Exploring Wave Motion: y(x,t) = 3e-(2x-4t)^2

[Faux Carnival]

Homework Statement

y(x,t) = 3e^{-(2x-4t)^2}

Consider the wave function which represents a transverse pulse that travels on a string along the horizontal x-axis.

a) Find the wave speed
b) Find the velocity of the string at x=0 as a function of time

Homework Equations

The Attempt at a Solution

I think, for b) I should take the derivative of the original wave function with respect to t.
Easy if that's the case.

I have no idea about part a.

 

[merryjman]

http://en.wikipedia.org/wiki/Wave_equation

http://mathworld.wolfram.com/WaveEquation.html

Looking at these pages, you can see that the second partial derivative of the wave function with respect to time is equal to the square of the speed multiplied by the second partial derivative of the wave function with respect to position.

 

[Faux Carnival]

Wow, thanks merry. I completely forgot about the linear wave equation.

And is my solution for part b correct? (Taking the derivative of the function with respect to time to find the string velocity function)

 

[merryjman]

I would say so. At x = 0 that wave function gives Y position as a function of time, so its time derivative would be the rate of change of the Y position.

 

[rwisz]

Hello! It's great to see that you are exploring wave motion and working on this problem. To begin, let's break down the wave function given in the problem. The function y(x,t) represents the displacement of the string at a certain point (x) and time (t). The term 3e represents the amplitude of the wave and the term -(2x-4t)^2 represents the phase of the wave. The phase of the wave determines the position and shape of the wave as it moves along the string.

a) To find the wave speed, we can use the equation v = λf, where v is the wave speed, λ is the wavelength, and f is the frequency. In this case, we can see that the wavelength is given by λ = 2x - 4t. Therefore, the wave speed can be calculated as v = λf = (2x - 4t)f. The frequency of the wave is not given in the problem, but it can be determined by looking at the period of the wave. The period is the time it takes for one complete wave to pass a certain point. In this case, the period is given by T = 2π/ω, where ω is the angular frequency. Since the wave function is in the form of a Gaussian, we can determine the angular frequency as ω = 2πf, where f is the frequency mentioned above. Therefore, the wave speed can be written as v = (2x - 4t)(2πf) = 4πf(x - 2t).

b) To find the velocity of the string at x=0 as a function of time, we can take the derivative of the wave function with respect to t. This will give us the velocity of the string at any given point (x) and time (t). Therefore, we can write the velocity as v(x,t) = ∂y/∂t = 3e-(2x-4t)^2(-8). This can be simplified to v(x,t) = 24e-(2x-4t)^2. Finally, to find the velocity at x=0, we can simply plug in x=0 into the equation, giving us v(0,t) = 24e-(4t)^2. This is the velocity of the string at x=0 as a function of time.

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