Standing Waves

Consider a string of total length l and mass m the

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Unformatted text preview: string. Consider a string of total length L and mass m. The ratio of the mass to length is called the linear density μ of the string: (1) Linear density characterizes the type of strings we are using. A fat string has a larger value μ than a skinny string made of the same material. Similarly, a steel wire has a larger value of μ than a plastic string of the same diameter. The units tell us that the numerical value μ is the mass in kg of a 1 m long section of string: that is, the linear density is the kilogram per meter of string, but because μ is a ratio, we can apply it to a segment of string of any length. Thus the mass of any length L of string is (2) 2.4 Wavelength and Frequency To completely describe a wave on a string (and the motion of any element along its length), we need a function that gives the shape of the wave. This means that we need a relationship on the form , (3) in which y is the transverse displacement of any string element as a function h of the time t and the position x of the element along the string. In general, a sinusoidal shape can be described with h being either a sine or cosine function: both give the same general shape for the wave. Now imagine a sinusoidal wave traveling in the positive direction of an x axis. As the wave sweeps through succeeding elements (that is, very short sections) of the string, the elements oscillate parallel to the y axis. At time t, the displacement of the element located at the position x is given by 2 , sin (4) Because this equation is written in terms of position x, it can be used to find the displacements of all the elements of the string as a function of time. Thus, it can tell us the shape of the wave at any given time and how that shape changes as the wave moves along the string. 2.4.1 Amplitude and Phase The amplitude ym of a wave is the magnitude of the maximum displacement of the elements from their equilibrium positions as the wave passes through them. (The subscript m stands for maximum.) Because ym is a magnitude, it is always a positive quantity, even if it is measured downward instead of upward. The phase of the wave is the argument kx-ωt of the sine in equation (4). As the wave sweeps through a string element at a particular position x, the phase changes linearly with time t. This means that the sine also changes, oscillating between +1 and -1. Its extreme positive value (+1) corresponds to a peak of wave moving through the element: at that instant the value of y at position x is ym. Its extreme negative value (-1) corresponds to a valley of the wave moving through the element: at that instant the value of y at position...
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