Question

a) A 1 meter long guitar string of linear mass density 2g/m3 is put under tension until it resonates with a fundamental frequency of 440 Hz. Determine the tension that produces this fundamental frequency. Also determine the other of the first four harmonic frequencies and draw diagrams illustrating what each of these oscillations looks like on the string.

b) This string will produce sound waves in the air, determine the wavelength of the sound waves.

c) Suppose you had two of these strings, placed 25 meters apart. How far from the midpoint between them must you stand to experience the first region of maximum destructive interference? How far would you have to travel to reach the second region of maximum deconstructive interference?

d) If you were to keep walking towards one of the strings, you would experience these regions of maximum destructive and constructive interference as pulses in the amplitude of the sound. If you were to walk at a speed of 1.75 m/s, at what frequency would these pulses occur? This is to say, how many regions of deconstructive interference do you pass per second?

e) To look at the problem another way, if you were to walk directly towards one of the strings and away from the other, both will undergo a Doppler shift. Calculate the Doppler shifted frequency of the string you’re walking towards, ft as well as the frequency of the string you’re walking away from, fa .

f) Given that you’re hearing these two slightly different frequencies at the same time, calculate the beat frequency that you hear from the slight difference between the two frequencies. What do you notice about this “beat frequency” between the Doppler shifted frequencies and the pulsing frequency we observed from passing through the standing wave pattern in part D?

Answer 1