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Equation 16.6 is the linear wave equation, which is one of the most important equations in physics and engineering. We derived it here for a transverse wave, but it is equally important when investigating longitudinal waves.
- Introduction
Figure 1.1 This image might be showing any number of things....
- Introduction
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 Figure, the y -position of the string between x = 0 and x = λ can be modeled as a sine function.
Use the trigonometric identities cos u + cos v = 2 cos\(\left(\dfrac{u − v}{2}\right)\) cos \(\left(\dfrac{u + v}{2}\right)\) and cos(−\(\theta\)) = cos(\(\theta\)) to find a wave equation for the wave resulting from the superposition of the two waves. Does the resulting wave function come as a surprise to you?
Examples include gamma rays, X-rays, ultraviolet waves, visible light, infrared waves, microwaves, and radio waves. Electromagnetic waves can travel through a vacuum at the speed of light, v = c = 2.99792458 x 10 8 m/s.
The general equation describing a wave is: y(x,t) = A sin(kx - wt) Let's say that for a particular wave on a string the equation is: y(x,t) = (0.9 cm) sin[(1.2 m-1)x - (5.0 s-1)t] (a) Determine the wave's amplitude, wavelength, and frequency. (b) Determine the speed of the wave.
A wave is modeled at time $$ t=0.00\,\text{s} $$ with a wave function that depends on position. The equation is $$ y(x)=(0.30\,\text{m})\text{sin}(6.28\,{\text{m}}^{-1}x)$$. The wave travels a distance of 4.00 meters in 0.50 s in the positive x-direction. Write an equation for the wave as a function of position and time.
In the next section, we show in more precise mathematical terms how Maxwell’s equations lead to the prediction of electromagnetic waves that can travel through space without a material medium, implying a speed of electromagnetic waves equal to the speed of light.
