## Part 2 – Transverse Waves on a Beaded String

Standing waves were excited on a beaded string using a mechanical driver.  A hanging mass was suspended over a pulley to produce tension in the system.  The length of the beaded string system from the driver to top of the pulley was measured.  Distance between beads was a constant 5 cm.  Frequency of oscillations was recorded from the driver (digital display).  Node separations were determined by dividing the length of the system by the order of the mode.  From this figure, wavelength was calculated.  Measurements were repeated for varying tensions, and a velocity vs tension relationship was determined.  Velocity dependence on frequency was examined to determine if the beaded string was dispersive.

Initially, results were compared to predictions for non-dispersive systems.  How does the experimental relationship between tension and velocity match up with the theoretical tension velocity relationship for waves on a string (this form only holds for nondispersive media)?  Consider the same formula from the analysis above:

V= sqrt(T/mass per unit length)

Does it matter that the mass on the string is not continuous?  How will that affect the accuracy of predictions?  Since the beaded string appears to be nondispersive over the range considered, we will assume that this formula is accurate in that respect.

The mass per unit length is .0115kg/.442m = 0.0260

 Tension (N) Predicted Value (m/s) Measured Value (m/s) 1.029 6.291 6.131 2.009 8.790 9.468 2.989 10.722 10.955 3.969 12.355 12.305 4.949 13.797 13.764 5.929 15.100 15.401

Most of these values are quite close.

The mass per unit length is .0213kg/.845m = 0.0252

 Tension (N) Predicted Value (m/s) Measured Value (m/s) 2.009 8.929 8.828 3.969 12.550 12.398 5.929 15.339 14.076

Again, most values are close.  It seems that the relation between velocity and tension is a reasonable estimate for this frequency and tension regime on the beaded string.

The data plot and regression of Velocity vs Tension for a 9-bead system is:

Since only two modes (1st and 2nd) were observed for this system, tension and velocity data for each mode is plotted separately.  The first mode is plotted in black, the second is plotted in magenta.

The data plot and regression of Velocity vs Tension for a 17-bead system is:

How do these results compare to the predicted velocity-tension relationship for standing waves on a string (assumes no dispersion)?  The regression equations were plotted along side the form

V= sqrt(T/mass per unit length)

using Mathematica:

In the regime studied, the results match remarkably well with estimates for nondispersive media.

Now, taking dispersion into account, results were analyzed again.  In theory, the dispersion relationship for a beaded string ought to be (Georgi.):

w^2 = 4(T/(ma)) sin^2(ka/2)

Where T is the tension, a is the separation between beads, m is the mass of each bead, k is the angular wave number (2pi/wavelength), and w is the angular frequency (2pi*frequency).  Notice that the dispersion relationship predicted by theory is independent of the total length of the oscillator.

m=0.00126 kg  and  a=0.05 m

 Tension (N) Omega Squared(k) Omega(k) 1.029 W^2=65333.33 sin^2 (.025k) W=255.60 sin (.025k) 2.009 W^2=127555.56 sin^2 (.025k) W=357.15 sin (.025k) 2.989 W^2=189777.78 sin^2 (.025k) W=435.63 sin (.025k) 3.969 W^2=252000 sin^2 (.025k) W=502.00 sin (.025k) 4.949 W^2=314222.22 sin^2 (.025k) W=560.56 sin (.025k) 5.929 W^2=376444.44 sin^2 (.025k) W=613.55 sin (.025k)

I did not plot this data in Excel because I only had two data points for each, which is really insufficient to show any sort of trend - or agreement with any given prediction.

m=0.00126 kg  and  a=0.05 m

 Tension (N) Omega Squared(k) Omega(k) 2.009 W^2=127555.56 sin^2 (0.025k) W=357.15 sin (.025k) 3.969 W^2=252000 sin^2 (.025k) W=502.00 sin (.025k) 5.929 W^2=376444.44 sin^2 (0.25k) W=613.55 sin (.025k)

Compare these predictions to the linear regression equations from the data (Excel) using Mathematica - plots of omega (w) vs k:

W=8.6049k+0.5099  (R^2=0.9994)

W=11.728k+1.4131  (R^2=0.9991)

W=13.627k+0.765  (R^2=0.9994)

The linear regression from the data matches very closely with theory’s prediction for the dispersion relationship.  Over the range of data taken, the velocity remains almost constant with changing velocity.  Only at much higher frequency (on the order of a few hundred Hertz) does the dispersion plot begin to curve sinusoidally.

The two-beaded system was observed at tensions produced by hanging masses of 105 grams and 205 grams.  In each case, resonance was clearly seen in the following modes in the following order (frequency ascending):

1st, a symmetric circular mode

2nd, a symmetric linear mode

3rd, an asymmetric circular mode

Linear and circular modes were a result of the phase difference between horizontal and vertical oscillations – in phase yields linear, out of phase yields circular.  The asymmetric mode is higher energy and therefore higher in frequency than the symmetric mode.  Horizontal oscillations may have been the result of slight misalignment of the rod, driver and pulley over which the beaded string was stretched.

Beyond these three modes of oscillation, there was a regime in which amplitude of oscillations decreased exponentially with distance from the driver.  At lower tension, this regime was reached at around 33 Hz.  At higher tension this regime was reached at around 41 Hz.  In each case, at around 50 Hz, vibrations ceased to be visible, though they were still perceptible if masses or the string were touched.

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