "vs X" plot, measure rough peak numerical values for each of the three energy densities, and fill these into the third column. What would you need to do to get the total energies?

Dimensional analysis has its limitations. Suppose we use a wiggly ``Packet'' instead of the nice smooth ``Pulse''? Run a packet, and in the fourth column, put the peak densities of the three kinds of energies you found. Why is the energy density so far from that predicted by dimensional analysis? Is there another way to estimate the energy densities, if the wavelength of the modulation is much smaller than the length of the packet?

Dimensional analysis doesn't work unless the objects really are described well using the dimensions: our funny packet has an internal length much smaller than the pulse width, so the local slope really isn't given by the height of the packet divided by the length of the packet.

(P2b) Relationship between Potential and Kinetic Energy: Traveling Wave

Using the fact that the pulse and packet are both traveling waves,
y(x,t) = f(x-vt)
derive a formula relating the potential energy density to the kinetic energy density, and add it to your writeup. Does the Pulse satisfy this? Does the Packet? Note each success and failure in your writeup.

(P2c) Relationship between Energy and Power: Traveling Wave

Again using the traveling wave form, derive a formula relating the energy density and the power, for your writeup. Using the shapes and the peak values, compare the energy density and power for a pulse moving to the right: is the ratio correct? Compare the energy density and power for a pulse moving to the left: is it still OK? Compare the energy density and power for a Packet. Note each success and failure in your writeup.

The relationships in parts (b) and (c) are very useful, and apply to all shapes of packets, but they only apply to traveling waves.

Links Back

Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).