Standing waves - Wave Equation

[elemis]

I don't completely understand how equation 4.4.4 was derived and determined. I understand the derivation behind the basic wave equation 4.3.4 but not what happened in 4.4.4. Why is there a need for all the negative signs ? Would a simple phase change suffice ?

Please do be a bit detailed in your explanation... Pretend I'm an idiot. Thank you !

 

[Redbelly98]

Since the reflected wave travels in the opposite direction as the incident wave, you need to flip the sign on either the kx or the ωt term***. The author chose to reverse the sign of kx to make it -kx. There was already a - sign with -ωt. For the phase term, the choice of sign is arbitrary.

Just changing the phase would express a wave traveling in the same direction as the incident wave, but shifted in phase.

Hope that helps.

EDIT added:
*** This is because a rightward-traveling wave has the form f(kx-ωt) or f(-kx+ωt). A leftward-traveling wave has the form f(kx+ωt) or f(-kx-ωt).

 

[elemis]

So I could equally have written kx+wt ? Why is there a need for the phase change ∅ ?

 

[Redbelly98]

So I could equally have written kx+wt ?

Yes; see the edit added to my earlier post.

Why is there a need for the phase change ∅ ?

As the book says, the type of boundary will determine ∅. Usually the boundary is either fixed or has a maximum amplitude. Also, the location of the boundary plays a role in what ∅ is.

Eg., for a fixed end located at x=x₀:

$$\cos(kx_0 + \phi /2) = 0$$

You'd solve that for ∅, given k and x₀. Your book is taking the fixed end to be at x=0, so that simplifies things somewhat.

 

[PJC01]

No problem, let me break it down for you. First, let's review the basic wave equation 4.3.4. This equation describes a traveling wave, where the wave moves in one direction and does not reflect back on itself. It is given by the equation:

y(x,t) = A sin(kx - ωt)

Where:
y(x,t) is the displacement of the wave at position x and time t
A is the amplitude of the wave
k is the wave number, which is related to the wavelength of the wave
ω is the angular frequency, which is related to the frequency of the wave

Now, in order to understand the standing wave equation 4.4.4, we need to first understand what a standing wave is. A standing wave is a wave that appears to be standing still, with no apparent motion. This is due to the interference of two waves traveling in opposite directions. The standing wave equation is given by:

y(x,t) = A sin(kx) cos(ωt)

Notice that in this equation, there is no time dependence in the sine term, only in the cosine term. This is what gives the illusion of a standing wave, as the sine term remains constant while the cosine term oscillates with time.

Now, let's take a closer look at the equation 4.4.4. You'll notice that there are negative signs in front of the sine and cosine terms. These negative signs are necessary in order to account for the interference of the two waves traveling in opposite directions. When these waves interfere, they can either reinforce or cancel each other out, depending on their relative phase. The negative signs account for this phase difference.

To understand this better, let's consider a simpler example. Imagine you have two waves traveling in opposite directions on a string. If the two waves are in phase, meaning they have the same amplitude and are at the same point in their cycle, they will add together to create a larger wave. However, if the two waves are out of phase, meaning they have opposite amplitude and are at opposite points in their cycle, they will cancel each other out and create a flat line. The negative signs in the standing wave equation account for this phase difference.

So, to answer your question, a simple phase change would not suffice because it does not account for the interference of the two waves traveling in opposite directions. The negative signs are necessary to accurately describe the standing wave and