Standing Waves

The goal for this experiment is to investigate and have a basic understanding of force driven standing waves.

A standing wave is a transverse wave traveling along a medium, in this case a string, and then reflected back and returns interfering with other waves. When interfering, at resonance, a standing wave is created with nodes and antinodes in the wave pattern.

Equations:

-Transverse wave in the positive x direction.

y1 = A sin(kx - wt),

where k = 2*pi/lambda & w = 2*pi*f

-Transverse wave reflected from fixed end

y2 = A sin(kx + wt)

-At resonance, the two waves combine and become the some of both waves.

                                y = y1 + y2 = (2A sin (kx))*cos(wt)

 Other equations:

-Wavelength for each harmonic of vibrating string.

Lambda = 2L/n

-If frequency is known, the wave speed can be determined with fundamental wave velocity equation.

v = f*lambda

-Substituting both.

f = vn/2L

-velocity of wave traveling on a string.

v = sqrt(T/mu),

where T is tension & and mu is mass per unity length.

Experiment:

We used mechanism that drives a wave on one side of the string, and the other side of the string was pulled by a hanging mass creating tension. We adjusted the frequency of the driving mechanism to find each harmonic and measured the wavelength.

For case 1 we had a string of length 140 cm and a hanging mass of 200 g.

-Data:

16.5 Hz, 1 loop,  2 nodes, lambda 140 cm

32.5 Hz, 2 loops, 3 nodes, lambda 67 cm

45.6 Hz, 3 loops, 4 nodes, lambda 45 cm

56.7 Hz, 4 loops, 5 nodes, lambda 37 cm

74.6 Hz, 5 loops, 6 nodes, lambda 28 cm

87.4 Hz, 6 loops, 7 nodes, lambda 23 cm

For case 2 we had a string of length 186.5 cm and a hanging mass of 50 g.