Strings

Author:
a.zampa
This applet simulates the motion of a string of length , mass , and tension (units of the international system of measurement) fixed at its ends. The string is decomposed into equally displaced points and it is assumed that its motion is only transversal (no longitudinal compones computed). Furthermore, the amplitude of oscillation is supposed to be sufficiently small in order to simplify the computation of the transverse component of acceleration, and the motion is assumed to be uniformly accelerated during a sufficiently small time interval (here ).
You have several controls:
  • button Initialise sets the initial profile of the string,
  • button Start starts the simulation,
  • button Stop ends the simulation,
  • label shows the time of simulation,
  • button Get initial time/Get final time allows to capture time lapses (shown on the right),
  • input boxes , , , and set the string parameters,
  • input box Initial profile of the string allows to define a personalised initial configuration of the string,
  • buttons Pulse, Fundamental, Second, Third, Fourth, Fifth, Sixth, and Signal set predefined profiles (respectively a gaussian pulse, the first seven harmonics, and a signal consisting of a superposition of the first seven harmonics with coefficients settable by means of the corresponding sliders).
Observe how the pulse propagates along the string, and using the tools of measurement of time provided, determine its speed. What is the relationship between the speed , the length , the mass and the tension of the string? [Hint: change the values of one parameter at a time, e.g. determine how the speed changes by doubling, tripling, quadrupling, ... (or halving, ... depending on what is convenient in terms of simulation speed) the tension of the string, then pass to mass and, subsequently, to length.] As a result of your investigations you should arrive at the following formula: . Confirm it. The strings are means for propagating mechanical waves, but you can also develop stationary waves that oscillate in amplitude over time but do not propagate in space. Naming wavelength the length of the section of the profile of the standing wave that repeats spatially, it is necessary that in the time taken by the wave to travel a length the complete oscillation occurs. As the starting point of the string must be a node of the wave, all the points at a distance multiple of must be nodes, and since the wave must have a node at the end point of the string, the length of the string must be an integer multiple of . In this way it turns out that the oscillation frequency of the standing wave must be a multiple of the frequency of the fundamental harmonic . Use the applet to confirm this result.