Standing Waves on a String

Introduction Consider a string, with its two ends fixed, vibrating transversely in one of its harmonic modes. Locations along the string where no vibration occurs, such as the end points,

Figure 1: Apparatus for Generating Standing Waves on a String are called nodes, whereas locations where maximum vibration occurs are called antinodes. The speed v with which a wave propagates along the string is given by [1] √ T v= , (1) µ where T is the tension in the string, and µ is the linear mass density. The speed is related to the wavelength λ and frequency f of the wave as follows: v = fλ .

(2)

Combining these two equations we obtain 1 √ λ= √ T. f µ 1

(3)

In this experiment a string is forced to vibrate at a known frequency. The tension in the string is varied so that the wavelength of waves propagating along the string also varies. A regression equation based on the form of Equation 3 is obtained using the tension and wavelength data derived from the experiment. The consistency of the theory underlying Eq. 3 is assessed by comparing the slope of the regression line to its theoretical value.

Procedure Record all measurements and calculations to the correct number of significant figures in the appropriate table. Perform the experiment and analysis as follows. 1. Add enough mass m to the mass holder so that the string supports a standing wave with one node in the interior of the string (The node will be located approximately one meter from the pulley.). 2. Measure the distance L from the node at the pulley to the node on the interior of the string. 3. Do the same for n = 2, 3, 4, n being the number of nodes in the interior of the string. In each case, measure the distance L from the node at the pulley to the interior node closest to the vibrator. 4. Calculate the tension T in the string for each case. The tension is the force exerted on the string by the mass hanging from the string: T = mg. Here the acceleration of gravity g = 9.80 m/s2 . 5. Calculate the wavelength λ of the standing wave in each case, where λ = 2L n . Note: the distance between consecutive nodes is λ2 . √ 6. Plot a graph of λ vs T . The quantity λ is considered the dependent variable and is plotted along the vertical axis (the ordinate). 7. Using Excel, perform a linear regression of the data, forcing the intercept to be zero. 8. Calculate the standard error in the slope using equation 8. Report the experimental value of the slope in the form Mest ± δM , where Mest is the value of the slope obtained from the regression. 1 9. Calculate the value of the slope according to theory, i.e. M = f √ µ . The frequency of vibration is f = 60 Hz, and the linear mass density of the string is µ = 2.98 × 10−4 (kg/m).

10. Compare the theoretical and experimental values of the slope for consistency using the criterion 10.

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Appendix Consider two sets of data yi and xi (i = 1 . . . N ) where the yi are assumed to be linearly related to the xi according to the relationship yi = M xi + B + ϵi .

(4)

The ϵi is a set of uncorrelated random errors. Estimates of the slope Mest and y– intercept Best can be obtained using a variety of techniques, such as the method of least squares estimation. Furthermore, it can be shown that the standard error of the slope δM is √ 1 − r 2 σy δM = , (5) N − 2 σx and the standard error of the intercept δB is √ δB = σy

1 − r2 N −2

√

N −1 x ¯2 + 2 , N σx

(6)

where r is the estimated correlation coefficient between the yi and xi , σy and σx are the estimated standard deviations of the yi and xi , and x ¯ is the estimated average of the xi .1 The quantities x ¯, σx , and σy can straightforwardly be calculated using function keys on a scientific calculator or defined functions in Excel. If, in the regression equation, Eq. 4, the quantity B, the y-intercept, is required to be zero, i.e. the regression equation is now yi = M xi + ϵi , (7) then the degrees of freedom in the random errors are N − 1 rather than N − 2 so that √ 1 − r 2 σy δM = . (8) N − 1 σx To reflect the statistical uncertainty in a quantity Q, where Q is either the slope M or y-intercept B, the quantity Q is typically reported as Qest ± δQ ,

(9)

which can be understood informally to mean that with high probability the true value of Q lies within the interval [Qest − 1.96 δQ, Qest + 1.96 δQ] ,

(10)

and its best estimate is Qest . 2 In general, any quantity expressed in the form of Eq. 9 can be interpreted according to Eq. 10. 1 Note:

in Equation 6 σx and σy are unbiased estimates. seemingly arbitrary value 1.96 has a precise statistical interpretation. For sufficiently large N it can be shown that the statistical distribution of Qest is a normal distribution of mean Q (the true value of Q) and standard deviation δQ. Thus, the probability that the true value of Q lies outside of the interval given by Equation 10 is 5%. 2 The

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References [1] Wikipedia. Vibrating string. http://en.wikipedia.org/wiki/ Vibrating_string, 2008. [Online; accessed 12-June-2008].

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Number of nodes interior to the string

m (kg)

T (nt)

λ (m)

1 2 3 4 Table 1: Data and Calculations I

Experiment Slope

± Table 2: Calculations II

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Theory

√ T (nt)