PHYS-302, Physics of Waves (Summer 2007)

W4 - Mechanical Waves in a Medium

Topical Outline for two-hour class:

• Mechanical Waves - We derived the equation of motion for longitudinal waves traveling in a 1-D medium consisting of masses and springs. We found that the propagation velocity for waves in a mechanical medium is the square root of the elastic modulus (restoring force) divided by the mass density.
• Hands-on Activity (Wave Speed) Three groups explored how the speed of transverse waves on a string depends on the tension in the string. The fourth group explored how the speed of longitudinal waves in a slinky varies with mass density and the elastic modulus of the spring. Here's a spreadsheet with the student results and with Dr. Russell's summary.
• Energy Carried by a Wave - We derived expressions for the kinetic energy and potential energy of the longitudinal waves in the mechanical mass-spring medium and found that the kinetic and potential energies are exactly identical at all times and at all locations. This is quite different from oscillators - where the potential and kinetic energies trade off back and forth. We discussed the average energy density, and power carried by a wave and found that the energy travels at the propagation velocity as a wave.
• Momentum Carried by a Wave - We discussed the momentum carried by a wave and found that momentum density is just the energy density divided by the wave speed.

Hands-on Activities

Three groups of three students each investigated the propagation velocity for transverse waves on a string. The wave speed for transverse waves on a string is the square root of tension (the elastic restoring force) divided by the linear mass density. The goal of this activity was to discover how the wave speed depended on the tension. This was done by observing the fundamental standing wave pattern for a fixed-fixed string, which has a frequency equal to f = c/(2L). Knowing the length of the string, and the frequency at which the standing wave pattern occurs, the propagation velocity cw may be found directly from cw = 2fL. Changing the tension changes the wave speed, which in turn changes the resonance frequency. Students changed the tension in the string by adjusting the amount of hanging mass attached to one end of the string after it passed over a pulley. Students were asked to plot their data to confirm the dependence of wave speed on tension. One group found that the frequency was proportional to the square root of the hanging mass, another group found that the frequency was proportional to the tension. The third group actually calculated the wave speed and found that it was proportional to the square root of the tension in the string as expected.
The other group of three students attempted to measure how the speed of longitudinal waves in a slinky depended on the mass density and spring constant of the slinky. They stretched the slinky to a specified length and measured the time for a pulse to travel down the slinky, reflect from the fixed end and return. Then they stretched the slinky to a longer length and measured the new travel time. Distance traveled divided by time yields the propagation velocity. We expected the wave speed to increase as the slinky was stretched longer. The mass density (mass per unit length) decreases as the slinky length increases (same mass spread over a larger length), and the elastic restoring force (kx) increases as the spring is stretched further. Both of these effects cause the wave speed to increase. Measuring the propagation time was difficult, but this group did indeed find that the wave speed increased as the length of the slinky increased due to the changes in mass density and elastic restoring force. Note: next time it would be interesting to measure the mass density as a function of length, and also the elastic restoring force in order to obtain a more quantitative result.