Sound Waves Legacy Problem #15 Guided Solution

It is the habit of an effective problem solver to read the problem carefully and develop a mental picture of what's going on to identify the known and the unknown quantities and then to think about what strategy can be used to relate the known to the unknown quantities. Here we read about an elastic cord which is vibrating in a so-called third harmonic wave pattern. We know the distance between the two poles to which the cord is mounted, it's 98 centimeters, I call that L or length, it's equal to 98 centimeters. I also know the frequency which causes this cord to vibrate in the third harmonic wave pattern, it's 84 hertz, I say F equal 84 hertz. What I'm looking for is I'm looking for the speed, the V of these waves. So V is my unknown. Now when I think about my topic of sound waves and particularly sound waves traveling through wires and cords and strings of that nature, I think of the wave equation. V equal F times lambda and I think, well, I could calculate V if I only knew the F and the lambda. Well, the F is easy, it's given as 84 hertz. The lambda or wavelength is something I need to calculate. It's not 98 centimeters, that's the length of the string. But within that string are waves and if we know it's vibrating as the third harmonic wave pattern, we should be able to define the wavelength. Now what I immediately do when I get a problem like this is I begin to sketch out the wave pattern. I draw myself a horizontal line, put dots on each end representing where the cord is mounted to the poles and then into the string I begin to draw a sine wave, which has three sections in it, vibrating up and down. In something I'll be able to get one and a half wavelengths within that length of string since each section is equivalent to half of a wavelength. So I can say that the 98 centimeters length equal 1.5 times lambda and I can divide each side of that equation by 1.5 and I end up calculating my lambda. Comes out to be 65.33 repeating centimeters. Now that I know the lambda and I'm given the frequency, I can calculate the speed using the wave equation. V equal F times lambda. When you do that you get 5488 centimeters per second or if you did it in units of meters it would be 54.88 meters per second and that can be rounded to two significant digits such that you get 55 meters per second or 5500 centimeters per second.