mathematical description of a wave
Now imagine drawing that same shape while your friend slowly pulled the piece of paper out from under your pencil – the line you would have drawn traces out the shape of a wave. A. Find the first time t1>0 when this is true.Express t1 in terms of ω, k, and necessary constants. The dotted line corresponds to a snapshot of the wave one second later, at \(t=1\text{s}\), when the wave has moved to the right by a distance \(vt=1\text{m}\). The ancient astronomer Hipparchus discovered that knowing one angle measure of a right triangle allows you to calculate. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. The equation above is written in terms of the wavelength, \(\lambda\), and period, \(T\), of the wave. k=\frac{2 \pi}{\lambda} \\ 5. Find out more about how we use your information in our Privacy Policy and Cookie Policy. Waves are familiar to us from the ocean, the study of sound, earthquakes, and other natural phenomenon. Since waves always are moving, one more important term to describe a wave is the time it takes for one wavelength to pass a specific point in space.

Thanks for watching. Taking the ratio and using the equation v = \(\frac{\omega}{k}\) yields the linear wave equation (also known simply as the wave equation or the equation of a vibrating string), \[\begin{split} \frac{\frac{\partial^{2} y(x,t)}{\partial t^{2}}}{\frac{\partial^{2} y(x,t)}{\partial x^{2}}} & = \frac{-A \omega^{2} \sin (kx - \omega t + \phi)}{-Ak^{2} \sin (kx - \omega t + \phi)} \\ & = \frac{\omega^{2}}{k^{2}} = v^{2}, \end{split}\], \[\frac{\partial^{2} y(x,t)}{\partial x^{2}} = \frac{1}{v^{2}} \frac{\partial^{2} y(x,t)}{\partial t^{2}} \ldotp \label{16.6}\]. What Are The Frequency And The Wavelength Of Jo This Wave? The module presents Cartesian coordinate (x, y) graphing, and shows how the sine function is used to plot a wave on a graph. 10 A. Similarly, any system, where the displacement of a particle as a function of position and time, \(D(x,t)\), satisfies the following equation: \[\frac{\partial ^{2}D}{\partial x^{2}}=\frac{1}{v^{2}}\frac{\partial ^{2}D}{\partial t^{2}}\]. In essence, what the modifier A does is increase (or amplify) the result of the function Sin(x), thus leading to larger resulting y values. Practice: The function for some transverse wave is = (0.5 ) sin [(0.8 −1 ) − 2(50 ) + 3 ]. None of the numbers change A omega K phi they're all the same in these equations as they would be in the irregular equation for the wave right where we have Y is A sine K X minus omega T plus phi it's all the same variables alright now cosine can get as big as positive 1 and as small as negative 1 so obviously the maximum transverse speed is just A omega likewise sine can get as big as positive 1 and as small as negative 1 so the largest transverse acceleration is just A omega squared just those coefficients of the trig functions and lastly we want to do one more example. Sophisticated mathematical equations and properties have been developed over the centuries to describe them.

The particles of the medium, or the mass elements, oscillate in simple harmonic motion for a mechanical wave.

where \(\phi=0\) corresponds to the displacement being zero at \(x=0\) and \(t=0\). As we saw earlier, the basic formula representing the sine function is: In this formula, y is the value on the y-axis obtained when one carries out the function Sin(x) for points on the x-axis. Use v = 340 m/s for the speed of sound and be sure to express your answer in degrees. To construct our model of the wave using a periodic function, consider the ratio of the angle and the position, \[\begin{align*} \dfrac{\theta}{x} &=\frac{2 \pi}{\lambda}, \\[4pt] \theta &=\frac{2 \pi}{\lambda} x.

For similar reasons, the initial phase is added to the wave function.

B. Time is measured in seconds and lengths are in meters. 2.8/2πD. Now on a transverse wave we have a transverse component in the velocity and because the wave is going up and going down going up and going down going up and going down obviously the velocity is changing that transverse velocity right initially it's going up so its a positive velocity then its coming down so it is a negative velocity since that's changing there must be a transverse acceleration. Adopted or used LibreTexts for your course? (B) Which of the expressions given is a mathematical expression for a wave of the same amplitude that is traveling in the opposite direction? What is the wavelength, in meters? A transverse harmonic wave travels on a rope according to the following expression:y(x,t) = 0.12sin(2.1x + 17.6t)1) What is the amplitude of the wave?2) What is the frequency of oscillation of the wave? About 55 degrees That displacement could correspond to the longitudinal displacement of a particle in a longitudinal wave. The phase angle tells us what that initial displacement is, the phase angle was determined by the initial displacement of the wave a wave that begins with no displacement has a phase angle of 0 degrees which is a pure sine wave. Imagine drawing a circle on a piece of paper. Question: (12) Mathematical Description Of A Wave (24pts) MA Below Is A History Graph Of A Wave At X 0 M. The (em) Wave Is Traveling To The Right At A Speed Of 2 M/s. The particle will move halfway to the crest due to reinforcement. The solid black line corresponds to a snapshot of the wave at time \(t=0\). 3. 3. 12.5 Hz. Mathematicians use the sine function (Sin) to express the shape of a wave.

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