# Vibrating string

In this case we are confronted with a system that presents several resonances corresponding to the different vibration modes of a string fixed at its two ends where a standing wave is formed. The fundamental mode is resonant at a frequency given by f 0 =v/(2L), v being the velocity of the waves in the string of length L. The other higher order modes resonate at frequencies given by nf 0 , n=2,3,4...

The three experiments we have ready to launch have been built on a string composed by 21 particles (m=2500, charge=0) linked by springs (k=50000, default length 10). The end particles are fixed. The period for the fundamental mode (free-undamped) has been measured to be 9.3906.

#### Excitation of the different modes

In all experiments we have included an extra particle (big mass = 10000, q=100), to visualize the Electric field value that will produce the external force on selected charged particles of the string.

It is highly recommended to have a look to the space distribution of the different modes previously presented (Section 3.3), because it is the base to feed a given mode.

To excite a mode we observe where are its crests located. There we introduce charges to be driven by an external Electric field. This field is set to a frequency close to the resonance frequency of the mode we want to excite.

To excite the fundamental mode (with a maximum amplitude, crest, at the center), the particle at this point has been given a charge (q=10000)

The alternating Electric field amplitude is Ez = 20, Period =9.3906.

For this mode (with two crests and a valley at the center), the particles number 5, from left and from right, are given equal and opposite charges (of magnitude 10000).

The frequency of the alternating field is set to twice the value of the frequency corresponding to the fundamental mode (Period T= 4.7)

The third higher mode is excited by reducing the period to 1/3 the value corresponding the fundamental mode of vibration (T= 3.13).

To excite this mode, a charge at the center of the string is enough, like for the fundamental mode. The difference between both cases is the frequency.

#### Assignments

• Reload the last experiment (Mode 3)

Change the electric field Period to 3 times the present value and observe/explain the result.
• Reload the same experiment and set the Electric field Period to the one corresponding to the 2nd. mode (T=4.7). Observe/explain the result.
• Try to excite the mode number 5.