Final answer:
To find the characteristics of a standing wave on a guitar string, one can use wave properties and string parameters. The wavelength for the third harmonic is 0.440 m, and the tension force in the string is approximately 366.08 N. The function of the wave can be represented mathematically, and the amplitude of a specific particle depends on its position along the string.
Step-by-step explanation:
The student's question pertains to the standing waves on a guitar string resonating in its third harmonic frequency. Let's solve each part of the problem one by one:
- The wavelength of the standing wave can be found by understanding that the third harmonic has three half-wavelengths along the length of the string. Therefore, wavelength (λ) = 2/3 * length (L), which gives: λ = 2/3 * 0.660 m = 0.440 m.
- The tension force in the string, FT, can be calculated using the formula FT = (n * f * μ * L)2, where n is the harmonic number (here equals to 3), f is the frequency, μ is the linear mass density, and L is the length of the string. Plugging the values, FT = (3 * 720 Hz * 1.90 g/m * 0.660 m)2 = 366.08 N (after converting g/m to kg/m).
- The function of the standing wave can be represented as y(x,t) = 2A * sin((n * π * x) / L) * cos((2 * π * f * t)), with A being the amplitude, n the harmonic number, x the position, L the length of the string, and t the time. This gives the function for the third harmonic.
- The amplitude of a particle located 5 cm from the left end would be A' = 2.5 mm * cos((3 * π * 0.05 m) / 0.660 m), since the displacement varies as the cosine of the position along the string. Therefore, A' reduces to approximately 2.24 mm.
- For E) the graph at t=0.1 s, you would plot the function y(x,0.1) across the length of the string, showing the characteristic sinusoidal pattern of a standing wave's third harmonic at a given instant.
Remember that in a standing wave, the amplitude of the displacement depends on the position along the string and is maximum at antinodes while zero at nodes.