Vibrations and Waves Legacy Problem #29 Guided Solution
Problem*
During a classroom demonstration, Mr. H uses a wave machine donated to the school by Bell Telephone Company. The wave machine consists of two 1-meter length sections of 50 steel rods. The steel rods are connected to each other so that when the first rod is disturbed from a rest position, the disturbance travels along the medium from rod to rod. One of the sections consists of longer steel rods; disturbances move at 20 cm/s in this section. The other section consists of shorter steel rods; disturbances move at 80 cm/s in this section. Mr. H connects the two sections together so that pulses can cross the boundary from one section to the other. He introduces a pulse with a length of 10 cm into the slower section. Determine the length of this pulse when it crosses the boundary into the faster section.
Audio Guided Solution
This question pertains to the concept of boundary behavior, the behavior of waves as they cross the boundary from one medium to another medium. If you're having difficulties picturing the situation, you might want to click the link here found on this page. It goes back to the boundary behavior page of the physics classroom tutorial. There you'll see some wonderful graphics and some links to animations that might help you picture the situation. So what we have is we have a wave that's traveling through one medium consisting of long steel rods. It travels through that medium at 20 centimeters per second. Once the wave gets to the end of that medium, it meets up with a second medium. There's a boundary between the two, and the pulses cross over the boundary and begin moving through the second medium. This is referred to as boundary behavior. One of the big ideas when it comes to boundary behavior is that as waves cross the boundary from one material to another material, the frequencies of the waves do not change. The speed will change because now the pulses or waves are traveling through a different medium, and the wavelength will change as the speed changes, but the frequency remains the same in both mediums. Because of this, we can say that whatever increase there is in the wave speed, there's a corresponding increase in the wavelength. So if the wave speed quadruples, then the wavelength must also quadruple. So in this question, the wave speed changes from 20 to 80 centimeters per second. That's a fourfold increase, and the length of the pulse or wavelength must change also by fourfold from 10 centimeters to 40 centimeters.
Solution
40 cm
Habbits of an Effective Problem Solver
- Read the problem carefully and develop a mental picture of the physical situation. If necessary, sketch a simple diagram of the physical situation to help you visualize it.
- Identify the known and unknown quantities and record in an organized manner, often times they can be recorded on the diagram itself. Equate given values to the symbols used to represent the corresponding quantity (e.g., \(\descriptive{v}{v,velocity} = 12.8 \unit{\meter\per\second}\), \(\descriptive{λ}{λ,wave length} = 4.52 \unit{m}\), \(\descriptive{f}{f,frequency} = \colorbox{gray}{Unknown}\)).
- Use physics formulas and conceptual reasoning to plot a strategy for solving for the unknown quantity.
- Identify the appropriate formula(s) to use.
- Perform substitutions and algebraic manipulations in order to solve for the unknown quantity.
Read About It!
Get more information on the topic of Vibrations and Waves at The Physics Classroom Tutorial.