Speed of wave from wave equation.

In summary: So the answer is (a).In summary, the function f(x,t)=A{ e }^{ \frac { 2abxt-{ a }^{ 2 }{ x }^{ 2 }-{ b }^{ 2 }{ t }^{ 2 } }{ { c }^{ 2 } } } represents a wave traveling in the positive x direction with a velocity of v = b/a. This can be obtained by expressing the function as a function of x-vt, which gives f(x-vt)=A{ e }^{ -\frac { a }{ c } { (x-\frac { b }{ a } t) }^{ 2 } }. Therefore, the answer to the question
  • #1
back2square1
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Homework Statement


A traveling pulse is given by [itex]f(x,t)=A{ e }^{ \frac { 2abxt-{ a }^{ 2 }{ x }^{ 2 }-{ b }^{ 2 }{ t }^{ 2 } }{ { c }^{ 2 } } }[/itex] where [itex]A, a, b, c[/itex] are positive constants of appropriate dimentions. The speed of pulse is:
a) [itex]\frac { b }{ a } [/itex]
b) [itex]\frac { 2b }{ a } [/itex]
c) [itex]\frac { cb }{ a } [/itex]
d) [itex]\frac { b }{ 2a } [/itex]

Homework Equations


[itex]\tilde { f } (x,t)=\tilde { A } { e }^{ i(kx-\omega t) }[/itex] and
[itex]\omega=kv[/itex]
[itex]\tilde { f }[/itex] is complex wave function.
[itex]\tilde { A }[/itex] is complex amplitude.
[itex]k[/itex] is the wave number.
[itex]\omega[/itex] is angular frequency.
[itex]v[/itex] is the velocity of pulse.

The Attempt at a Solution


The strategy was reduce the equation into the standard wave equation and see which part corresponds with [itex]\omega[/itex] and [itex]k[/itex]. Once we find the expression for [itex]\omega[/itex] and [itex]k[/itex], its not very difficult to find the velocity using the 2nd equation.

The equation doesn't contain [itex]i[/itex]. But reducing the equation, we get something like this,
[itex]f(x,t)={ e }^{ { (i\frac { ax-bt }{ c } ) }^{ 2 } }[/itex]. The only problem is how to get rid of the square? Using logarithms wasn't helpful.
 
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  • #2
back2square1 said:

Homework Statement


A traveling pulse is given by [itex]f(x,t)=A{ e }^{ \frac { 2abxt-{ a }^{ 2 }{ x }^{ 2 }-{ b }^{ 2 }{ t }^{ 2 } }{ { c }^{ 2 } } }[/itex] where [itex]A, a, b, c[/itex] are positive constants of appropriate dimentions. The speed of pulse is:
a) [itex]\frac { b }{ a } [/itex]
b) [itex]\frac { 2b }{ a } [/itex]
c) [itex]\frac { cb }{ a } [/itex]
d) [itex]\frac { b }{ 2a } [/itex]
Your wave function is a function of position and time. What is speed, with respect to these two variables? :tongue:
 
  • #3
back2square1 said:
[itex]\tilde { f } (x,t)=\tilde { A } { e }^{ i(kx-\omega t) }[/itex]
This equation is for a "sinusoidal" or "harmonic" wave that extends from -∞ to ∞.

More generally, any well-behaved function f(x,t) of the form f(x-vt) would represent a wave traveling in the positive x direction with speed v. (You do not need to have ##i## in the expression).

Can you express your function in the form f(x-vt)?
 
  • #4
TSny said:
You do not need to have ##i## in the expression.

I don't understand. ##{ e }^{ i\theta }=cos(\theta )+i sin(\theta )## is the reason why we can write sinusoidal wave in Euler's from. Without ##i##, how can we say its a sinusoidal wave?

TSny said:
Can you express your function in the form f(x-vt)?
Expressing it as a function of ##x-vt##, it looks like this: ##f(x-vt)=A{ e }^{ -\frac { a }{ c } { (x-\frac { b }{ a } t) }^{ 2 } }##, which indicates that velocity of pulse is ##v=\frac { b }{ a }##. So, are you trying to say that is the answer?
 
  • #5
back2square1 said:
I don't understand. ##{ e }^{ i\theta }=cos(\theta )+i sin(\theta )## is the reason why we can write sinusoidal wave in Euler's from. Without ##i##, how can we say its a sinusoidal wave?

Note that you are talking about "sinusoidal" waves. Non-sinusoidal waves do not have to be in that form, naturally. Yet they must still have a propagation velocity, so that can't depend on some property of trigonometric functions.

The only reason why we study sinusoidal waves extensively is because they are the simplest solutions of the wave equation and because practically any wave can be represented as an (infinite) sum of sinusoidal waves.
 
  • #6
back2square1 said:
Expressing it as a function of ##x-vt##, it looks like this: ##f(x-vt)=A{ e }^{ -\frac { a }{ c } { (x-\frac { b }{ a } t) }^{ 2 } }##, which indicates that velocity of pulse is ##v=\frac { b }{ a }##. So, are you trying to say that is the answer?

Yes, that's essentially right. I think the factor of a/c should also be squared, but that doesn't change the answer.
 
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1. What is the wave equation?

The wave equation is a mathematical formula that describes the behavior of waves, including their speed, frequency, and amplitude. It is commonly used in physics and engineering to study various types of waves, such as sound, light, and water waves.

2. How is the speed of a wave calculated from the wave equation?

The speed of a wave can be calculated by using the wave equation: v = λf, where v is the speed, λ (lambda) is the wavelength, and f is the frequency of the wave. This equation shows that the speed of a wave is directly proportional to its wavelength and frequency.

3. What factors affect the speed of a wave?

The speed of a wave can be affected by several factors, including the medium through which it travels, the temperature of the medium, and the frequency and wavelength of the wave. For example, sound waves travel faster in solids than in liquids or gases, and light waves travel faster in a vacuum than in any other medium.

4. Can the speed of a wave be greater than the speed of light?

No, according to Einstein's theory of relativity, the speed of light is the maximum speed at which any object or information can travel in the universe. Therefore, the speed of any type of wave, including electromagnetic waves, cannot exceed the speed of light.

5. How is the speed of a wave related to its energy?

The speed of a wave is not directly related to its energy. However, the energy of a wave is directly proportional to its amplitude (the height of the wave). This means that a higher amplitude wave will have more energy, but its speed will remain the same as a lower amplitude wave with the same frequency and wavelength.

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