advertisement

PY1006 Study Problems – Week 8 Traveling waves, Standing waves, Resonance (1) The equation of a transverse wave traveling along a wire is y = 6.0 sin(0.020πx + 4.0πt) where x and y are in centimeters and t is in seconds. Determine the (a) amplitude, (b) wavelength, (c) frequency, (d) speed and (e) direction of propagation of the wave. (f) What is the transverse displacement at x = 3.5 cm when t = 0.26 s? (g) Find an expression for the transverse speed u of the string; what is the maximum transverse speed umax ? (Hint: the transverse speed at some point x will be given by u = dy/dt, treating x as a constant) Suppose the wire has a radius of 0.455 mm and is made of copper (ρ = 8.96 g/cm3 ). What are the (h) linear density of the wire and the tension in the wire? Answers: (a) A = 6.0 cm; (b) λ = 100 cm; (c) f = 2.0 Hz; (d) v = 200 cm/s; (e) the wave is traveling in the negative x direction; (f) −2.03 cm; (g) u = 24π cos(0.020πx + 4.0πt), umax = 24π cm/s = 75.4 cm/s;(h) 0.0583 g/cm, 0.0233 N (2) Two sinusoidal waves with the same period and with amplitudes A1 = 6.0 mm and A2 = 7.5 mm travel in the same direction. They produce a resultant (net) wave with an amplitude of Anet = 12.0 mm. The phase constant of wave 1 is zero and the phase constant of wave 2 is an unknown angle φ between 0 and 90◦ . (a) Draw a phasor diagram showing the two waves and the resultant wave (for convenience, draw wave 1 along the horizontal axis). (b) Find the value of the phase constant φ of wave 2. Answer: (a) this will be similar to the phasor diagram we drew for the example we solved in lecture; (b) φ = 54.9◦ (3) A string is stretched between fixed supports separated by 75.0 cm. It is found that two successive frequencies, fn and fn+1 , are equal to 420 Hz and 315 Hz. (a) Find the values of n and n + 1 that correspond to these frequencies. (b) Find the lowest resonance frequency. To what wavelength does this correspond? (c) Find the speed of the waves on the string. Answer: (a) n = 3, n + 1 = 4; (b) 105 Hz, λ = 1.50 m; (c) 158 m/s