# 16.2 Mathematics of waves  (Page 3/11)

 Page 3 / 11

## Characteristics of a traveling wave on a string

A transverse wave on a taut string is modeled with the wave function

$y\left(x,t\right)=A\phantom{\rule{0.2em}{0ex}}\text{sin}\left(kx-wt\right)=0.2\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(6.28\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1}x-1.57\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}t\right).$

Find the amplitude, wavelength, period, and speed of the wave.

## Strategy

All these characteristics of the wave can be found from the constants included in the equation or from simple combinations of these constants.

## Solution

1. The amplitude, wave number, and angular frequency can be read directly from the wave equation:
$y\left(x,t\right)=A\phantom{\rule{0.2em}{0ex}}\text{sin}\left(kx-wt\right)=0.2\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(6.28\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1}x-1.57\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}t\right).$
$\left(A=0.2\phantom{\rule{0.2em}{0ex}}\text{m;}\phantom{\rule{0.2em}{0ex}}k=6.28\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1};\phantom{\rule{0.2em}{0ex}}\omega =1.57\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}\right)$
2. The wave number can be used to find the wavelength:
$\begin{array}{}\\ k=\frac{2\pi }{\lambda }.\hfill \\ \lambda =\frac{2\pi }{k}=\frac{2\pi }{6.28\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1}}=1.0\phantom{\rule{0.2em}{0ex}}\text{m}.\hfill \end{array}$
3. The period of the wave can be found using the angular frequency:
$\begin{array}{}\\ \\ \omega =\frac{2\pi }{T}.\hfill \\ T=\frac{2\pi }{\omega }=\frac{2\pi }{1.57\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}}=4\phantom{\rule{0.2em}{0ex}}\text{s}.\hfill \end{array}$
4. The speed of the wave can be found using the wave number and the angular frequency. The direction of the wave can be determined by considering the sign of $kx\mp \omega t$ : A negative sign suggests that the wave is moving in the positive x -direction:
$|v|=\frac{\omega }{k}=\frac{1.57\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}}{6.28\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1}}=0.25\phantom{\rule{0.2em}{0ex}}\text{m/s}.$

## Significance

All of the characteristics of the wave are contained in the wave function. Note that the wave speed is the speed of the wave in the direction parallel to the motion of the wave. Plotting the height of the medium y versus the position x for two times $t=0.00\phantom{\rule{0.2em}{0ex}}\text{s}$ and $t=0.80\phantom{\rule{0.2em}{0ex}}\text{s}$ can provide a graphical visualization of the wave ( [link] ). A graph of height of the wave y as a function of position x for snapshots of the wave at two times. The dotted line represents the wave at time t = 0.00 s and the solid line represents the wave at t = 0.80 s . Since the wave velocity is constant, the distance the wave travels is the wave velocity times the time interval. The black dots indicate the points used to measure the displacement of the wave. The medium moves up and down, whereas the wave moves to the right.

There is a second velocity to the motion. In this example, the wave is transverse, moving horizontally as the medium oscillates up and down perpendicular to the direction of motion. The graph in [link] shows the motion of the medium at point $x=0.60\phantom{\rule{0.2em}{0ex}}\text{m}$ as a function of time. Notice that the medium of the wave oscillates up and down between $y=+0.20\phantom{\rule{0.2em}{0ex}}\text{m}$ and $y=-0.20\phantom{\rule{0.2em}{0ex}}\text{m}$ every period of 4.0 seconds. A graph of height of the wave y as a function of time t for the position x = 0.6 m . The medium oscillates between y = + 0.20 m and y = −0.20 m every period. The period represented picks two convenient points in the oscillations to measure the period. The period can be measured between any two adjacent points with the same amplitude and the same velocity, ( ∂ y / ∂ t ) . The velocity can be found by looking at the slope tangent to the point on a y -versus- t plot. Notice that at times t = 3.00 s and t = 7.00 s , the heights and the velocities are the same and the period of the oscillation is 4.00 s.

Check Your Understanding The wave function above is derived using a sine function. Can a cosine function be used instead?

Yes, a cosine function is equal to a sine function with a phase shift, and either function can be used in a wave function. Which function is more convenient to use depends on the initial conditions. In [link] , the wave has an initial height of $\text{y}\left(0.00,0.00\right)=0$ and then the wave height increases to the maximum height at the crest. If the initial height at the initial time was equal to the amplitude of the wave $\text{y}\left(0.00,0.00\right)=\text{+}A,$ then it might be more convenient to model the wave with a cosine function.

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