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2. The dispersion relation for oscillation of a string (It has a length L with linear mass density μ and under tens...
5) A string with mass density μ and tension T is fixed at the points z=0...
5) A string with mass density μ and tension T is fixed at the points z=0 and z-L, and supports standing waves of the form f (z, t) = A sin kz coswt. Since these standing waves are produced by counterpropagating travelling waves (due to all of the multiple reflections), the relationship ω-ku still applies, where u is the velocity of the travelling waves. a) In terms of L, what are the allowed, discrete values of the wavenumber kn which...
A certain string has a linear mass density of 0.25 kg/m and is stretched with a...
A certain string has a linear mass density of 0.25 kg/m and is stretched with a tension of 25 N. One end is given a sinusoidal motion with frequency 5 Hz and amplitude 0.01m. At time t=0, the end has zero displacement and is moving in the positive y-direction. a) Find the position of the point at x= 0.25 m, and t= 0.1 s b )Find the transverse velocity of the point, x=0.25 m at time t=0.1s. c) Find the...
The high E-string of a steel string guitar is about L = 0.7m long. It has a mass per unit length of p = 5.3 *10^-4 kg/m....
The high E-string of a steel string guitar is about L = 0.7m
long. It has a mass per unit length of p = 5.3 *10^-4 kg/m. Find
the tension T required to properly tune the string, given that a
high E has a frequency of f = 1318.51 hertz
(a) The high E-string of a steel string guitar is about L 0.7 m long. It has a mass per unit length of ρ= 5.3 x 10-4 kg/m. Find the...
u(x, t) represents the vertical displacement of a string of length L = 16 with wave...
u(x, t) represents the vertical displacement of a string of length L = 16 with wave equation 25uxx = uft at position x along the string and at time t Find u(x, t) if a. the initial velocity of the string is 0 and the rightmost position b. the initial velocity is a constant 5 and the vertical displacement is 0. c. the initial velocity is a constant 5 and the rightmost position is held at a vertical displacement of...
Normal modes and resonance frequencies Normal Modes and Resonance Frequencies Learning Goal: To understand the concept...
Normal modes and resonance frequencies
Normal Modes and Resonance Frequencies Learning Goal: To understand the concept of normal modes of oscillation and to derive some properties of normal modes of waves on a string A normal mode of a closed system is an oscillation of the system in which all parts oscillate at a single frequency In general there are an infinite number of such modes each one with a distinctive frequency f, and associated pattern of oscillation. Consider an...
A uniform string of length L = 1 is described by the one-dimensional wave equation au...
A uniform string of length L = 1 is described by the one-dimensional wave equation au dt2 dx where u(x,t) is the displacement. At the initial moment t = 0, the displacement is u(x,0) = sin(Tt x), and the velocity of the string is zero. (Here n = 3.14159.) Find the displacement of the string at point x = 1/2 at time t = 2.7.
segments over the length L of the string, where the length of each vibrating segment equals...
segments over the length L of the string, where the length of each vibrating segment equals one-half wavelength. Use this fact to show that the fr of the allowed standing waves on this string are given by fn-nfi, where n 1,2,3, 4,5,... and fi is the fundamental frequency. In other words, derive an expression relating the nth harmonic to the fundamental frequency. Yo may use the fact that the wave velocity is the same for all modes. 1. For a...
(The wave equation) Consider a string with fixed zero ends of length L with speed parameter...
(The wave equation) Consider a string with fixed zero ends of length L with speed parameter c, with initial position -X u(x,0) = x € (0, L/2] c [L/2, L] C L and zero initial velocity. (a) Find the normal modes of the solution and specify the spatial and temporal frequencies for each. (You do not need to derive the general solution to the wave equation with fixed ends.) (b) Describe how the tension Th, density p and length L...
Consider an elastic string of length L whose ends are held fixed. The string is set in motion fro...
Answer needed in form summation from n=1 to infinity:
Consider an elastic string of length L whose ends are held fixed. The string is set in motion from its equilibrium position with an initial velocity ut(x, 0) = g(x). Let L-12 and a = 1 in parts (b) and (c). (A computer algebra system is recommended.) 8x 2 (a) Find the displacement u(x, t) for the given g(x). (Use a to represent an arbitrary constant.)
Consider an elastic string of...
12. A longitudinal standing wave can be created in a long, thin aluminum rod by stroking...
12. A longitudinal standing wave can be created in a long, thin aluminum rod by stroking the rod with very dry fingers. This is often done as a physics demonstration, creating a high-pitched, very annoying whine. From a wave perspective, the standing wave is equivalent to a sound standing wave in an open-open tube. In particular, both ends of the rod are anti-nodes. What is the fundamental frequency of a 2.50 m -long aluminum rod? The speed of sound in...
5) A string with mass density μ and tension T is fixed at the points z=0 and z-L, and supports standing waves of the form f (z, t) = A sin kz coswt. Since these standing waves are produced by counterpropagating travelling waves (due to all of the multiple reflections), the relationship ω-ku still applies, where u is the velocity of the travelling waves. a) In terms of L, what are the allowed, discrete values of the wavenumber kn which...
The high E-string of a steel string guitar is about L = 0.7m
long. It has a mass per unit length of p = 5.3 *10^-4 kg/m. Find
the tension T required to properly tune the string, given that a
high E has a frequency of f = 1318.51 hertz
(a) The high E-string of a steel string guitar is about L 0.7 m long. It has a mass per unit length of ρ= 5.3 x 10-4 kg/m. Find the...
u(x, t) represents the vertical displacement of a string of length L = 16 with wave equation 25uxx = uft at position x along the string and at time t Find u(x, t) if a. the initial velocity of the string is 0 and the rightmost position b. the initial velocity is a constant 5 and the vertical displacement is 0. c. the initial velocity is a constant 5 and the rightmost position is held at a vertical displacement of...
Normal modes and resonance frequencies
Normal Modes and Resonance Frequencies Learning Goal: To understand the concept of normal modes of oscillation and to derive some properties of normal modes of waves on a string A normal mode of a closed system is an oscillation of the system in which all parts oscillate at a single frequency In general there are an infinite number of such modes each one with a distinctive frequency f, and associated pattern of oscillation. Consider an...
A uniform string of length L = 1 is described by the one-dimensional wave equation au dt2 dx where u(x,t) is the displacement. At the initial moment t = 0, the displacement is u(x,0) = sin(Tt x), and the velocity of the string is zero. (Here n = 3.14159.) Find the displacement of the string at point x = 1/2 at time t = 2.7.
segments over the length L of the string, where the length of each vibrating segment equals one-half wavelength. Use this fact to show that the fr of the allowed standing waves on this string are given by fn-nfi, where n 1,2,3, 4,5,... and fi is the fundamental frequency. In other words, derive an expression relating the nth harmonic to the fundamental frequency. Yo may use the fact that the wave velocity is the same for all modes. 1. For a...
(The wave equation) Consider a string with fixed zero ends of length L with speed parameter c, with initial position -X u(x,0) = x € (0, L/2] c [L/2, L] C L and zero initial velocity. (a) Find the normal modes of the solution and specify the spatial and temporal frequencies for each. (You do not need to derive the general solution to the wave equation with fixed ends.) (b) Describe how the tension Th, density p and length L...
Answer needed in form summation from n=1 to infinity:
Consider an elastic string of length L whose ends are held fixed. The string is set in motion from its equilibrium position with an initial velocity ut(x, 0) = g(x). Let L-12 and a = 1 in parts (b) and (c). (A computer algebra system is recommended.) 8x 2 (a) Find the displacement u(x, t) for the given g(x). (Use a to represent an arbitrary constant.)
Consider an elastic string of...
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