Physics 4BL Experiment 5.docx

# Here v g is the group velocity and ω k ck one could

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Here, v g is the group velocity and ω ( k ) = ck . One could use some oscillating wave on a non-perfect string as an example of a dispersive dispersion relation, where the group velocity could (for instance) be expressed as: v g = δω ( k ) δk = 4 ( T μ k 4 + Lk 7 ) 3 ( 4 T μ k 3 + 7 Lk 6 ) and: ω ( k ) = ( T μ k 4 + Lk 7 ) 4 . Here, T is the tension in the string where μ is the linear mass density. Note that the angular frequency of the wave changes with k, so thus, velocity also changes with angular frequency. (iii) (iv)

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Introduction The purpose of this experiment is to experimentally obtain a value for the speed of sound in air, and compare it to the accepted value of 343 m/s. This is done via two different methods: the first method involves travelling sound waves, and the second method involves the use of standing waves. The first method of measuring the speed of sound with travelling waves makes use of the property that air is a non-dispersive medium for sound waves. This means that all frequencies of sound travel with the same phase velocity, v p , which is given as: v p = where f is the (cyclical, temporal) frequency and λ is the wavelength. The phase velocity can be equivalently expressed as: v p = ω k Where ω = 2π f is the angular frequency and k = 2π/λ is the wavevector. To measure the speed of sound, a Rigol wave function generator is used to make a speaker produce sound waves of frequencies ranging from 4kHz to 20kHz. The sound waves from the speaker are then received by a microphone mounted on a linear track. Both the signals from the Rigol and the microphone are measured by an oscilloscope. A plot of the dispersion relationship of ω versus k is made with the data acquired, and linear regression of the data is then used to obtain an experimental value for the speed of sound, which is given by the slope of the regression (by Equation 2). The second method for obtaining the speed of sound uses the superposition of standing sound waves. Once again, the Rigol wave generator causes sound waves to be produced by a speaker to be measured by a microphone. Unlike the first method, a movable reflector plate is set up across from the speaker in order to reflect the incoming sound waves from the speaker. These reflected waves will then interfere with the incident waves, and when the incident and reflected waves are (1) (2)
in phase, a standing waves is created with nodes and antinodes (due to destructive and constructive wave interference) occurring between the plate and the microphone. Measuring the distance between respective nodes and antinodes as a function of reflector position can be used to obtain the wavelengths of the sound waves. The position of reflector and the signal of the microphone are collected via the myDAQ software and the oscilloscope.

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