Wave equation and graphing wave instance

In summary, the wave equation is a mathematical formula used in various fields of science to describe the behavior of waves. It is typically graphed as a sine or cosine function and the different components represent characteristics of the wave. There are two types of waves, transverse and longitudinal, which differ in the direction of particle movement. The wave equation is dependent on the properties of the medium and can be applied to all types of waves, although the specific form may vary.
  • #1
sapiental
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If the end of a string is given a single shake, a wave pulse
propagates down the string. A particular wave pulse is described by the function

y(x,t) = (A^3/(A^2 + (x - vt)^2))


where A = 1.00 cm, and v = 20.0 m/s.
a) Sketch the pulse as a function of x at t = 0. How far along the string does the pulse
extend?
b) Sketch the pulse as a function of x at t = 0.001 s.
c) At the point x = 4.50 cm, at what time t is the displacement maximum?
d) At which two times is the displacement at x = 4.50 cm equal to half its maximum value?
e) Show that y(x, t) satisfies the wave equation.


a) y(x, t = 0) = (A^3/(A^2 + (x - v(0))^2))

i used the command plot(x,y) I let x = 0 through 1 (this is MATLAB a simple graphing program)

plot(x, (.01m^3/(.01m^2 + (x)^2)

(Please see my attached graph a)

b) y(x, t = .001) = (A^3/(A^2 + (x - vt)^2))

Please see my attached graph1

c) y(x = .045m, t) = (A^3/(A^2 + (x - vt)^2))

I want to graph this as a function of t( t being the x axis) what range should I use for t ( i.e 1s, 2s..)?

My attempt is to let t = 0:.1:1 making the x-axis values (0s, .1s, .2s, ... 1s)

so to plot t vs (A^3/(A^2 + (x - vt)^2))

d) Can someone give me an iedea of how to graph the equation to get the actual wave on mt graph?

e) I think I have to take the partial derivative of either the wave equation or this particular pulse equation but I'm very bad at calc and havn't covered partial derivatives. If somebody could do this out for me step by step I would really learn alot.
 

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  • #2



a) The pulse extends along the string until it reaches the end, so it extends infinitely in the x direction. However, its amplitude decreases as it propagates down the string.

b) At t = 0.001 s, the pulse will have moved a distance of 20 cm (20 m/s * 0.001 s) along the string. So the pulse will be centered at x = 20 cm, and its amplitude will be smaller compared to the pulse at t = 0.

c) To find the time at which the displacement is maximum at x = 4.50 cm, we need to set x = 4.50 cm and solve for t in the equation y(x, t) = (A^3/(A^2 + (x - vt)^2)) = A^3/(A^2 + (4.50 cm - 20 m/s * t)^2). This gives t = 0.225 s.

d) We can find the times at which the displacement at x = 4.50 cm is half its maximum value by setting y(x, t) = 0.5 * A^3/(A^2 + (4.50 cm - 20 m/s * t)^2) and solving for t. This gives two solutions: t = 0.1125 s and t = 0.3375 s.

e) To show that y(x, t) satisfies the wave equation, we need to take the partial derivatives of y with respect to x and t and then substitute them into the wave equation, which is ∂^2y/∂x^2 = (1/v^2) * ∂^2y/∂t^2. This will show that the second derivatives are equal, thus proving that y(x, t) satisfies the wave equation.
 

FAQ: Wave equation and graphing wave instance

1. What is the wave equation and how is it used in science?

The wave equation is a mathematical formula that describes the behavior of waves. It is used in various fields of science, such as physics, engineering, and oceanography, to understand and predict the motion and properties of waves.

2. How is the wave equation graphed and what do the different components represent?

The wave equation is typically graphed as a sine or cosine function, with the horizontal axis representing time and the vertical axis representing displacement or amplitude. The different components, such as wavelength, frequency, and amplitude, represent different characteristics of the wave.

3. What is the difference between a transverse wave and a longitudinal wave?

A transverse wave is a wave in which the particles of the medium move perpendicular to the direction of the wave, while a longitudinal wave is a wave in which the particles of the medium move parallel to the direction of the wave.

4. How does the wave equation relate to the properties of a medium?

The wave equation is dependent on the properties of the medium the wave is traveling through, such as density and elasticity. These properties affect the speed at which the wave travels and the behavior of the wave as it interacts with the medium.

5. Can the wave equation be applied to all types of waves?

Yes, the wave equation can be applied to all types of waves, including electromagnetic waves, sound waves, and water waves. However, the specific form of the equation may vary depending on the properties and behavior of the specific type of wave being studied.

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