3/26/2014 Master Ithttp://www.webassign.net/v4cgi/questions/tutorial_popup.tpl 1/3Master itThe wave function for a traveling wave on a taut string is (in SI units)(a) What are the speed and direction of travel of the wave?(b) What is the vertical position of an element of the string at t = 0, x = 0.131 m?(c) What is the wavelength of the wave?(d) What is the frequency of the wave?(e) What is the maximum transverse speed of an element of the string?Part 1 of 6 ConceptualizeAs a function of two independent variables, the wave function represents a twodimensional graph varying intime. The position in space of a point on the string at one instant in time is an example of the informationcontained in the wave function. The constants for amplitude, wave speed, wavelength, and frequency in thegeneral form of the wave equation completely describe the wave.Part 2 of 6 CategorizeWe use the traveling wave model. We compare the given equation with the general form of the equation Part 3 of 6 AnalyzeWe note that sin(θ) = −sin(−θ) = sin(−θ + π) so the given wave function can be written asThus, we find that the wave constant isk = 2 π rad/m,and the angular speed isω = 14 π rad/s.We will use the wave function to determine the answers to parts (a)through (e).(a) For constant phase, x must increase as t increases, so the wave travels in the positive x direction.Comparing the specific form to the general form, we haveThe speed and direction of the wave are both specified by the vector wave velocity.= fλ = y(x,t) = 0.395 sin 14πt − 2πx + π4y(x,t) = 0.395 sin (14πt − 2πx + π/4)y = A sin (kx − ωt + ).y(x,t) = (0.395) sin (−14πt + 2πx + π + π/4).y(x,t) = 0.395 sin (14πt − 2πx + π/4)14πt − 2πx + = kx − ωt + .π4ωk3/26/2014 Master Ithttp://www.webassign.net/v4cgi/questions/tutorial_popup.tpl 2/3= = 7 m/sPart 4 of 6 Analyze(b) Substituting t = 0, x = 0.131 m, we havey= = = = 0.0149 m = 1.49 cm.Note that when we take the sine of a quantity with no units, the quantity is not in degrees, it is in radians. Sowe must make sure that the calculator is properly set for entering values in radians.Part 5 of 6 Analyze(c) The wavelength isλ= = = 1 m,(d) and the frequency isf= = = 7 Hz.Part 6 of 6 Analyze(e) The particle speed isThe maximum speed occurs when the cosine factor is 1, so we have the following.vy,max= = 17.4 m/s 14 π rad/s 2 π rad/m0.395 m sin − 0.262 π + 0.25 π0.395 m sin − 0.012 π0.395 m 0.0377 2π radk2π rad 2 π rad/mω2π rad 14 π rad/s2π radvy = = 0.395 m 14 π rad/s cos 14 πt − 2 πx + π/4 .∂y∂t 14 π rad/s 0.395 m 3/26/2014 Master Ithttp://www.webassign.net/v4cgi/questions/tutorial_popup.tpl
View Full Document
Unlocking...