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By the end of this section, you will be able to:
  • Model a wave, moving with a constant wave velocity, with a mathematical expression
  • Calculate the velocity and acceleration of the medium
  • Show how the velocity of the medium differs from the wave velocity (propagation velocity)

In the previous section, we described periodic waves by their characteristics of wavelength, period, amplitude, and wave speed of the wave. Waves can also be described by the motion of the particles of the medium through which the waves move. The position of particles of the medium can be mathematically modeled as wave function     s , which can be used to find the position, velocity, and acceleration of the particles of the medium of the wave at any time.

Pulses

A pulse    can be described as wave consisting of a single disturbance that moves through the medium with a constant amplitude. The pulse moves as a pattern that maintains its shape as it propagates with a constant wave speed. Because the wave speed is constant, the distance the pulse moves in a time Δ t is equal to Δ x = v Δ t ( [link] ).

Figure a shows a pulse wave, a wave with a single crest at time t=0. The distance between the start and end of the wave is labeled lambda. The crest is at y=0. The vertical distance of the crest from the origin is labeled A. The wave propagates towards the right with velocity v. Figure b shows the same wave at time t=t subscript 1. The pulse has moved towards the right. The horizontal distance of the crest from the y axis is labeled delta x equal to v delta t.
The pulse at time t = 0 is centered on x = 0 with amplitude A . The pulse moves as a pattern with a constant shape, with a constant maximum value A . The velocity is constant and the pulse moves a distance Δ x = v Δ t in a time Δ t . The distance traveled is measured with any convenient point on the pulse. In this figure, the crest is used.

Modeling a one-dimensional sinusoidal wave using a wave function

Consider a string kept at a constant tension F T where one end is fixed and the free end is oscillated between y = + A and y = A by a mechanical device at a constant frequency. [link] shows snapshots of the wave at an interval of an eighth of a period, beginning after one period ( t = T ) .

Figure shows different stages of a transverse wave propagating towards the right, taken at intervals of 1 by 8 T. Dots mark points on the wave. These move up and down from – A to +A. A dot that is at the equilibrium position at time t=T, moves to +A at time t=T plus 2 by 8 T. It then moves back to the equilibrium position at time t= T plus 4 by 8 T. It moves to –A at time t=T plus 6 by 8 T and back to the equilibrium position at time t=2T. Similarly, all dots move to their original positions at time t=2T.
Snapshots of a transverse wave moving through a string under tension, beginning at time t = T and taken at intervals of 1 8 T . Colored dots are used to highlight points on the string. Points that are a wavelength apart in the x -direction are highlighted with the same color dots.

Notice that each select point on the string (marked by colored dots) oscillates up and down in simple harmonic motion, between y = + A and y = A , with a period T . The wave on the string is sinusoidal and is translating in the positive x -direction as time progresses.

At this point, it is useful to recall from your study of algebra that if f ( x ) is some function, then f ( x d ) is the same function translated in the positive x -direction by a distance d . The function f ( x + d ) is the same function translated in the negative x -direction by a distance d . We want to define a wave function that will give the y -position of each segment of the string for every position x along the string for every time t .

Looking at the first snapshot in [link] , the y -position of the string between x = 0 and x = λ can be modeled as a sine function. This wave propagates down the string one wavelength in one period, as seen in the last snapshot. The wave therefore moves with a constant wave speed of v = λ / T .

Recall that a sine function is a function of the angle θ , oscillating between + 1 and −1 , and repeating every 2 π radians ( [link] ). However, the y -position of the medium, or the wave function, oscillates between + A and A , and repeats every wavelength λ .

Practice Key Terms 4

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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