Waves: Analysis/Speed of Disturbance

[Desh627]

Homework Statement

A disturbance can be written as: y(x,t)= (e^-(x/b)2e^2xt/be^-t2)^a

This disturbance is:
(A) not a traveling wave
(B) a traveling wave with speed v = a
(C) a traveling wave with speed v = a/b
(D) a traveling wave with speed v = b

Homework Equations

y(x,0) = f(tx)
y(x,t) = f'(x) = f(x-vt)
$$\partial$$²f = 1$$/$$v² $$\partial$$²y$$/$$dt²
dz²

The Attempt at a Solution

We haven't really dealt with waves using e, and I'm not quite sure exactly how to attack this problem: plug it into the wave equation, trying to simplify to equate what's in the parenthesis to compare with x-vt, etc., so I'm a touch lost overall in trying to attack this problem.

 

[anathema]

your first step should be to analyze the given disturbance and try to understand its properties and behavior. In this case, the disturbance is given by a mathematical equation in the form of a Gaussian function. This function has a characteristic bell-shaped curve and is often used to describe physical phenomena such as light, sound, and heat.

To determine if this disturbance is a traveling wave, we need to look at the equation and see if it contains any terms that involve both x and t. If there are such terms, then the disturbance is a traveling wave.

In this case, we can see that the equation contains both x and t in the exponent of the first term, e-(x/b)2, and in the second term, e2xt/be-t2. This means that the disturbance is indeed a traveling wave.

Next, we need to determine the speed of this wave. The speed of a traveling wave is given by the ratio of the distance traveled by the wave to the time taken to travel that distance. In this case, the distance traveled by the wave is given by the parameter a, and the time taken is given by the parameter b. Therefore, the speed of this wave is v = a/b.

Therefore, the correct answer is (C) a traveling wave with speed v = a/b.

 

[kerimek]

I would approach this problem by first identifying the key components and variables involved. The disturbance is represented by the function y(x,t) and is dependent on both space (x) and time (t). The disturbance also has a coefficient (a) and a parameter (b) in the equation.

Next, I would use the given equations and information to analyze the speed of the disturbance. The first step is to determine if the disturbance is a traveling wave or not. A traveling wave is a disturbance that moves through a medium, maintaining its shape and speed. In this case, the disturbance is a traveling wave because it has a dependence on both space and time.

To calculate the speed of the disturbance, I would use the equation v = λf, where v is the speed, λ is the wavelength, and f is the frequency. However, since the disturbance is not described in terms of wavelength or frequency, I would use the equation v = d/t, where v is the speed, d is the distance traveled, and t is the time it takes to travel that distance.

Using this equation, I would first determine the distance traveled by the disturbance in one unit of time (t=1). This can be done by setting t=1 in the given equation and simplifying to find the distance traveled.

Next, I would determine the speed of the disturbance by dividing the distance traveled by the time taken (t=1). This speed would then be compared to the given options to identify the correct answer.

Overall, the disturbance is a traveling wave with a speed of v = a/b. This can be seen by comparing the given equation to the wave equation, where v = a/b and λ = b. Therefore, the correct answer is (C) a traveling wave with speed v = a/b.