Determine the direction and speed of the wave from a given wave equation

In summary, to determine the direction of propagation and speed of a wave described by the equation \psi(x,t) = A e^{-a(bx+ct)^{2}}, one can rearrange the expression in the exponent and read off the speed as v = ±c/b. However, the direction of propagation (positive or negative x direction) depends on the signs of b and c. Another method involves setting bx + ct = C and taking the derivative with respect to time, yielding v = -c/b as the unique solution. Both methods are valid approaches.
  • #1
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Homework Statement


Given an equation for a wave [itex]\psi(x,t) = A e^{-a(bx+ct)^{2}}[/itex] determine the direction of its propagation if you know [itex]\psi(x,t) = f(x \pm vt)[/itex] and use this to find its speed.

Homework Equations



The Attempt at a Solution


I figured I would just rearrange the expression in the exponent, so as to yield [itex]x + \frac{c}{b}t[/itex], and then just read off [itex]v = \pm\frac{c}{b}[/itex]. However, if we don't know whether [itex]\psi(x,t) = f(x + vt)[/itex] or [itex]\psi(x,t) = f(x - vt)[/itex], can we really determine the direction of its propagation?

Also, I found somewhere the answer to this question would uniquely be [itex]v = -\frac{c}{b}[/itex] by letting [itex]bx + ct = C[/itex], and then after solving for x, [itex]x = \frac{C}{b} - \frac{c}{b}[/itex], taking the derivative with respect to time, yielding the above unique solution with the minus sign. Is this the proper way of doing things instead of just rearranging the expression like I did?

Thanks!
 
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  • #2
Either way is fair game!
 
  • #3
Ryker said:
I figured I would just rearrange the expression in the exponent, so as to yield [itex]x + \frac{c}{b}t[/itex], and then just read off [itex]v = \pm\frac{c}{b}[/itex]. However, if we don't know whether [itex]\psi(x,t) = f(x + vt)[/itex] or [itex]\psi(x,t) = f(x - vt)[/itex], can we really determine the direction of its propagation?
The only difference between f(x+vt) and f(x-vt) is the direction that the wave propagates. f(x-vt) represents a wave moving in the +x direction, and f(x+vt) represents a wave moving in the -x direction, where v>0 is the speed of the wave. Whether v=±c/b depends on the signs of b and c.
 
  • #4
klawlor419 said:
Either way is fair game!

vela said:
The only difference between f(x+vt) and f(x-vt) is the direction that the wave propagates. f(x-vt) represents a wave moving in the +x direction, and f(x+vt) represents a wave moving in the -x direction, where v>0 is the speed of the wave. Whether v=±c/b depends on the signs of b and c.
Thanks for the replies! After thinking about it some more, that's what I figured, as well, as I just couldn't justify why there would necessarily be a minus sign.
 
  • #5


I would say that your approach of rearranging the expression in the exponent to yield x + \frac{c}{b}t and then reading off v = \pm\frac{c}{b} is a valid method for determining the direction and speed of the wave. However, it is important to note that this method assumes that \psi(x,t) = f(x \pm vt) is the correct form of the wave equation. If we are not sure about the sign in front of vt, then we cannot determine the direction of propagation with certainty.

The alternative method you mentioned, using the equation bx + ct = C and taking the derivative with respect to time, is also a valid approach. This method is based on the fact that the wave equation can be written as a function of the form f(x \pm vt), where v is the speed and the sign in front of vt determines the direction of propagation. Therefore, by finding the derivative of this function with respect to time, we can determine the speed and direction of the wave.

Both methods are valid and can be used depending on the situation. It is important to carefully consider the form of the wave equation and the given information before deciding on the appropriate method to use.
 

1. What is a wave equation?

A wave equation is a mathematical formula that describes the motion of a wave over time and space. It typically includes variables such as amplitude, wavelength, and frequency.

2. How do you determine the direction of a wave from a given wave equation?

The direction of a wave can be determined by looking at the sign of the coefficient in front of the variable representing the direction, such as x or y. A positive coefficient indicates a wave moving in the positive direction, while a negative coefficient indicates a wave moving in the negative direction.

3. How do you determine the speed of a wave from a given wave equation?

The speed of a wave can be determined by calculating the square root of the ratio of the coefficients in front of the variables representing time and space, typically denoted as c. This is known as the wave speed or propagation velocity.

4. Can the direction and speed of a wave change over time?

Yes, the direction and speed of a wave can change over time due to various factors such as changes in the medium or interactions with other waves.

5. How is the direction and speed of a wave related to its frequency and wavelength?

The direction and speed of a wave are directly related to its frequency and wavelength. As frequency increases, the wave will typically travel faster, while a longer wavelength generally results in a slower speed. The direction of the wave is also influenced by the wavelength, with longer wavelengths often causing the wave to bend or diffract more than shorter wavelengths.

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